Paper Title: Capacities of quantum Markovian noise for large times.



Vera Verto: Multimodal Hijacking Attack
    
Minxing Zhang1  Ahmed Salem2  Michael Backes1  Yang Zhang1




1CISPA Helmholtz Center for Information Security    
2Microsoft

    
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§ ABSTRACT 
Given a quantum Markovian noise model, we study the maximum dimension of a classical or quantum system that can be stored for arbitrarily large time. We show that, unlike the fixed time setting, in the limit of infinite time, the classical and quantum capacities are characterized by efficiently computable properties of the peripheral spectrum of the quantum channel. In addition, the capacities are additive under tensor product, which implies in the language of Shannon theory that the one-shot and the asymptotic i.i.d. capacities are the same. We also provide an improved algorithm for computing the structure of the peripheral subspace of a quantum channel, which might be of independent interest.





§ INTRODUCTION





Consider a quantum system that we would like to use for storing quantum or classical information. This system is affected by noise that we assume is Markovian. It is natural to ask what is the minimum error that can be achieved for storing log D (qu)bits of information for some fixed time t. This is a typical question studied in Shannon theory, but here we focus on the limit t →∞, i.e., the information should remain in the system for arbitrarily long times.





 Building a quantum system able to store quantum information for large times is one of the important goals of quantum information theory and it has been studied from different angles. Quantum error correction gives a mechanism to actively preserve the quantum information in a system undergoing local noise <cit.>. Such methods can achieve much more than a memory and can be used for fault-tolerant quantum computation <cit.>. A related important area of research is the study of self-correcting (or passive) quantum memories <cit.>, i.e., physical systems that are robust to different forms of imperfections including thermal noise.
 
 In this paper, we study the question of quantum memory from an abstract Shannon theory perspective where given a noise model, the objective is to characterize the fundamental limits that can be achieved without placing computational restrictions on the encoding/decoding operations. As an example, we would like to characterize how many qubits can be stored reliably for time t as a function of the noise model and t? Such questions were studied in the recent work <cit.> and they obtained, among other results about scrambling, conditions for the classical capacity to be zero as well as lower bounds on the classical capacity of ergodic channels.



  
Our results In this work, we focus on the setting of arbitrarily large time, i.e., t →∞ and we characterize both the classical and quantum capacities for a fixed error δ in terms of the peripheral spectrum of the noise model (Theorem <ref>). In addition, we show that such capacities are additive for the tensor product of channels (Theorem <ref>) and can be computed in polynomial-time in the dimension (Theorem <ref>). This algorithm for computing the structure of the peripheral subspace (or the fixed point subspace) of a quantum channel improves on previous works and might of independent interest. We note that the fact that we can efficiently characterize the infinite-time capacities of noisy channels is in contrast with other settings for which capacities or optimal success probabilities correspond to hard problems, such as maximum independent set <cit.> or maximum coverage problems <cit.>.



§ CAPACITIES OF QUANTUM NOISE MODELS


For a Hilbert space H, we use the notation B(H) for the bounded linear operators on H. When we do not need to make the Hilbert space explicit and when it has dimension d, we denote the algebra of linear operators (or d × d complex matrices) by M_d(C) or simply M_d for short. We recall that a quantum channel E :  M_D(C) → M_d(C) is a completely positive and trace preserving linear map. We refer to <cit.> for more detailed definitions and properties of quantum channels.

We start with the standard definition of an error correcting code for a noisy quantum channel T. Note that throughout the paper d ≥ 1 is an integer and T is a quantum channel from _d( C) to _d( C).
    
        A quantum channel E:  M_D( C)→ M_d( C) is a (D,δ) classical code for a channel T:  M_d( C)→ M_d( C) if there exists a recovery channel R:  M_d( C)→ M_D( C) such that the average fidelity of the channel R∘ T∘ E over all diagonal density matrices in M_D is at least 1-δ, i.e.,
        
    1/D∑_i=1^D  i|  R∘ T∘ E(|i i|) |i≥ 1-δ,

        where {|i⟩}_i=1^D is a fixed orthonormal basis of C^D. We say that E is a (D,δ) quantum code if there exist channel R such that the entanglement fidelity of R∘ T∘ E is at least 1-δ, i.e.,
        
    F_E( R∘ T∘ E) = ⟨Φ^+|(I⊗ ( R∘ T∘ E) )(|Φ^+Φ^+|) |Φ^+⟩≥ 1-δ,

        where |Φ^+ = 1/√(D)∑_i=1^D |i⟩⊗|i⟩∈ ( C^D)^⊗ 2 and I is the identity quantum channel on the reference system of dimension D.
    
    
    Then, we introduce the capacity as the dimension of the largest code for the channel T for a fixed error parameter δ.
    
        The (one-shot) classical capacity for a channel T is defined as 
                
    C_δ(T) = sup_Elog D,

        where the supremum is over all possible (D,δ) classical codes E for T. Similarly, the (one-shot) quantum capacity for the channel T is defined as 
        
    Q_δ(T) = sup_Elog D,
 
        where the supremum is taken over all possible (D,δ) quantum codes.
    

    We now see  as the noise that is applied for one time unit. Thus, by the Markov assumption, after t time steps, the noise that is applied to the system is given by composing  t times, which we denote by ^t = ∘⋯∘. As mentioned in the introduction, in this paper, we are interested in the limit of large time t, which motivates the following definition.
     
        The infinite-time classical or quantum capacity of a quantum channel T is defined as
        
        
    C_δ^∞ ( T)= lim_t→∞ C_δ( T^t)   and    Q_δ^∞( T) = lim_t→∞ Q_δ( T^t).

    
    We make a few remarks about this definition.
    
    In this definition, the encoding could in principle depend on the time t. However, all the achievability results (i.e., lower bounds of the capacities) we obtain, the same encoder  works for ^t for all times t ≥ 1. In contrast, it is important for the results to hold that the recovery channel  does depend on the time t.
    
    
    This definition is concerned about passive error correction where no recovery is allowed at regular intervals. One could define an active capacity as follows:
    
    Q_δ^active, t( T) = log{max D : ∃, _1, …, _t s.t.  F_E(_t ∘∘⋯∘∘_1 ∘∘) ≥ 1-δ},

    where : _D →_d and _i : _d →_d for 1 ≤ i ≤ t-1 and _t : _d →_D are quantum channels. Note that the active capacity can be larger than the passive one; in particular, it is simple to see that for any t ≥ 1, Q_0^active, t( T) = Q_0( T) by choosing the recovery maps _i to decode and re-encode information. The works <cit.> have studied some Shannon-theoretic aspects of active error correction.
    
    
    Note that in general T^t does not have a limit as t →∞, but for the capacities C_δ ( T^t) and Q_δ ( T^t), the limits exist as they are nonnegative nonincreasing sequences.        
    

    The characterization of both the classical and quantum capacities will be in terms of the peripheral subspace χ_ T of the quantum channel T. The following proposition summarizes the definition and key properties about the peripheral subspace that will be used in this paper. A proof of these statements and more properties can be found in <cit.>.
    
    
    The peripheral subspace of a quantum channel T is defined by
    
    χ_T = span{ X ∈M_d(C) : T(X) = λ X,  for some  |λ| = 1}.

    It satisfies the following:
    
    
  * There exists a direct sum decomposition of the Hilbert space C^d = H_0 ⊕⊕_k=1^K (H_k,1⊗ H_k,2) for some nonnegative integer K such that 
    
    χ_T = 0 ⊕⊕_k=1^K M_d_k(C) ⊗ω_k,

    where M_d_k(C) is the full matrix algebra on H_k,1, with d_k =  H_k,1, and ω_k is a density operator on H_k,2.
    
  * There exists unitaries U_k acting on H_k,1 and a permutation π which permutes systems {1, …, K} having the same dimension such that for every X ∈χ_T of the form X = 0 ⊕⊕_k=1^K x_k ⊗ω_k, we have
    
    T(X) = 0 ⊕⊕_k=1^K U_k x_π^-1(k) U_k^†⊗ω_k.

    
    
    
    
    We are now ready to state our main theorem.
    
        Let T:  M_d → M_ d be a quantum channel and let K and {d_k}_k=1^K be the integers from the peripheral subspace decomposition in (<ref>).
        
        
        For any δ∈ [0,1), the infinite time classical capacity of T is given by
        
    C_δ^∞(T) = log(⌊∑_k d_k/1-δ⌋),

        and the quantum capacity can be bounded as
        
    log(max_k d_k/√(1-δ)) ≤  Q_δ^∞ ( T) ≤log(max_k d_k)+log(1/1-δ).

        
        In addition, as shown in Theorem <ref> below, the values {d_k}_k determining the structure of χ_ can be computed in time polynomial in d given the description of .
    
    We note that, in independent work, the special case δ = 0 of (<ref>) and (<ref>) were proved in <cit.>. Before proving the Theorem, we make a few remarks.

    
    For the quantum capacity, we do not have an exact expression for the capacity but upper and lower bounds that differ by roughly 1/2log(1-δ). The upper and lower bounds we establish in the proof is slightly stronger and it is simpler to express in terms of the optimal error δ for a fixed code size D:
    
    max_(γ_k)_k∑_k=1^Kγ_k^2/D^2≤sup{1- δ : ∃ (D, δ)  quantum code for }≤max_(γ_k)_k∑_k=1^K d_k γ_k/D^2

    where the supremum is taken over integers γ_k satisfying γ_k ≤ d_k and ∑_k=1^K γ_k ≤ D. Note that it is simple to see that the optimal choice for γ_k is simply to take γ_1 = d_1, …, γ_s = d_s and γ_s+1 = D - ∑_k=1^s γ_k where s = { s : ∑_k=1^s d_k < D }. When D is the sum of the largest r elements of (d_k)_k, then the upper and lower bounds match and we obtain an exact characterization. Observe that (<ref>) follows from (<ref>) by observing that max_(γ_k)_k∑_k=1^Kγ_k^2 ≥min(D^2, max_k d_k^2) and 
    ∑_k=1^K d_k γ_k≤ (max_k d_k) D. We leave the problem of computing the exact optimal fidelity as an open problem.
    

    
    Note that a classical stochastic d × d matrix M can be seen as a special case of a quantum channel ( |i⟩⟨i'|) = 0 if i ≠ i' and ( |i⟩⟨i|) = ∑_j M_j,ij. It is straightforward to verify that the eigenvalues of M are the eigenvalues of  and that the quantity ∑_k=1^K d_k for such a channel is the number of eigenvalues of M of modulus 1 counted with multiplicity or the dimension of the peripheral subspace. This quantity even has a combinatorial interpretation as the sum of the periods of the bottom strongly connected components of the directed graph associated with the Markov chain M, as shown in <cit.>. Note that this combinatorial interpretation holds more generally for nonnegative matrices and such a decomposition into bottom strongly connected components is sometimes called the Frobenious normal form of the matrix <cit.>, see also e.g., in <cit.> for the period of each component. Using the combinatorial interpretation, one can find an algorithm running in linear time in the size of the graph for computing the capacity by using e.g., Tarjan's algorithm <cit.> to find the bottom strongly connected components and then find the periods of each component using <cit.>.
    
    
    
    
    
    
    
    
    
    
    

    
    
    We start with the achievability statement, i.e., lower bounds on the capacities. Using the decomposition of the space in Proposition <ref>, C^d = H_0 ⊕⊕_k=1^K (H_k,1⊗ H_k,2), let {|e_k,j⟩}_j=1^d_k be orthonormal bases of H_k,1. For different H_k,1 having the same dimension, we will identify the orthonormal bases.


    We define the classical code E as follows.  Let σ : {1, …, ∑_k=1^K d_k}→{(k,j) : k ∈{1, …, K}, j ∈{1, …, d_k}} be an arbitrary bijection, then 
    
    E(i) = 
        {[              e_σ(i)⊗ω_k  if  1 ≤ i ≤∑_k=1^K d_k;                 e_1⊗ω_k if ∑_k=1^K d_k < i ≤ D. ].

    For 1 ≤ i ≤∑_k=1^K d_k, we have (i) = e_k,j⊗ω_k ∈ B(H_k,1⊗ H_k,2) with (k,j) = σ(i). As a result, applying , we get
    
    ((i)) = U_π(k)e_k,j⊗ω_π(k) U_π(k)∈ B(H_π(k),1⊗ H_π(k),2),

    using Proposition <ref>.
    Composing t times the map , we get  
    
    ^t((i)) = U_π^t(k)⋯ U_π(k)e_k,j U^†_π(k)⋯ U^†_π^t(k)⊗ω_π^t(k)∈ B(H_π^t(k),1⊗ H_π^t(k),2).

    
    Note that we always have ^t((i)) ∈χ_. For a time t, we choose a recovery map  defined by 
    
    (X) = ∑_k=1^K V_k U^†_π(k)⋯ U^†_π^t(k)_H_π^t(k),2(P_π^t(k) X P_π^t(k)) U_π^t(k)⋯ U_π(k) V_k^†,

    where P_k is the orthogonal projection onto H_k,1⊗ H_k,2 and V_k = ∑_j=1^d_k|σ^-1(k,j)⟩⟨e_k,j|. Clearly, an error occurs only if i > ∑_i=1^K d_K and as such the success probability is given by
    
    1/D∑_i=1^D ⟨i|∘^t ∘( i ) |i⟩ = min(1, ∑_k=1^D d_k /D).

    As a result, if we choose D = ∑_k=1^K d_k/1-δ, we obtain the desired achievability.

    Now let us construct a quantum code of dimension D satisfying the condition max_(γ_k)_k{∑_k=1^Kγ_k^2/D^2}≥ 1-δ.
    
    Let γ_k achieve the maximum. It is convenient to label the basis of C^D by elements in S ∪ S', where S = {(k,j) : k ∈{1, …, K}, j = 1 ∈{1, …, γ_k}} and S' = {1, …, D-∑_k=1^Kγ_k}. By the condition ∑_k = 1^K γ_k ≤ D, the set S is of size at most D.
    
    
    In order to define our encoding map , we first define the Kraus operators E_k = ∑_j=1^γ_k|e_k,j⟩⟨(k,j)| for k = 1 to K, and F_i = |e_(1,1)⟩⟨i| for i ∈ S'.
    
    Then for X ∈_D(C) we define (X) = ∑_k=1^K (E_k X E_k^†) ⊗ω_k + ∑_i ∈ S' F_i X F_i^†⊗ω_1. As a result, we have
    
    (|(k,j)⟩⟨(k,j')|) = |e_k,j⟩⟨e_k,j'|⊗ω_k,

    and (|(k,j)⟩⟨(k',j')|) = 0 for k ≠ k'. For i, i' ∈ S' we get
    
    (|(k,j)⟩⟨i|) = 0,

    
    (|i⟩⟨i|) = |e_1,1⟩⟨e_1,1|⊗ω_1    and   (|i⟩⟨i'|) = 0  for  i≠ i'.
        
    Thus, applying ^t, we get
    
    (^t ∘)(|(k,j)⟩⟨(k,j')|) = U_π^t(k)⋯ U_π(k)|e_k,j⟩⟨e_k,j'| U_π(k)^†⋯ U_π^t(k)^†⊗ω_π^t(k).

    Note that ^t ∘(|(k,j)⟩⟨(k',j')|) ∈χ_. We choose  as
    
    (X) = ∑_k=1^K V_k  U^†_π(k)⋯ U^†_π^t(k)_H_π^t(k),2(P_π^t(k) X P_π^t(k)) U_π^t(k)⋯ U_π(k) V_k^†,

    where P_k is the orthogonal projection onto H_k,1⊗ H_k,2 and V_k = ∑_j=1^γ_k|(k,j)⟩⟨e_k,j|.
    

    
    
    
    Then we can compute the entanglement fidelity as
    
    ⟨Φ^+|(I⊗ R∘ T^t ∘ E )(|Φ^+Φ^+|) |Φ^+⟩
       ≥1/D^2∑_(k,j), (k',j') ∈ S⟨(k,j)|∘^t ∘( |(k,j)⟩⟨(k',j')| ) |(k',j')⟩
       = 1/D^2∑_k=1^K∑_j,j'=1^γ_k 1 
       = ∑_k=1^K γ_k^2 /D^2
       ≥ 1-γ,

    which proves the desired result.




    











    
    
    
    We now move to the converse bound. For this, it is convenient to consider the peripheral projection channel T_P <cit.> which satisfies the following properties:  
        <cit.>
        
            
  * T_P( M_d)= χ_ T
            
  * there exist an increasing sequence {t_i} such that lim_i→∞ T^t_i =  T_P
            
  * T_P(X) = X for any X ∈χ_ T.
        

    Note that if  is a (D, δ) code for , and - '_♢≤η, where the diamond norm is defined as - ' _♢ = sup_ρ(I⊗ ( - '))(ρ) _1, then  is also a (D, δ + η) code for '. 
    
    Consider a sequence t_i such that lim_i →∞^t_i = _P and let η_i = ^t_i - _P _♢. Let _i be a (D, δ) code for the channel ^t_i. Then _i is also a (D, δ + η_i) code for the channel _P. Thus C_δ(^t_i) ≤ C_δ + η_i(_P).
    
    Taking the limit i →∞, we have that sup_δ' < δ C_δ'(_P)≤ C^∞_δ() ≤inf_δ' > δ C_δ'(_P). The exact same result holds for the quantum capacity as well. It now suffices to find upper bounds on the capacities of the channel _P.

    
    First, let us show that the classical and quantum capacities of _P and ∑_k=0^K _k ∘_P are the same, where we define _k as follows.
    Define P_0 to be the orthogonal projector onto H_0 and _0(X) = P_0 X P_0. Then define P_k to be the orthogonal projector onto H_k,1⊗ H_k,2 and _k : B(C^d) → B(H_k,1) by _k(X) = _H_k,2(P_k X P_k). Then ∑_k=0^K _k ∘_P is clearly a quantum channel. As _P(_d) = χ_, ∑_k=0^K _k ∘_P = ∑_k=1^K _k ∘_P so  ∑_k=1^K _k∘_P is a quantum channel mapping B(C^d) to ⊕_k=1^K B(H_k,1).  
    The inequality C_δ(∑_k=1^K _k ∘_P) ≤ C_δ(_P) is clear. For the other inequality, let  be a code for _P and  be a corresponding recovery map. We define the recovery map ' = ∑_k=1^K ∘_k where _k : B(H_k,1) → B(H_k,1⊗ H_k,2) and maps _k(x_k) = x_k ⊗ω_k. It is easy to see that
    
    ' ∘∑_k=1^K _k ∘_P ∘ = ∘_P ∘,

    which proves the desired statement. For this reason, in what follows, we assume that H_0 = 0 and H_k,2 = 1 and the space decomposes as C^d = ⊕_k=1^K H_k,1.
    

    Let E be a (D,δ) classical code for T_P and R a corresponding recovery channel. Note that because _P(M_d) = χ_, for any density operator ρ∈ B(C^d), there exists positive operators ρ_k on H_k,1, _P(ρ) = ⊕_k=1^K ρ_k. In addition, as _P is trace-preserving ∑_k(ρ_k) = 1 and ρ_k ≤ I_H_k,1, where I_H_k,1 is the identity on H_k,1. As a result, _P(ρ) ≤⊕_k=1^K I_H_k,1. Thus,
    
    1/D∑_i=1^D⟨i| R ∘_P ∘ E(i) |i⟩   ≤1/D∑_i=1^D ⟨i|( ⊕_k=1^K I_H_k,1) |i⟩
       = 1/D( ⊕_k=1^K I_H_k,1)
        = ∑_k=1^K d_k/D.

    Thus, for any (D, δ) code, we should have ∑_k=1^K d_k/D≥ 1-δ which implies D ≤∑_k=1^K d_k/1-δ. This implies that 
    C_δ(_P) ≤∑_k=1^K d_k/1-δ. As a result C^∞_δ() ≤∑_k=1^K d_k/1-δ' for any δ' > δ which gives the desired result.

    Let us now move to the quantum capacity. Let E be a (D,δ) classical code for T_P and R a corresponding recovery channel. 
    
    
    In order to compute the entanglement fidelity, recall that the  entanglement fidelity of channel E with Kraus operators {E_i} is given by F_E( E) = ∑_i |(E_i)|^2. Let us introduce Kraus operators for {E_j}_j for _P ∘, {R_i}_i for . As _P(_d) = χ_, the operator {P_k E_j}_k, j are also Kraus operators for the map _P ∘, where we recall that P_k is the projector onto H_k,1.
    
    

    Then we have
            
    F_E( R ∘_P ∘ E)    = 1/D^2∑_i,j,k|(R_i P_k E_j)|^2.

    Let us denote α_k^2 = ∑_i,j|(R_iP_k E_j)|^2. We show two properties on α_k: α_k^2 ≤ d_k β_k with ∑_kβ_k = D and α_k^2 ≤ d_k^2. We start with the first property
            
    α_k^2    ≤∑_i,j( R_i P_k P_k^† R_i^† ) (E_j^† P_k^† P_k E_j)
       = (P_k) ∑_j( E_j^† P_k E_j  ).
    
For the first inequality, we used the Cauchy-Schwarz inequality |(A^† B)|^2 ≤(A^† A) (B^† B). Then we used the fact that  is trace preserving. Now note that because _P ∘ is a quantum channel, we have ∑_k,j(  E_j^† P_k^† P_k E_j ) = D. Calling β_k = ∑_j( E_j^† P_k^† P_k E_j ), we proved the first claimed inequality. 

To show the second inequality α_k^2 ≤ d_k^2, we write
            
    α_k^2    ≤∑_i,j( E_j R_i P_k P_k^† R_i^† E_j^† ) ( P_k^† P_k )
       = (P_k) d_k 
       = d_k^2 .
 
where we used again the Cauchy-Schwarz inequality and the fact that _P ∘∘ is trace-preserving. Defining γ_k = min(d_k, β_k), we have that ∑_k=1^K γ_k ≤ D and γ_k ≤ d_k and the entanglement fidelity can be bounded as 

    F_E( R ∘_P ∘ E) 
                       ≤1/D^2∑_k=1^K α_k^2 
       ≤1/D^2∑_k=1^K d_k γ_k ,

which concludes the proof of (<ref>).
    

    

 §.§ Continuous-time dynamical system


    We now consider a quantum Markovian evolution with a continuous time parameter. Such an evolution is called a quantum dynamical semigroup or quantum Markov semigroup and is a family of quantum channels (_t)_t ≥ 0 satisfying the property _t+s = _t∘_s for t, s ≥ 0. As we show below, the infinite-time capacities of such evolutions can be easily related to the discrete time case considered above. We note that a quantum Markov semigroup (_t)_t ≥ 0 is characterized by a linear map L (called the generator) satisfying T_t= e^ L t: (_t)_t ≥ 0 forms a quantum Markov semigroup if and only if there exists a Hermitian matrix H and matrices {L_j}_j such that <cit.><cit.>
    
    L(ρ)= i[ρ, H]+∑_j L_j ρ L_j^† -1/2( L_j^† L_j ρ + ρ L_j^† L_j ).

    
    
    

    

     
     Let (_t)_t ≥ 0 be a quantum Markov semigroup. Then we have
     
    lim_t →∞ C_δ(_t) = C^∞_δ(_1) 
    lim_t →∞ Q_δ(_t) = Q^∞_δ(_1).

     In addition, the peripheral subspace χ__1 can be expressed in terms of the spectrum of generator L as follows:
             
    χ__1 = span{ X ∈M_d(C) | ∃θ∈ R ,   L(X) = iθ X }.
 
     

     

    
            Let ℰ be a (D,δ) classical (or quantum) code for 𝒯_t, and ℛ be its corresponding recovery channel. If t' ≤ t, then ℰ is a (D,δ) classical (or quantum) code for 𝒯_t' with recovery operator ℛ∘𝒯_t-t'. Therefore, we have the following inequalities:
            
    C_δ(𝒯_⌊ t ⌋) ≥ C_δ(𝒯_t) ≥ C_δ(𝒯_⌈ t ⌉)

            and
            
    Q_δ(𝒯_⌊ t ⌋) ≥ Q_δ(𝒯_t) ≥ Q_δ(𝒯_⌈ t ⌉).

            In the limit as t →∞, both C_δ(𝒯_⌊ t ⌋) and C_δ(𝒯_⌈ t ⌉) converge to C_δ^∞(𝒯_1). Therefore, we have
            
    lim_t →∞ C_δ(𝒯_t) = C_δ^∞(𝒯_1)
                  and  lim_t →∞ Q_δ(𝒯_t) = Q_δ^∞(𝒯_1).

        
            Next, we examine the peripheral subspace of  𝒯_1. It is straightforward to see that if ℒ(X) = iθ X for some operator X and real number θ, then  𝒯_t(X) = e^tℒ(X) = e^iθ tX, which implies X ∈χ_ T_t. Consequently, if X is spanned by eigenoperators of ℒ corresponding to imaginary eigenvalues, then X belongs to χ_𝒯_1.
            
                
           
    


      To show the converse, we use the “super-operator form" which represents the linear map ℰ as a matrix Ê acting on the vector space of operators. Specifically, for a quantum channel ℰ and a density matrix ρ, the action of ℰ can be written as vec(ℰ(ρ)) = Êvec(ρ), where vec(·) denotes the column-stacking vectorization of a matrix.
            Let  L be the super-operator form of ℒ, with a Jordan decomposition given by
            
    L = V ( ⊕_ℓ=1^s J_ℓ(λ_ℓ) ) V^-1,

            where J_ℓ(λ_ℓ) is a Jordan block corresponding to an eigenvalue λ_ℓ of L, i.e., J_ℓ(λ_ℓ) = λ_ℓ I + J_ℓ(0) with J_ℓ(0) = [ 0 1 0 … 0; 0 0 1 … 0; ⋮ ⋮ ⋱ ⋮ ⋮; 0 … … … 0 ]∈_d_ℓ, where d_ℓ is the size of the block.
            Thus the super-operator of 𝒯_1 = e^ℒ has the form
            
    e^L = V ( ⊕_ℓ=1^s e^J_ℓ(λ_ℓ)) V^-1,

           with 
              













    e^J_ℓ(λ_ℓ) = e^λ_ℓ·[          1          1       1/2!          ⋯ 1/(d_ℓ-1)!;          0          1          1          ⋯ 1/(d_ℓ-2)!;          0          0          1          ⋯ 1/(d_ℓ-3)!;          ⋮          ⋮          ⋮          ⋱          ⋮;          0          0          0          ⋯          1 ].

Thus the eigenvalues of _1 are {e^λ_ℓ}_ℓ. Now let X be an eigenvector of _1 with eigenvalue e^λ_ℓ with |e^λ_ℓ| = 1. We know from <cit.> that for such eigenvalues d_ℓ = 1. Therefore, X is also an eigenvector of L with eigenvalue λ_ℓ and λ_ℓ is pure imaginary.








        

    

    
    
    
    
    
        
        

        
        
        
        

        
         
        
        
        
        
        

        
        
    



§ TENSOR PRODUCT OF QUANTUM CHANNELS


In this Section, we show the the infinite-time capacities are basically additive under tensor product. To show that, we start with the following lemma about the peripheral subspace.
    
        Consider two quantum channels, T:  M_d( C)→ M_d( C) and S: M_d'( C)→ M_d'( C), with their respective peripheral projections denoted by T_P and S_P, and their respective peripheral subspaces denoted by χ_ T and χ_ S. For the tensor product of these channels, T ⊗ S, the peripheral projection is given by T_P ⊗ S_P, and the corresponding peripheral subspace is χ_ T ⊗χ_ S.
    

    
        By definition  there exist increasing sequences {m_i}, {n_i}, and {k_i} such that
        
    lim_i→∞ T^m_i =  T_P,  lim_i→∞ S^n_i =  S_P, and  lim_i→∞ ( T⊗ S)^k_i = ( T⊗ S )_P.
 
        
        Since any power of a peripheral projection is itself, we have
        
    lim_i→∞ T^m_in_ik_i =  T_P,  lim_i→∞ S^m_in_ik_i =  S_P, and  lim_i→∞ ( T⊗ S)^m_in_ik_i = ( T⊗ S )_P.

        Therefore, (  T⊗ S)_P =  T_P ⊗ S_P.

        Next, we consider the peripheral subspaces. The peripheral subspace is the fixed-point subspace of the peripheral projection. If X∈χ_ T⊗χ_ S, then it can be written as X= ∑_i Y_i⊗ Z_i where Y_i∈χ_ T and Z_i∈χ_ S. So we have
        
    T_P⊗ S_P (X)= ∑_i  T_P(Y_i)⊗ S_P(Z_i)= ∑_i Y_i⊗ Z_i=X,

        showing that X is a fixed point of  T_P⊗ S_P . Therefore χ_ T ⊗χ_ S ⊆χ_ T⊗ S.

        For the other direction, let X∈ M_d( C)⊗ M_d'( C) belongs to peripheral subspace of T⊗ S, and decompose X as X= ∑_i Y_i⊗ Z_i  where Y_i∈ M_d( C) and Z_i∈ M_d'( C). Since X is a fixed point of T_P ⊗ S_P, we have 
        
    X=  T_P ⊗ S_P(X)= ∑_i  T_P(Y_i) ⊗ S_P(Z_i).

        Because T_P(Y_i)∈χ_ T and S_P(Z_i)∈χ_ S ,  it follows that  X∈χ_ T ⊗χ_ S. Thus  χ_ T⊗ S ⊆χ_ T ⊗χ_ S. 
        Therefore, we conclude that χ_ T⊗ S = χ_ T ⊗χ_ S.
    

    

    
    
    
        For any two quantum channels T and S, the infinite-time zero-error classical and quantum capacities are additive under the tensor product. Specifically, we have:
        
    C_0^∞( T ⊗ S)    =  C_0^∞( T) +  C_0^∞( S), 
    
                  Q_0^∞( T ⊗ S)    =  Q_0^∞( T) + Q_0^∞( S).

       
    

    
        By Lemma <ref>, the  peripheral subspace of T⊗ S is the tensor product of the peripheral subspace of T and S. So if χ_ T= 0 ⊕⊕_k=1^K M_d_k(C) ⊗ω_k for the decomposition C^d = H_0 ⊕⊕_k=1^K H_k,1⊗ H_k,2 and χ_ S= 0 ⊕⊕_k'=1^K'M_d'_k'(C) ⊗ω'_k' for the decomposition C^d' = H'_0 ⊕⊕_k'=1^K' H'_k',1⊗ H'_k',2, then we can decompose
        
    C^d⊗C^d' = H_0 ⊕⊕_k ∈{1, …, K}, k' ∈{1, …, K'} H_k,1⊗ H'_k',1⊗ H_k,2⊗ H'_k',2,

        where 
        
    H_0 = (H_0 ⊗ H'_0) ⊕(H_0 ⊗ (⊕_k'=1^K' H'_k',1⊗ H'_k',2) ) ⊕((⊕_k=1^K H_k,1⊗ H_k,2) ⊗ H'_0)

        and get for this decomposition
        
    χ_ T⊗ S= 0 ⊕⊕_k,k'M_d_k× d'_k'(C) ⊗ω_k⊗ω'_k'.


        By Theorem <ref>, C_0^∞(⊗S) = log(∑_k,k' d_k d'_k') = C_0^∞() + C_0^∞(S) and Q_0^∞(⊗S) = log(max_k,k' d_k d'_k') = Q_0^∞() + Q_0^∞(S).
        
        
    


    


    A simple corollary is that when taking tensor powers of a channel, as is common in Shannon theory, the classical and quantum capacities are given by the zero-error capacities at infinite time.
    
        Let δ∈ [0,1) and T:  M_d → M_d be a quantum channel. Then 
        
    lim_m→∞1/m C_δ^∞( T^⊗ m)= C_0^∞( T),

        and
        
    lim_m→∞1/m Q_δ^∞( T^⊗ m)= Q_0^∞( T).

    

    

    
By Theorem <ref> and Theorem <ref>, we have
        
    C_δ^∞ (  T^⊗ m)
               = log(⌊2^C_0^∞ (  T^⊗ m)/1-δ⌋) 
       = log(⌊2^mC_0^∞ (  T)/1-δ⌋).

        But 
        
    m C_0^∞() + log(1/1-δ) - 1 ≤log(⌊2^m C_0^∞ (  T)/1-δ⌋) ≤ m C_0^∞() + log(1/1-δ),

    which proves the desired result.

    For the quantum capacity, the same argument gives
        
        
        
        
        
        
        
    m Q_0^∞()+ 1/2log(1/1-δ)-1 
                ≤  Q_δ^∞( T^⊗ m) 
                ≤ m Q_0^∞()+ log(1/1-δ),

        which proves the desired result.  
        
        
        
        
    

    




    
    





    












    




    
    








        

    



§ AN EFFICIENT ALGORITHM FOR THE STRUCTURE OF THE PERIPHERAL SUBSPACE


    In this section, we describe an algorithm to efficiently determine the structure of the peripheral subspace of a quantum channel. An algorithm using O(d^8) operations for this task was previously described in <cit.> and a related algorithm in <cit.>. Here, we improve it and show that it can be done in O(d^6 log d). The paper <cit.> also describes an algorithm to represent the peripheral subspace as a linear span but relies on <cit.> for the structure, and for the latter, it is not clear whether it is efficient. 
    Note that here, we work under a simplifying assumption that standard operations on a d × d matrix can be done in time O(d^3) (e.g., matrix multiplication, spectral decomposition, Jordan normal form). We leave for future work a more detailed complexity analysis for finding the structure of the peripheral subspace.

    
    
        Let 𝒯: ℳ_d(ℂ) →ℳ_d(ℂ) be a quantum channel, and let χ_𝒯 = 0 ⊕⊕_k=1^K ℳ_d_k(ℂ) ⊗ω_k be the decomposition of its peripheral subspace as described in Proposition <ref>. Algorithm <ref>, takes the super-operator form of 𝒯 as input and returns a representation of the Hilbert space decomposition and the fixed density matrices ω_k in time O(d^6 log d). 
        
        
    

    
    
    
    
    
    
        An implementation of this algorithm 
        with examples can be found in <cit.>.
    


    

    The first step is to compute the peripheral projection channel _P. Let T̂∈_d^2 be the super-operator form of T. Using the Jordan normal form, we can express T̂ as
    
    
    
    
    T̂ = ∑_ℓ=1^s λ_ℓ P_ℓ +N_ℓ, where λ_ℓ are the eigenvalues of T̂, P_ℓ are projections and N_ℓ are nilpotent. The super-operator form of T_P is given by <cit.>
    
    T̂_P= ∑_ℓ: |λ_ℓ|=1 P_ℓ.

    Thus, T̂_P can be computed using the Jordan normal form of T̂ which can be obtained in time O((d^2)^3) = O(d^6).
    
    

    Recall that χ_ = Fix(_P), where Fix() is the fixed point space of a map , i.e., Fix() = span{X ∈_d : (X) = X }. The rest of the algorithm computes the structure of the fixed point space of the quantum channel _P.
    
    
    
     
    

    
    
    
    
    
    
    


    


    Let χ_𝒯 = Fix(_P) = 0 ⊕⊕_k=1^K ℳ_d_k⊗ω_k for the Hilbert space decomposition 
    
    C^d = H_0 ⊕⊕_k=1^K H_k,1⊗ H_k,2.

    Instead of working directly with χ_𝒯, it is convenient to work with the matrix algebra ⊆ B(C^d) which is defined as
        
    𝒜 = ⊕_k=1^K ℳ_d_k⊗ I_d_k'.

    for the Hilbert space decompostion (<ref>) with d_k =  H_k,1 and d'_k =  H_k,2. Then, χ_𝒯 is called a distortion of the matrix algebra 𝒜 = ⊕_k=1^K ℳ_d_k⊗ I_d_k', i.e., there is a completely positive map  such that () = χ_𝒯.
    
    

    In <cit.>, an algorithm is given to find operators A_1, …, A_N ∈_d such that = span(A_1, …, A_N). This is done by computing left and right eigenvectors of T̂ and thus can be done in time O(d^6).

    

    What remains is now to determine the structure of . Let us call P_k the orthogonal projector on H_k,1⊗ H_k,2. The first step is to compute the projectors P_k and the second step is to find the dimensions d_k and d'_k.
    
    
    We start with the first step, i.e., computing P_k. For that we use that the projectors P_k are the minimal projections in the center of . Recall that the center Z() = { X ∈ : X Y = YX  ∀ Y ∈} and that Z() = ∩' where ' is the commutant of  defined by ' = { X ∈ B(C^d) : XY = YX   ∀ Y ∈}. 

    Note that for the matrix algebra  in (<ref>), we have ' = ⊕_k=1^K I_H_k,1⊗_d_k' and Z() = ⊕_k=1^K C I_H_k,1⊗ I_H_k,2. Given an algebra , a minimal projection in  is an orthogonal projection P such that P  P = C P <cit.>. The minimal projections of Z(), that are also called the minimal central projections, are exactly the projectors P_k we are looking for.
    
    
    Thus, in order to compute the projectors {P_k}_k=1^K, we first compute a representation of Z() as a linear span. We do this by computing a representation of the commutant ' as the linear span of some operators B_1, …, B_N', which can be done by computing the kernel of the matrix Γ described in Lemma <ref>. Then, the center can be computed by taking the intersection of subspaces  and '; see Algorithm <ref>. Then Algorithm <ref> described a general algorithm to find all minimal projections of an algebra in B(C^d) in time O(N d^3 log d), where N is the number of operators describing the algebra as a linear span. Note that the center Z() has dimension at most d and so we may assume N ≤ d. As such, we have computed all the {P_k}_k=1^K in time O(d^5 log d).

    Now, the second step is within each block k, we want to compute d_k =  H_k,1 and d'_k =  H_k,2. For that we now compute minimal projections P_k,j in  satisfying P_k,j P_k = P_k,j in algebra  by using Algorithm <ref> with inputs P_k and {A_1, …, A_N}. From the form of the algebra , we know that any minimal projection Q with QP_k = Q is of the form Q = ψ⊗ I_H_k,2 for some |ψ⟩∈ H_k,1. As a result, there exists orthogonal unit vectors |e_k,j⟩ such that P_k,j = e_k,j⊗ I_H_k,2. We have that d'_k = (P_k,j) and d_k is the number of minimal projections found {P_k,j}_j. For each k, the runtime for finding the minimal projections {P_k,j}_j=1^k O(d_k × d^5 log d) (as the dimension N of the algebra  can be up to d^2). As ∑_k=1^K d_k ≤ d, the runtime of this step is O(d^6 log d).

    
    
         
    
    
    








    





    





    

    









    












   



         


    
Below we provide a lemma that was used in the above proof. For this lemma we need the concept of operator-vector correspondence: given a matrix X∈_n(ℂ), it represents a vector,  |X⟩⟩∈ℂ^n⊗ℂ^n.     


    
            Let 𝒜 be a matrix algebra generated by {A_1, ⋯, A_N}, and 𝒜' be the commutant of 𝒜. Then X belongs to 𝒜' if and only if |X⟩⟩ belongs to the kernel of Γ, where 
            
    Γ = ∑_i=1^N ( A_i ⊗ I - I ⊗ A_i^T )^†( A_i ⊗ I - I ⊗ A_i^T ).

    

    
        By the fact that A⊗ B|C = |ACB^T (see <cit.>) applies if the dimensions of A, B, and C indicate ACB^T is a valid matrix. So X⊗ I |Y= |XY and I⊗ X^T|Y =|YX. Therefore if we define 
        
    Γ_A_i:=( A_i ⊗ I - I ⊗ A_i^T )^†( A_i ⊗ I - I ⊗ A_i^T ),

         then an operator X commutes with A_i if and only if Γ_A_i|X= 0. As Γ_A_i is positive,  the kernel of Γ is the intersection of kernels of all Γ_A_i. Thus, X ∈' if and only if | X ∈Γ.
         
    





    



    
    
































    

    


    




    
    




    


























    
    




































































        






                
























            







                    




        











        






                

    




                    

    




    


    

§ CONCLUDING REMARKS


    In summary, we have shown that in the setting of infinite time, the classical and quantum capacities are essentially given by the zero-error capacities and can be efficiently computed, unlike the fixed time setting. We see this result as a first step towards understanding channel capacities for quantum evolutions. It would be interesting to study the behaviour of capacities for finite t both for active and passive error correction.







    

§ ACKNOWLEDGEMENTS

    We acknowledge support from the European Research Council (ERC Grant AlgoQIP, Agreement No. 851716). MR is supported by the Marie Skłodowska- Curie Fellowship from the European Union’s Horizon Research and Innovation programme, grant Agreement No. HORIZON-MSCA-2022-PF-01 (Project number: 101108117).
    We would like to thank Nilanjanna Datta for a talk at the “Quantum information" conference at the SwissMAP Research Station in Les Diablerets on <cit.> which stimulated this work.
    




    plain

    







 Paper Title: The dynamical alpha-Rényi entropies of local Hamiltonians grow at most linearly in time.








[]d.toniolo@ucl.ac.uk
[]danielet@alumni.ntnu.no
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United
Kingdom


Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United
Kingdom













§ ABSTRACT 
We consider a generic one dimensional spin system of length L, arbitrarily large, with strictly local interactions, for example nearest neighbor, and prove that the dynamical α-Rényi entropies, 0 < α < 1, of an initial product state grow at most linearly in time. We extend our bound on the dynamical generation of entropy to systems with exponential decay of interactions, for values of α close enough to 1. We provide a non rigorous argument to extend our results to initial low-entangled, meaning O(log L) states. This class of states includes many examples of spin systems ground states, and also critical states. This implies that low entanglement states have an efficient MPS representation that persists at least up to times of order log L. 
The main technical tools are the Lieb-Robinson bounds, to locally approximate the dynamics of the spin chain, a strict upper bound of Audenaert on α-Rényi entropies and a bound on their concavity. Such a bound, that we provide in an appendix, can be of independent interest.



The dynamical α-Rényi entropies of local Hamiltonians grow at most linearly in time
    Sougato Bose 0000-0001-8726-0566
    August 5, 2024
===================================================================================



















§ INTRODUCTION






α-Rényi entropies are generalized quantum entropies that are good quantifiers of entanglement <cit.>. Their properties are listed, for example, in <cit.>.


Several experimental breakthroughs <cit.> have shown how quantum entropies can be detected and measured making them not only crucial quantities to characterized the behavior of many body systems from a theoretical point of view but also susceptible of experimental tests.  


The scaling of α-Rényi entropies, 0 < α < 1, of the reduced density matrix of a state, of a one-dimensional system, have been shown to determine whether such a state could be efficiently represented by a matrix product state (MPS) <cit.>. In higher dimensional systems, instead, a non volume law for the Rényi entropy does not imply an efficient MPS representation <cit.>. 


The variation of the von Neumann entropy of the reduced density of a state evolved in time according to the unitary evolution of a system has attracted a lot of attention starting from the SIE (small incremental entangling) and SIM (small incremental mixing) conjectures attributed by Bravyi in <cit.> to Kitaev. See <cit.> also for early references on this topic. A recent review covering these themes is for example <cit.>. These conjectures roughly say that the rate of increase in time of the von Neumann entropy and the entropy of mixing are upper bounded by a constant independent from the overall system's size. Audenaert <cit.>, Van Acoleyen et al. <cit.> and Marien et al. <cit.> have given proofs of such conjectures. In particular as a consequence of the (at most) linear growth in time of the entanglement entropy the authors of <cit.> were able to prove, using the formalism of the quasi-adiabatic continuation <cit.>, that states within the same phase of matter have the same scaling of the entanglement entropy, implying, in particular, the stability of the area law in dimension larger than one. A phase of matter in this context is identified by the set of gapped eigenstates of local Hamiltonians that can be smoothly connected. Recently a concept of quantum phase has emerged in the more general context of Lindbladian evolution <cit.>. The area law for one-dimensional gapped state has already been proven directly <cit.>. A proof for higher dimensions is still lacking, nevertheless under additional mild conditions Masanes has proven a low entropy law <cit.>.


In our work we are able to upper bound, as a function of time and α, the variation of the α-Rényi entropy, 0 < α < 1,  of the partial trace of the time evolution of a generic product state, when the time evolution is induced by a nearest neighbor Hamiltonian H = ∑_j=-L^L-1 H_j,j+1 in one dimension. With r the local Hilbert space dimension, the bound reads:

    Δ S_α(t) ≤ K   r^1/α-1 max_j{H_j,j+1}  t  log r + K'

with both K and K' constants of order 1.
This is the first rigorous result, as far as we know, on the dynamical evolution of α-Rényi entropy, with 0<α < 1. The upper bound on the von Neumann entropy (entanglement entropy) of the same reduced state follows as the  α→ 1 limit.


We are also able to generalize (<ref>), despite only with a non-rigorous argument, see section <ref>, to the evolution of an initial pure state with α-Rényi entropy (of bipartition) up to O ( log L ), in this case the constant K' becomes of the order of the initial entropy of the state.


The linear dependence on t of the upper bound on  Δ S_α(t) that we have established is the best possible for large t. In fact in <cit.> a lower bound on the dynamical generation, starting from a product state, of the entanglement entropy for the Ising model was proven: Δ S(t) ≥4/3π t -1/2ln t -1.
The Ising model falls into the class of models, 1-dimensional nearest-neighbor, that we are considering here. Δ S(t) denotes the variation of the von Neumann entropy.  α-Rényi entropies, α < 1, are upper bounds of the von Neumann entropy, therefore they cannot growth slower than t for large t.


The bound (<ref>) is easy to generalize to the case of a k-neighbors Hamiltonian. We explicitly consider the case of interactions decreasing exponentially in section <ref>, proving a bound as (<ref>) but only for values of α sufficiently close, from below, to 1. 




Independently from the existence of an energy gap, the bound (<ref>) shows that a state of a one dimensional local system with interactions decaying fast enough, that has an efficient MPS representation, continues upon time-evolution to have an efficient MPS representation up to times at least of the order of log L.


Our work is organised as follows: in the section <ref> we present the physical setting and discuss how the unitary dynamics is approximated making use of Lieb-Robinson bounds. In here we follow the approach  considered in <cit.>. A short proof of the Lieb-Robinson bound, following <cit.>, is in the appendix <ref>. In the section <ref> we present the upper bound on α-Rényi entropies, 0 < α < 1, proven by Audenaert in <cit.> and discuss how it reduces to the more familiar Fannes-Audenaert-Petz bound <cit.> on the von Neumann entropy in the limit α→ 1. In the section <ref> we set up and sum up the series that gives rise to the upper bound on the Rényi entropies (<ref>), discussing how the faster than exponential decrease of the spatial part of the Lieb-Robinson bound of a strictly local Hamiltonian enables the summation of the series for all 0 <  α≤ 1, for arbitrarily large one-dimensional systems. We eventually state the final bound on the dynamical variation of the Rényi entropies (<ref>). In the section <ref> we extend our bound to systems with exponential decay of interactions for values of 0 <  α≤ 1 close enough to 1. We then show in <ref> the generalization of the theory to pure states with entanglement of order log L. In the section <ref> we discuss  possible extensions of our work and relations with other results in the literature. A set of appendices close the paper collecting proofs and technicalities.









§ PHYSICAL SETTING
 





The physical setting is that of a one-dimensional spin chain with sites in the interval [-L,L]. The local Hilbert space is C^r. The Hamiltonian is a sum of nearest neighbors terms:

    H=∑_j=-L^L-1 H_j,j+1

H_j,j+1 is a short hand for 1_r ⊗ ... ⊗ H_j,j+1⊗ ...  ⊗1_r, to make clear  that the support of H_j,j+1 is on the sites {j,j+1}.  We define J := max{  H_j,j+1} < ∞, with · denoting the operatorial norm, that is the maximum singular value.
Our final goal is the evaluation of the variation, associated to time evolution, of the α-Rényi entropies, with 0 < α≤ 1, of a reduced density matrix. Namely we want to upper bound as a function of time:

    Δ S_α (t) :=  | S_α( _[1,L]ρ(t) ) - S_α( _[1,L]ρ) |
 
_[1,L] denotes partial tracing on the Hilbert space ⊗_j=1^L ℋ_j, while ρ(t) is the evolution, at time t of the state ρ, ρ(t) := e^-itHρ e^itH.
We decompose the operator of unitary evolution e^-itH, following the approach of <cit.>, to put in evidence that the term H_0,1 of the Hamiltonian (<ref>) is responsible for the “spread” of entanglement among the two halves of the system. We define

    V(t):= e^it (H_[-L,0]+H_[1,L]) e^-itH = e^it (H-H_[0,1]) e^-itH

Moreover: H_[-L,0] := ∑_j=-L^-1 H_j,j+1 and H_[1,L] := ∑_j=1^L-1 H_j,j+1, that collect the terms in the Hamiltonian (<ref>) with support contained respectively in the intervals [-L,0] and [1,L]. The term H_0,1 has instead support on both these intervals.  
It is easy to verify that replacing e^-itH with V(t) leaves Δ S_α (t) in equation (<ref>) invariant.
A crucial idea from <cit.> is to write V(t) as a recursive product:

    V(t)= V_Λ_L   V^*_Λ_L-1 V_Λ_L-1  ... V^*_Λ_l  V_Λ_l =  ( ∏_j=l+1^L V_Λ_j V^*_Λ_j-1) V_Λ_l

The notation used in (<ref>) is as follows. Λ_j := [-j,j], V_Λ_j is the V-evolution associated to the Hamiltonian  H_[-j,j]:=∑_k=-j^j-1 H_k,k+1, namely:

    V_Λ_j(t):= e^it (H_[-j,0]+H_[1,j])  e^-itH_[-j,j] = e^it (H_[-j,j]-H_[0,1]) e^-itH_[-j,j]

According to (<ref>) it holds: V_Λ_L(t)=V(t). The value of l in the RHS of (<ref>) is a variational parameter, that we will tune, see appendix <ref>, to optimize  the upper bound on the dynamical variation of Rényi entropy. In particular since the LHS of (<ref>) is  l-independent it means that at different instants of time t the value of l that optimizes the bound can be different, therefore the value of l will be in general time-dependent.

Intuitively each factor V_Λ_jV^*_Λ_j-1 in (<ref>) is close to the identity. To compare such  dynamics we will make use of an equivalent formulation of the Lieb-Robinson bounds through the quantity Δ_j(t) that is defined, following <cit.>, and discussed in the appendix <ref>. 










§ AUDENAERT BOUND ON Α-RÉNYI ENTROPIES, 0 < Α < 1 
 





In the paper <cit.> Audenaert presents a sharp form of the Fannes bound on the difference among the von Neumann entropy of a pair of states (density matrices) ρ and σ. The same bound also appears in the book of Petz <cit.>, theorem 3.8, where the proof is attributed to Csiszár.


As a byproduct, Audenaert was able to obtain in <cit.> a sharp bound on the difference among the α-Rényi entropies, with 0 < α < 1, of ρ and σ. The α-Rényi entropies, with α≠ 1, are defined as:

    S_α (ρ) = 1/1-αlogρ^α

It holds S_α (·) ≤ S_β (·) with α≥β. This means that our attention is focused on quantum entropies that upper bound the von Neumann entropy, that is the α→ 1 limit of the α-Rényi entropies. 

    S (ρ) =  -ρlogρ

With  T := 1/2ρ-σ_1 the trace distance among the two states ρ and σ, and d the dimension of the Hilbert space where the states are acting upon, the bound of Audenaert reads as equation A3 of <cit.>: 

    |S_α(ρ)-S_α(σ)| ≤1/1-αlog[ ( 1-T )^α + (d-1)^1-αT^α]

This inequality is sharp, in the sense that for every value of T, 0 ≤ T ≤ 1, there is a pair of states that saturates the bound. These are given by:

    ρ=Diag(1,0, ... , 0), σ=Diag( 1-T,T/d-1, ... ,T/d-1)

The inverse is also true, namely that these are the only states that saturates the bound <cit.>.
We see that with α = 0 the upper bound (<ref>) becomes T independent and therefore trivial, being always equal to log d, that is the maximal value of the α-Rényi entropy with α=0, called Hartley, or max, entropy <cit.>. In what follows we explicitly assume 0 < α≤ 1, with  α = 1 taken as a limit.  
The states (<ref>) play the same role also in the Audenaert-Fannes-Petz inequality on the difference of von Neumann entropy:

    |S(ρ)-S(σ)| ≤ T log (d-1) + H_2(T,1-T)

H_2(T,1-T) denotes the binary Shannon entropy: H_2(T,1-T) := -Tlog T -(1-T) log (1-T).
It can be easily shown that the bound (<ref>) is the limit for α→ 1 of the bound  (<ref>). The Rényi entropies, with the von Neumann entropy being the continuation at α = 1, are decreasing in α for all positive α. This can be shown, for example, using the concavity of the logarithm and the Jensen inequality. The upper bound (<ref>) is also decreasing with 0 ≤α < 1, we explicitly show that in appendix <ref>. 
Both the bounds (<ref>) and (<ref>) are increasing in T only with T ∈ [0,1-1/d]. With T = 1- 1/d the maximum value is reached, that for both entropies is equal to log d. As a consequence, since it is usually only possible to obtain upper bounds on the trace  distance the following slight modification of the bound (<ref>), that is given in Theorem 3.1 of <cit.> is useful. With T ≤ R ≤ 1 it is: 



    |S_α(ρ)-S_α(σ)| ≤1/1-αlog ( (1-R)^α + (d-1)^1-α R^α)    with  R ≤ 1-1/d
    log d    with  R ≥ 1 - 1/d

In (<ref>) Δ S_α is increasing in R. 
For the analogous resetting of the bound given in (<ref>), we refer to Lemma 1 of <cit.> and <cit.>. Recently a new upper bound on the variation of von Neumann entropy that makes use of the trace distance and the norm distance among states has appeared <cit.>.


A further upper bound to (<ref>) and (<ref>), that is not tight but increasing in R, for all 0≤ R ≤ 1, is: 

    |S_α(ρ)-S_α(σ)| ≤1/1-αlog ( 1- α R + (d-1)^1-α R^α)

This follows from (1-u)^α≤ (1-α u) for u> -1 and 0 < α < 1, and the fact that the logarithm is an increasing function.


An important remark about the bounds (<ref>) and (<ref>) is that their derivatives in the trace distance, when evaluated at T=0, diverge. It means that increasing T, from T=0 where the two states coincide, the difference in entropy increases at a diverging rate. For the von Neumann entropy and the α-Rényi entropy, with α > 1, it has been established <cit.> that, for local Hamiltonians systems, the dynamically generated entropy actually grows at a finite rate from t=0. 


In this work we will be interested in the dynamical generation of entropy over any time interval, proving that with strictly local Hamiltonians is always independent form the system's size for all α∈ (0,1) , while when interactions decay exponentially there always exist, for any finite system, a set of α < 1 close to 1 such that the scaling is system-size independent.









§ UPPER BOUNDING Δ S_Α (T) FOR A NEAREST NEIGHBOR HAMILTONIAN
 






We have already defined in (<ref>), Δ S_α (t) :=  | S_α( _[1,L]ρ(t) ) - S_α( _[1,L]ρ) | as the object of our investigation. We now set an upper bound to  Δ S_α (t) in the form of a series making use of the decomposition of V(t) given in equation (<ref>). 


We write  Δ S_α (t) first as a telescopic sum, as shown below, and then use the triangular inequality:

    Δ S_α (t)     =  | S_α( _[1,L] V(t) ρ V^*(t) ) - S_α( _[1,L]ρ) | 
        = | S_α( _[1,L] V_Λ_L(t) ρ V_Λ_L^*(t) ) - S_α( _[1,L]   V_Λ_L-1(t) ρ V_Λ_L-1^*(t) ) + 
        + S_α( _[1,L]   V_Λ_L-1(t) ρ V_Λ_L-1^*(t) ) -  ... + S_α( _[1,L]   V_Λ_l(t) ρ V_Λ_l^*(t) ) - S_α( _[1,L]ρ)  | 
       ≤∑_k=l+1^L | S_α( _[1,L]   V_Λ_k(t) ρ V_Λ_k^*(t) ) - S_α( _[1,L]   V_Λ_k-1(t) ρ V_Λ_k-1^*(t) )| + 
     
        + | S_α( _[1,L]  V_Λ_l(t) ρ V_Λ_l^*(t) ) - S_α( _[1,L]ρ)  |

At this point a crucial observation  follows from the assumption of ρ being a product state

    ρ = ⊗_j=-L^L ρ_j

with ρ_j : C^r →C^r. This  implies that each _[1,L]   V_Λ_k(t) ρ V_Λ_k^*(t) factorizes: the unitary V_Λ_k is supported on the interval [-k,k], with a slight abuse of notation we write V_Λ_k = V_Λ_k⊗1_[-k,k]^c, where [-k,k]^c denotes the complement of [-k,k] in [-L,L]. Moreover with 

    ρ_[-k,k]:= ⊗_j=-k^k ρ_j

we have that:

    _[1,L]( V_Λ_kρ V^*_Λ_k) = _[1,k]_[k+1,L]( V_Λ_kρ_[-k,k] V^*_Λ_k⊗ρ_[-k,k]^c) = _[1,k]( V_Λ_kρ_[-k,k] V^*_Λ_k) ⊗_[k+1,L]ρ_[-k,k]^c

It follows that:

    | S_α( _[1,L]   V_Λ_k(t) ρ V_Λ_k^*(t) ) - S_α( _[1,L]   V_Λ_k-1(t) ρ V_Λ_k-1^*(t) )| 
        = 
     | S_α(_[1,k]( V_Λ_kρ_[-k,k] V^*_Λ_k) ⊗_[k+1,L]ρ_[-k,k]^c) -  S_α( _[1,k]( V_Λ_k-1ρ_[-k,k] V^*_Λ_k-1) ⊗_[k+1,L]ρ_[-k,k]^c) | 
        = | S_α( _[1,k]( V_Λ_kρ_[-k,k] V^*_Λ_k) ) -  S_α( _[1,k]( V_Λ_k-1ρ_[-k,k] V^*_Λ_k-1) ) |

In (<ref>) we have embedded V_Λ_k-1 in the Hilbert space over the interval [-k,k] simply with a tensorization with the identity, that means, with a slight abuse of notation: V_Λ_k-1 = 1_r⊗ V_Λ_k-1⊗1_r. In equation (<ref>) we have used the additivity of Rényi entropies: S_α(ρ⊗σ) = S_α(ρ) + S_α(σ). We then realize that the density matrices in equation (<ref>) are supported on the Hilbert space over the interval [-k,0] that has dimensionality r^k+1. 
We define:

    T_k(t):= 1/2|̱_[1,k]( V_Λ_kρ_[-k,k] V^*_Λ_k) - _[1,k]( V_Λ_k-1ρ_[-k,k] V^*_Λ_k-1)   |̱_1

Using the upper bound (<ref>) for each term of the sum (<ref>) and taking into account (<ref>) we have that:

    Δ S_α (t) ≤1/1-α∑_k=l+1^Llog[ 1-α T_k(t) + (r^k+1-1)^1-αT_k(t)^α] + (l + 1)log r

We notice that we have upper bounded the last term in (<ref>), | S_α( _[1,L]ρ_l(t) ) - S_α( _[1,L]ρ)  |, with the trivial bound for density matrices supported on a Hilbert space of dimension r^l+1, that is (l + 1)log r. This choice will be justified a posteriori after the minimization of (<ref>) with respect to l.


The upper bound on the trace distance that is developed in the appendix <ref>, being J := sup{H_j,j+1} and the rescaled time t':= 4   J   t, reads:

    T_k(t) ≤1/2  V_Λ_kρ_[-k,k] V^*_Λ_k  -V_Λ_k-1ρ_[-k,k] V^*_Λ_k-1_1 ≤1/4(4 J t)^k/k! = 1/4t'^k/k!

We have used the fact that for a generic matrix A defined on a bipartite Hilbert space ℋ_1 ⊗ℋ_2, it holds _ℋ_j A _1 ≤  A _1, with j ∈{1,2}, as a reference see, for example, equations 7 and 17 of <cit.>. This also generalizes to all CPTP maps 𝒩, namely 𝒩(ρ) -𝒩(σ) _1 ≤ρ - σ_1. 


The faster than exponential decrease in k of the upper bound (<ref>) on T_k(t), ensures that, when coupled in (<ref>) with  (r^k+1-1)^1-α,  the sum will converge. We will pick up the L →∞ limit.
Inserting (<ref>) into (<ref>), and defining u_k(t'):= 1/4( e t'/k)^k > 1/4t'^k/k!, see the brief discussion in <ref> on u_k(t'), we get:

    Δ S_α (t) ≤1/1-α∑_k=l+1^Llog[ 1- α u_k(t') + r^(k+1)(1-α) u_k(t')^α] + (l + 1)log r



In the appendix <ref> is proven that (<ref>) has a unique minimum with respect to l.  
A positive integer l that minimizes (<ref>) satisfies the following pair of conditions. Denoting f(l,t') the RHS of (<ref>), the minimum is reached for that value of l such that f(l,t') is smaller of both f(l+1,t') and f(l-1,t').

    f(l,t') ≤ f(l+1,t') 
       f(l,t')  ≤ f(l-1,t')

These equations must be thought at a fixed α and at a fixed time t', with l determined as a function of α and t'. Then the equations for the minimization of Δ S_α(t) are:

    1/1-α log[ 1- α u_l+1(t') + r^(l+2)(1-α) u_l+1(t')^α] ≤log r  
       1/1-α log[ 1- α u_l(t') + r^(l+1)(1-α) u_l(t')^α] ≥log r

 This implies that despite we are not able to solve explicitly the system (<ref>), that can be done numerically, the approximate solution for u_l+1(t') that we are going to provide simply gives rise to a further upper bound to (<ref>). We prefer to provide a loser upper bound but with a transparent physical meaning, than a tighter one. Nevertheless, as discussed after (<ref>), for a generic local (nearest neighbor in this case) Hamiltonian the linear dependence in t' of the upper bound to Δ S_α is the best possible.  



We choose to provide an approximate solution to (<ref>) solving, again approximately, with α far enough from 1, the following equation

    1/1-α log[ r^(l+1)(1-α) u_l(t')^α] ≈log r

That leads to:

    u_l(t) ≈ r^-l1-α/α
       t'^l/4(e/l)^l≈ r^-l1-α/α
       t'/4^1/l(e/l) ≈ r^-1-α/α
        e   r^1-α/α  t' ≈ l   4^1/l

It is: 1 < 4^1/l≤ 4 with l ≥ 1. As the approximate solution of (<ref>) we pick: 

    l_α(t') := c   r^1-α/α  t'

with c > e. The reason for c > e in our approach is to ensure exponential decrease in t of the sum in  (<ref>), as discussed in appendix <ref>.
We stress again that this despite being an approximate solution of (<ref>), when l is replaced with l_α(t') in (<ref>) still provides an upper bound to Δ S_α(t) because of the uniqueness of the minimum in l of (<ref>).



We now plug (<ref>) into the sum (<ref>). We expect that for all t' it will be at most of order 1 because of the fast decrease of the u_k(t'), at a fixed t', with k ≥ l_α(t') +1. The upper bound to (<ref>)  is performed, consistently with the limit α→ 1, in appendix <ref>. Recollecting that t':=4Jt, this leads to 

    Δ S_α(t) ≤ K   r^1/α-1  J   t  log r + K'

with K=4c and K' ≤(1+e/4c(e-1))log r, see appendix <ref>, both of order 1.

It is interesting to compare our upper bound with the entanglement entropy rate obtained in <cit.> in the case of von Neumann entropy, α = 1. Their equation (133), being an upper bound on the rate for all times, implies, in their notations:

    Δ S_1(t) ≤ 2^ν +1 c (log d) A ∑_r r^2νh(r)  t

For comparison we write (<ref>) in the case of a one dimensional system, ν=1, for a strictly local Hamiltonian, in our notations: Δ S_1(t) ≤ 8 (log r) J   t. Looking at the coefficients of proportionality with t in (<ref>), with α=1, that is 4cJ log r, it appears that we do slightly worse than them. 












§ UPPER BOUNDING  Δ S_Α (T) FOR A SYSTEM WITH EXPONENTIAL DECAY OF INTERACTIONS
 






We will extend the results about the dynamical generation of α-Rényi entropy of a strictly local Hamiltonian to the case of an Hamiltonian with exponentially decaying interactions. 


We are able to generalize the approach that we set up in section <ref> for nearest-neighbour Hamiltonians to Hamiltonians that are sum of terms that are supported on the whole system but with decaying interactions. 
The scheme that we develop works for interactions with decay-rate at least that is exponential, f denotes such decay, and potentially could be extended to the case of sub-exponential decay.
Following <cit.>, a function f is defined to have sub-exponential decay if, for any γ <1, |f(t)| ≤ C_γexp(-t^γ), for some C_γ which depends on γ.


Denoting H = ∑_r H_r the Hamiltonian, each term H_r is supported on the whole lattice, we will refer to H_r as “centered” in r. H_r is an interaction decaying according to the function f , meaning that there exist constants J and ξ, that are independent from r, such that with r inside the region X, that for simplicity is assumed connected, it holds:

    1/2^|X^c|( _X^c H_r ) ⊗1_X^c - H_r ≤  J    f ( dist(r,X^c)/ξ)

For simplicity in (<ref>) we are considering the case of qubits, namely, the local Hilbert space dimension is equal to 2.
The physical interpretation of (<ref>) is straightforward: the approximation, in norm, of H_r with an operator that is supported on the region X containing r, improves enlarging the distance of r from the complement of X. The definition (<ref>) is employed, for example, in  <cit.>. We observe that if H_r was strictly local with support contained in X, then the LHS of (<ref>) would be exactly vanishing. Also taking f with compact support the RHS of (<ref>)
vanishes with X large enough, this corresponds to Hamiltonians with k nearest neighbors, a generalization of the case studied in section <ref>.


A Hamiltonian with decaying interactions does not provide an immediate way to identify the terms H_r responsible for the spread of entanglement across the two halves of the system, that was given for a generic nearest-neighbor Hamiltonian by H_I :=H_[0,1]. We introduce a distance b, that will be determined by minimization of the upper bound to the dynamical Rényi entropy, such that, the terms of H mostly responsible for the connection among the left and right halves of the system are those centered within the distance b from the origin: ∑_|r| < b H_r. The physical intuition is that b will be  in relation with the decaying length ξ of definition (<ref>).


With this in mind we set up a different telescopic sum than (<ref>). The first step is to rewrite:

    e^-itH  = e^-it ( H_[-L,-b] + H_[b,L]) e^it ( H_[-L,-b] + H_[b,L]) e^-itH

Where we have introduced, with 0 < b ≤ j ≤ L:

    H_[b,j] := 1/2^|([1,2j] ∩ [1,L])^c|( _([1,2j] ∩ [1,L])^c∑_r=b^j H_r ) ⊗1_([1,2j] ∩ [1,L])^c

H_[b,j] is supported on [1,2j]  ∩ [1,L]. We also define: 

    H_[-j,j] := 1/2^|([-2j,2j] ∩ [-L,L])^c|( _([-2j,2j]∩ [-L,L])^c∑_r=-j^j H_r ) ⊗1_([-2j,2j]∩ [-L,L])^c




We notice that H_[-L,L] = H. 
In equation (<ref>) we have put in evidence at the exponent the sum of the terms centered on the left half and right half of the system outside the interval (-b,b) that, as we said, represents the region that we assume to “connect” the left and right halves and be responsible of the entanglement spread.  

The first factor of (<ref>) is not supported across the two halves of the system therefore it does not contribute to the entropy. Looking at the second and third factor in (<ref>) we realize that, imagining the values of b and -b independent and setting them respectively equal to b=1 and -b=0, if the Hamiltonian H was a nearest neighbor Hamiltonian, then it would coincide with the V operator that we introduced in (<ref>), following <cit.>. 

We define 

    V_Λ_j(t) := e^it ( H_[-j,-b] + H_[b,j])   e^-it H_[-j,j]

that is supported on [-2j,2j]  ∩  [-L,L]. Then:

    Δ S_α (t)     =  | S_α( _[1,L] e^-itHρ e^itH) - S_α( _[1,L]ρ) | 
        = | S_α( _[1,L]V_Λ_L(t)  ρV_Λ_L^*(t)  ) - S_α( _[1,L]ρ)  |  
        = | S_α( _[1,L]V_Λ_L(t)  ρV_Λ_L^*(t)   ) -  S_α( _[1,L]V_Λ_L-1(t)   ρV_Λ_L-1^*(t) ) + 
        +  ... + S_α( _[1,L]V_Λ_l(t)  ρV_Λ_l^*(t) ) - S_α( _[1,L]ρ) | 
       ≤∑_k=l^L-1  | S_α( _[1,L]V_Λ_k+1(t)  ρV_Λ_k+1^*(t)   ) -  S_α( _[1,L]V_Λ_k(t)   ρV_Λ_k^*(t) )| + 
        + | S_α( _[1,L]V_Λ_l(t)  ρV_Λ_l^*(t) ) - S_α( _[1,L]ρ) |

At this point we again assume that ρ is a product state, therefore according to (<ref>) and the  following equations, we have:

    Δ S_α (t)    ≤∑_k=l^L-1  | S_α( _[1,2(k+1)]V_Λ_k+1(t)  ρ_[-2(k+1),2(k+1)]V_Λ_k+1^*(t)   ) -  S_α( _[1,2(k+1)]V_Λ_k(t)   ρ_[-2(k+1),2(k+1)]V_Λ_k^*(t) )| + 
        + | S_α( _[1,2l]V_Λ_l(t)  ρ_[-2l,2l]V_Λ_l^*(t) ) - S_α( _[1,L]ρ_[-2l,2l]) |

In the above we have used the fact that  V_Λ_k(t) is supported on [-2k,2k]. The dimensional reduction associated with product states allows the application of the Audenaert upper bound (<ref>). We then need to provide an upper bound to the trace distance that according to appendix <ref>, equations (<ref>) and (<ref>), reads:

    R_k(t)     := 1/2_[1,2(k+1)]( V_Λ_k+1(t)  ρ_[-2(k+1),2(k+1)]V_Λ_k+1^*(t) ) - _[1,2(k+1)](  V_Λ_k(t)  ρ_[-2(k+1),2(k+1)]V_Λ_k^*(t) ) _1 
       ≤1/2V_Λ_k+1(t)  ρ_[-2(k+1),2(k+1)]V_Λ_k+1^*(t) - V_Λ_k(t)  ρ_[-2(k+1),2(k+1)]V_Λ_k^*(t) _1 ≤   t   J   O (  g(b,J,t)   f ( k/4 ξ) )

With g(b,J,t):= b  t   J((e^vt-1) + a) + c, with a and c of O(1), and with v the Lieb-Robinson velocity of the Hamiltonian. We can now apply to (<ref>) the Audenaert bound (<ref>), that reads:

    Δ S_α (t) ≤1/1-α∑_k=l+1^Llog[ (1-R_k(t) )^α + (2^2(k+1)+1-1)^1-αR_k(t)^α] + 2l +1

We now consider explicitly the case of an Hamiltonian that is a sum of terms with exponential tails f(k/ξ) = e^-k/ξ. In this case the upper bound to  Δ S_α (t) is proven in (<ref>) of appendix  <ref>. The final result, for 1/1+1/8ξln2 < α≤ 1, is:

    Δ S_α (t) ≤β t + A(t)

with β > v/1/4ξ-(2ln2)1-α/α and A(t) decreasing exponentially fast for large t and A(0)=0. This implies that 

    lim_t→∞Δ S_α (t)/t <  2v/1/4ξ- (2ln2)1-α/α

The limit α→ 1, that gives the von Neumann entropy, is:

    lim_t→∞Δ S_1 (t)/t < 8vξ














§ DYNAMICAL Α-RÉNYI ENTROPY FOR INITIAL LOW ENTANGLED STATES
 






In this section we extend the theory that have developed in section <ref> for product states to initial low entangled states. This will allow us to show that, for one dimensional systems, states with an efficient MPS representation continue to have such efficient representation, upon time evolution, at least up to times of the order of log L. 


The main technical tools employed in this section are: an important result about the existence of an efficient MPS representation for low entangled states <cit.> in one dimension and an upper bound on the concavity of α-Rényi entropies, 0 <α <1, proven using the theory of majorization, in appendix <ref>, this result can be of independent interest.  


The results of this section will be, contrarily to all the previous results, based on an unproven assumption about the price paid, in terms of entropy, by replacing a product state with its time evolution, we clearly state this in equation (<ref>).  


Let us consider a low entangled state vector |v⟩∈ (C^2)^⊗ (2L+1), namely we assume that S_α(_[1,L] |v⟩⟨ v| ) = O(log L). The results of <cit.> ensure that a low entangled state |v⟩, together with an additional assumption on the distribution of the tails of the distribution of the Schmidt coefficients of |v⟩⟨ v|, see after equation (4) of <cit.>, has an efficient representation as a matrix product state (MPS). This means that the bond dimension (D) of the MPS is of order poly(L). In fact the rank of the state is upper bounded by D^2 <cit.>.  See, for example, <cit.> for reviews on MPS and related topics. We express this fact saying that |v⟩⟨ v|  has rank of O(poly(L)) or equivalently that is the linear combination of O(poly(L)) terms:

    |v⟩⟨ v| := k_-L,...,k_Lj_-L,...,j_L∑ c_(j_-L,...,j_L)c_(k_-L,...,k_L)   |j_-L⟩⟨ k_-L|  ⊗ ... ⊗ |j_L⟩⟨ k_L|

The states { |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| } in (<ref>) are all orthonormal. The number of elements of this basis is 2^2L+1. 



In section <ref> we upper bounded Δ S_α(t) for initial product states. Restricting our attention to pure product states, it is  S_α(0)=0. As show in appendix <ref>, equation (<ref>), a convex combination of O(poly(L)) pure product states has dynamical α-Rényi entropy upper bounded by K_α t + K' + O(log L). 

    S_α( _[0,L] U(t) ( ∑_j q_j | j ⟩⟨ j | ) U^*(t) ) ≤ K_α t + K' + O(log L)

(<ref>) implies that, for times t ≤ O(log L), the state U(t) ( ∑_j q_j | j ⟩⟨ j | ) U^*(t) has an efficient MPO (matrix product operator) representation.


Let us consider now a state vector | v ⟩  with low entanglement, as above. We claim that such a state, that has as efficient MPS representation continues to have an efficient representation under time evolution for time O(log L). This claims follows from the following assumption that we think is intuitively justified, but nevertheless unproven:


  for times t ≤ O(log L) we assume:

    S_α( _[0,L]j_-L,...,j_L∑( U(t)  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| U^*(t) ) U(t)  | v ⟩⟨ v | U^*(t) ( U(t)  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| U^*(t) ) ) ≃
        S_α( _[0,L]j_-L,...,j_L∑(  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L|  ) U(t)  | v ⟩⟨ v | U^*(t) ( |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| ) ) + O(poly(log L))


Let us discuss the meaning of <ref>. In equation (<ref>) we have replaced U(t)  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| U^*(t) with |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L|, considering this is a product state we know, because of our bound (<ref>), that its entropy increases at most linearly in time, so our reasoning is that overall within a time scale O(log L) the error in entropy in the LHS of (<ref>) would be of the same order. The upper bound of Audenaert, (<ref>), that we have employed to prove (<ref>), turns out to be inefficient in this context because the trace distance that we need to employ that bound is not small enough. At the same time since Audenaert's bound is also designed for two completely generic states while here our states are related by unitary evolution. From assumption <ref>, it follows that:


    S_α( _[0,L] U(t)  | v ⟩⟨ v |  U^*(t) ) ≤
        S_α( _[0,L]j_-L,...,j_L∑(  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L|  ) U(t)  | v ⟩⟨ v | U^*(t) ( |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| ) ) 
       ≃  S_α( _[0,L]j_-L,...,j_L∑  U(t) (  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| )  | v ⟩⟨ v | ( |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| ) U^*(t)  ) + O(poly(log L)) 
        = S_α( _[0,L]j_-L,...,j_L∑ |c_(j_-L,...,j_L)|^2  U(t) (  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| ) U^*(t)  ) + O(poly(log L)) 
       ≤ K_α t + K' + O(poly(log L))

The inequality in (<ref>) follows from pinching, see for example section II.5 of <cit.>, and the fact that pinching with respect to a product basis and partial tracing commute. In (<ref>) we made use of our assumption <ref>, and in (<ref>) we employed the definition (<ref>), where following by the assumption  S_α(_[1,L] |v⟩⟨ v| ) = O(log L), the number non vanishing coefficients c_(j_-L,...,j_L) in (<ref>) is of order O(poly(L)).



The conclusion from (<ref>), and lemma 2 of <cit.>, is that a state |v⟩ with an efficient MPS representation will continue to have, under time evolution, such efficient representation for times t ≤ O(log L).  














§ OUTLOOK
 






In this work we have made the first step towards a general theory that allows to obtain from the Lieb-Robinson bounds of local Hamiltonians an upper bound to the dynamical generation of α-Rényi entropies. In here we were mainly concerned with systems with linear lightcones, in <cit.> we present an application to systems with logarithm lightcones, mostly in relation with the many-body-localization phenomenology. 


A natural way to extend our theory is to look into higher dimensions and to consider explicitly the Hamiltonians with interactions decreasing slower than exponentially. This would allow to address the stability of the area law for α-Rényi entropies in local systems using the formalism of the quasi-adiabatic continuation, in analogy to what done for the von Neumann entropy by the authors of <cit.>.


An intriguing relation of our upper bound on dynamical generation of entropy with the topology of one dimensional spin systems, as quantified by the GNVW index <cit.>, looks possible: on one hand the authors of <cit.> have established that the dynamics of local Hamiltonians has a vanishing GNVW index, on the other the authors of <cit.> have put this index in relation with the Rényi entropy of such evolution via the Choi-Jamiołkowski mapping. We could expect that a tight bound on Δ S_α (t) should imply the vanishing of the GNVW index for the unitary evolution of local Hamiltonians. 










§ ACKNOWLEDGMENTS

 Daniele Toniolo and Sougato Bose acknowledge support from UKRI grant EP/R029075/1. D. T. is glad to acknowledge discussions about topics related to this work with Álvaro Alhambra, Ángela Capel, Angelo Lucia, Dylan Lewis, Emilio Onorati and Lluís Masanes.  






§ APPENDICES









§ DECREASE IN Α OF THE AUDENAERT UPPER BOUND ON Δ S_Α
 






α-Rényi entropies converge to the von Neumann entropy both in the limit α↑ 1 and α↓ 1. Therefore they are continued for  α = 1 into the von Neumann entropy.
A simple application of the Jensen inequality in relation with the concavity of the  log shows that with 0 <  α < 1, S_α is decreasing in α. In the same spirit we show that the Audenaert upper bound on Δ S_α of equation (<ref>), and therefore (<ref>), is decreasing in α.


    d/d  α[ 1/1-αlog( (1-T)^α + (d-1)^1-α T^α) ] = 
        = 1/(1-α)^2log( (1-T)^α + (d-1)^1-α T^α) + 
        + 1/1-α (log(1-T)) (1-T)^α - (log(d-1)) (d-1)^1-α T^α + (d-1)^1-α  (log T) T^α/(1-T)^α + (d-1)^1-α T^α
        = 1/(1-α)^2log( (1-T)^α + (d-1)^1-α T^α) + 
        + 1/(1-α)^2 (log(1-T)^1-α) (1-T)^α  +  log((d-1)^α-1 T^1-α) (d-1)^1-α T^α/(1-T)^α + (d-1)^1-α T^α

Defining 

    p_1 := (1-T)^α/(1-T)^α + (d-1)^1-α T^α≥ 0 
        p_2 := (d-1)^1-α T^α/(1-T)^α + (d-1)^1-α T^α≥ 0 
        x_1 := (1-T)^1-α
        x_2 := (d-1)^α-1 T^1-α

We have  p_1 + p_2 = 1, therefore since the logarithm is a concave function we can apply Jensen inequality:

    ∑_j p_j log x_j ≤log( ∑_j p_j  x_j  )

to the second term in (<ref>). Noticing that:

    ∑_j p_j  x_j = 1/(1-T)^α + (d-1)^1-α T^α

this results in:

    d/d  α[ 1/1-αlog( (1-T)^α + (d-1)^1-α T^α) ] ≤ 0












§ LIEB-ROBINSON BOUNDS FOR STRICTLY LOCAL HAMILTONIANS
 







The results of this section partially appeared in <cit.>, we include it because the faster than exponential decrease of the spatial part of the Lieb-Robinson bounds in (<ref>) is the crucial ingredient for the summation of the series (<ref>). H is as in (<ref>), A is any operator supported in [0,1]. The idea that motivates the definition of Δ_j(t) as below is to quantify how much the evolution in time of an operator is affected by terms in the Hamiltonian that are far away from the support of such operator.



    Δ_j(t)    :=  e^iH_Λ_j+1tAe^-iH_Λ_j+1t - e^iH_Λ_jtAe^-iH_Λ_jt
        = ∫_0^t ds  d/ds[ e^iH_Λ_j+1(t-s)(  e^iH_Λ_jsAe^-iH_Λ_js) e^-iH_Λ_j+1(t-s)] 
        = ∫_0^t ds    e^iH_Λ_j+1(t-s)[  H_Λ_j+1 - H_Λ_j , e^iH_Λ_jsAe^-iH_Λ_js] e^-iH_Λ_j+1(t-s)

We notice that H_Λ_j+1 - H_Λ_j = H_j,j+1 + H_-j-1,-j and H_Λ_j-1 have disjoint supports, therefore they commute, therefore we can insert e^iH_Λ_j-1sAe^-iH_Λ_j-1s in the RHS of the commutator in (<ref>) provided [0,1] that is the support of A does not overlap with [j,j+1] and [-j-1,-j], the smallest j that ensures this to be true is j=2.

    Δ_j(t)    = ∫_0^t ds    e^iH_Λ_j+1(t-s)[  H_j,j+1 + H_-j-1,-j, e^iH_Λ_jsAe^-iH_Λ_js - e^iH_Λ_j-1sAe^-iH_Λ_j-1s] e^-iH_Λ_j+1(t-s)
       ≤ 4 max{  H_j,j+1  ,   H_-j-1,-j}∫_0^t ds   e^iH_Λ_jsAe^-iH_Λ_js - e^iH_Λ_j-1sAe^-iH_Λ_j-1s

We recognize that the integrand in (<ref>) is Δ_j-1(t), meaning that we have set up a recursive equation.
Defining J := max_j { H_j,j+1}, (<ref>) is rewritten as 

    Δ_j(t)    ≤ 4 J ∫_0^t ds Δ_j-1(s) 
       ≤ (4 J)^j-1∫_0^t dt_1 ∫_0^t_1 dt_2 ... ∫_0^t_j-2 dt_j-1Δ_1(t_j-1)

At this point we observe that, based on equations (<ref>)-(<ref>)

    Δ_1(t) ≤ 4 J A  t

In conclusion:

    Δ_j(t) ≤A  (4 J)^j∫_0^t dt_1 ∫_0^t_1 dt_2 ... ∫_0^t_j-2 t_j-1 dt_j-1≤A  (4 J)^jt^j/j!

that renormalizing time as t':= 4 J t, reads:

    Δ_j(t) ≤A t'^j/j!



It turns out that an estimate on Δ_j(t) is equivalent to the usual form of the Lieb-Robinson bounds. Let us sketch this equivalence.


Given A and B operators defined for simplicity on a unidimensional lattice system Λ_L:=[-L,L], with A supported on [0,1] and the support of B on [l+1,l+1+|b|], implying ((A),(B) ) = l, we get:

    [A(t),B]    = [A(t)- e^iH_Λ_ltAe^-iH_Λ_lt + e^iH_Λ_ltAe^-iH_Λ_lt ,B] = [A(t)- e^iH_Λ_ltAe^-iH_Λ_lt ,B]
       ≤  2BA(t)- e^iH_Λ_ltAe^-iH_Λ_lt

With A(t) = e^iH_Λ_LtAe^-iH_Λ_Lt, we get:

    A(t)- e^iH_Λ_ltAe^-iH_Λ_lt =     e^iH_Λ_LtAe^-iH_Λ_Lt - e^iH_Λ_L-1tAe^-iH_Λ_L-1t + 
        + e^iH_Λ_L-1tAe^-iH_Λ_L-1t - e^iH_Λ_L-2tAe^-iH_Λ_L-2t  + 
        + ...  + 
        + e^iH_Λ_l+1tAe^-iH_Λ_l+1t - e^iH_Λ_ltAe^-iH_Λ_lt

Then:

    A(t)- e^iH_Λ_ltAe^-iH_Λ_lt≤A∑_j=l^∞t'^j/j!


    ∑_j=l^∞t'^j/j!    = t'^l  ∑_j=l^∞t'^j-l/j! = t'^l  ∑_k=0^∞t'^k/(k+l)! =  t'^l/l!∑_k=0^∞t'^k/k+l kk!≤ t'^l/l!∑_k=0^∞( t'/l+1)^k

In the last step of (<ref>) we have used k+l k≥(l+1)^k/k!. With t' < l+1 it follows:

    ∑_j=l^∞t'^j/j!≤ t'^l/l!1/1-t'/l+1

With t' ≤l+1/2, it is 1/1-t'/l+1≤ 2, then overall, given that l:= ((A),(B)), it holds:

    [A(t),B]≤ 4 AB t'^l/l!

We stress that the condition  t' ≤l+1/2 is not restrictive, in the sense that the trivial bound to [A(t),B], for generic A and B, is 2 AB, then let us determined the time t' such that t'^l/l! = 1/2, that with l large enough is

    t'= ( l!/2)^1/l≤ (el)^1/ll/e≤l+1/2

For a more general approach that applies also to any dimension, see the derivation of Lieb-Robinson bound in appendix C of <cit.>. 








We conclude this section with two remarks. If the operator A is positive, A ≥ 0, then the rescale time becomes t'=2 J t, in fact we can use A - A/21 = A/2, from <cit.>, for a better upper bound. With B is a generic operator, we have:

    [A,B]  =  [A - A/21,B] ≤ 2A - A/21  B  =  A    B



A more general relation is actually true, namely, if A ≥ 0, denoting λ_max and λ_min the largest and smallest eigenvalues of A, then A - λ_min1 - λ_max-λ_min/21 = λ_max-λ_min/2. With λ_min=0 we recover the previous one.


The second remark regards the extension of (<ref>) to any Schatten norm, that, with |A|:=√(A^*A), is defined as:

    A_p := ( |A|^p )^1/p

Using Hölder's inequality, we have A   B_p ≤A _p  B. This implies:

    e^iH_Λ_j+1tAe^-iH_Λ_j+1t - e^iH_Λ_jtAe^-iH_Λ_jt_p ≤A _p  t'^j/j!

This is relevant when considering quantities like the so called out-of-time-correlators (OTOC): ( σ  A(t)BA(t)B ). Denoting σ=1/d the maximally mixed state, with A=A^*, B=B^*, A^2=B^2=1, and with j the distance among the supports of A and B we get:

    1/d( [A(t),B]^* [A(t),B] )=: 1/d [A(t),B] _2^2 = 2 - 2/d(  A(t)BA(t)B ) ≤1/d(4 A _2  t'^j/j!)^2 =  (  4 t'^j/j!)^2
       1/d(  A(t)BA(t)B ) ≥ 1 - 8 (   t'^j/j!)^2 ≥ 1 -  (  8( e t'/j)^j )^2

At t=0, A and B commutes, moreover, under the assumption A^2=B^2=1, it holds: 1/d(  ABAB ) = 1/d(  A^2B^2 ) = 1, then the bound (<ref>) is saturated. Also:

    1/d | (  A(t)BA(t)B ) | ≤1/d |  A(t)BA(t)B | ≤1/d B A(t)BA(t) _1  ≤1/d B A^2(t)B _1  ≤1/d B ^2 A^2(t) _1 = 1

In (<ref>) we have used the fact that for two matrices C and D such that the product CD is Hermitean then: CD _1  ≤DC_1, that is theorem 8.1 of <cit.>.


The interpretation of (<ref>) is straightforward: the decrease of (  A(t)BA(t)B ) signals the increase of the overlap among the supports of A(t) and B starting from a time scale of the order of the distance among the supports of A and B. 
We stress that with A ≥ 0 and B ≥ 0 both positive then (  A(t)BA(t)B ) is positive as well, is fact a product of positive operators is positive.



 














§ EVALUATING THE TRACE DISTANCE (<REF>)
 





We now upper bound

    T_k(t) :     = 1/2_[1,k]  V_Λ_k(t) ρ_[-k,k]  V^*_Λ_k(t) - _[1,k] V_Λ_k-1(t) ρ_[-k,k]  V^*_Λ_k-1(t) _1 
        = 1/2_[1,k]  V_Λ_k(t) ρ  V^*_Λ_k(t) - _[1,k] V_Λ_k-1(t) ρ  V^*_Λ_k-1(t) _1

To do so we employ the following upper bound where U(t) is a generic time-dependent unitary operator with 
U(0)= 1_ℋ and σ a generic density matrix.

    U(t) σ U^*(t) - σ_1    = ∫_0^t ds  d/ds( U(s)σ U(s)^* ) _1 =  ∫_0^t ds  [ ( d/ds  U(s) ) σ U(s)^* + U(s) σ( d/ds  U^*(s) ) ] _1 
       = ∫_0^t ds  [ ( d/ds  U(s) ) σ U(s)^* - U(s) σ U(s)^* ( d/ds  U(s) ) U^*(s) ] _1 
       = ∫_0^t ds   U(s) [ U^*(s) ( d/ds  U(s) ) σ  -  σ U(s)^* ( d/ds  U(s) ) ] U^*(s)  _1 
       ≤∫_0^|t| ds [ U^*(s)  d/ds  U(s)  , σ] _1 
     ≤ 2 ∫_0^|t| ds   U^*(s)  d/ds  U(s)   
       ≤ 2|t| sup_s∈[0,|t|] U^*(s)  d/ds U(s)



We now apply (<ref>) to (<ref>).



    V_Λ_k+1(t) ρ  V^*_Λ_k+1(t) - V_Λ_k(t) ρ  V^*_Λ_k(t) _1 =    V_Λ_k^*(t)  V_Λ_k+1(t) ρ  V^*_Λ_k+1(t) V_Λ_k(t)  - ρ_1 
       ≤ 2 ∫_s=0^|t| ds  V^*_Λ_k+1(s) V_Λ_k(s) d/ds(  V_Λ_k^*(s)  V_Λ_k+1(s) ) 
        = 2 ∫_s=0^|t| ds  V^*_Λ_k+1(s) V_Λ_k(s) d/ds(  V_Λ_k^*(s) ) V_Λ_k+1(s) + V^*_Λ_k+1(s) d/ds  V_Λ_k+1(s)

Recalling that V_Λ_k(s):= e^is (H_Λ_k -H_I)  e^-is H_Λ_k, where we have defined H_I := H_[0,1], we have that:

    i d/ds  V_Λ_k(s) = e^is (H_Λ_k -H_I) H_I e^-is (H_Λ_k -H_I)V_Λ_k(s)

Then:

    T_k+1(t) ≤∫_s=0^|t| ds  e^is (H_Λ_k -H_I) H_I e^-is (H_Λ_k -H_I)  -  e^is (H_Λ_k+1 -H_I) H_I e^-is (H_Λ_k+1 -H_I)  =  ∫_s=0^|t| ds Δ_k(s) 
       ≤∫_s=0^|t| ds H_I(4 J s)^k/k!≤1/4t'^k+1/(k+1)!

We remark that the dynamics of Δ_k(t) is here generated by H_Λ_k - H_I, but all the steps that lead to (<ref>) are equally valid. To obtain (<ref>) we have used (<ref>), the definition of J := sup{H_j,j+1}, that implies H_I ≤ J, and finally the definition of t':= 4Jt.


 

 
 
 
 
 


§ BRIEF DISCUSSION OF U_K(T)
 






We have defined u_k(t) := 1/4(e   t/k)^k ≥1/4t^k/k!. u_k(t) upper bounds the trace distance. With a fixed k, a sufficient condition for u_k(t) ≤ 1 is given by t ≤k/e. 

It is easy to see that, with a fixed t, u_k(t) is a decreasing function of k, in fact:

    u_k+1(t)/u_k(t) = (e   t/k+1)^k+1/(e   t/k)^k=e   t/k+1( k/k+1)^k < e   t/k+1

The condition t ≤k/e, that implies u_k(t) < 1, also ensures that u_k(t) is a decreasing function of k.




 
 
 
   
   
 



§ ALGORITHM FOR THE MINIMIZATION OF THE SUM THAT UPPER BOUNDS Δ S_Α(T)
 





In this section we discuss the existence and uniqueness of the solution l_α(t) that satisfies the two equations (<ref>). We consider α∈ (0,1), the value α=1 can be dealt with analogously. 



  * We fix t  and consider a value l̅ of l such that both the LHSs of the equations in (<ref>) are larger equal than 1 such that u_l(t) ∈ [0,1-1/2^l̅ +2].  We will discuss the minimal t that allows the existence of such an l̅ at the end of this section.


  * We increase l from l̅ to l̅ +1. We remark the crucial fact that the value taken from the LHS of the second equation is now the value that the LHS of the first equation in (<ref>) had at the previous step. Therefore they cannot jump in one step from being both larger than one to be both smaller than one. Since u_l(t) < 1 is a decreasing function of l and both the LHSs of (<ref>) are increasing in u with u_l(t) ∈ [0,1-1/2^l̅ +2] they  will both decrease. 


  * We will repeat increasing l till the LHS of the first equation will become smaller than 1. This will certainly happen at a certain point, in fact with u of the order of 2^-(l+2)1-α/α  the LHS of (<ref>) will be of order 1. This is the value of l that minimizes (<ref>). 


What can almost certainly be improved of our approach is the constant of proportionality of t in the upper bound (<ref>). That is left for future research.   







 


§ ESTIMATE OF THE SERIES IN (<REF>)
 






We want to estimate the series in (<ref>), that is copied below (<ref>),  using  l_α(t') := c   r^1-α/α  t', with c > e as in (<ref>), and u_k(t'):= 1/4( e t'/k)^k. The result will be that for every t'>0 and α∈ (0,1] (<ref>) is upper bounded by a constant of order 1. 



    1/1-α∑_k=l_α+1^Llog[ 1 - α u_k(t') + r^(k+1)(1-α) u_k(t')^α]



Let us discuss the fact that the argument of the log in (<ref>) is an increasing function of 0 ≤ u ≤ 1.

    0 = ∂/∂ u[ 1- α u + r^(k+1)(1-α) u^α] = -α + α r^(k+1)(1-α) u^α-1

That has the solution, corresponding to a maximum, as it is seen by the always negative second derivative, u = r^k+1. r^k+1 has minimal value, reached for k=1 (corresponding to t=0), equal to r^2. The minimal value of r is 2, therefore, in the interval 0 ≤ u ≤ 1, the RHS of (<ref>) is an increasing function of u. It means that replacing u with an upper bound (smaller than r^2) we still obtain an upper bound, despite the negative term - α u in the argument of the log in (<ref>). 


We observe that:

    r^(k+1)(1-α) u_k(t')^α = r^(k+1)(1-α)/4^α( et'/k)^k α = r^(1-α)/4^α r^kα1-α/α( et'/k)^k α = r^(1-α)/4^α( r^1-α/αet'/k)^k α = r^(1-α)/4^α( e/c l_α/k)^k α

We then upper bound (<ref>):

    1/1-α∑_k=l_α +1 ^Llog[ 1- α u_k(t') + r^(k+1)(1-α) u_k(t')^α] 
        = 1/1-α∑_k=l_α +1 ^Llog[ 1- α1/4( e/c r^1-α/αl_α/k)^k + r^(1-α)/4^α( e/c l_α/k)^k α] 
       ≤1/1-α∑_n=1 ^∞[ - α1/4(e/c r^1-α/α)^l_α + n(  l_α/l_α + n)^l_α + n + r^(1-α)/4^α(e/c)^α(l_α + n)(  l_α/l_α + n)^α(l_α + n)]



In the last step we have changed the variable of summation from k to n, with k=l_α +n,  moved the upper limit of the sum to ∞, and used the inequality log(1+x) ≤ x with x ≥ -1, as discussed above.


It is well known that the function (x/x+n)^x+n = (1-n/x+n)^x+n is increasing in x, with lim_x→∞(1-n/x+n)^x+n = e^-n. With x=l_α this corresponds to the t →∞ limit. Then going back to  (<ref>) we can further upper bound it with:

    ≤1/1-α[ - α1/4(e/c r^1-α/α)^l_α +1∑_n=1 ^∞ e^-n + r^(1-α)/4^α(e/c)^α(l_α + 1)∑_n=1 ^∞  e^-α n] 
        = 1/1-α[ - α1/4(e/c r^1-α/α)^l_α +11/e-1 + r^(1-α)/4^α(e/c)^α(l_α + 1)1/e^α-1] 
        = 1/1-α[ - α1/4(e/c r^1-α/α)^cr^1-α/αt' +11/e-1 + r^(1-α)/4^α(e/c)^α(cr^1-α/αt' + 1)1/e^α-1]

It is easy to see that for all t > 0, in the limit α→ 0 the upper bound (<ref>) tends to zero. To achieve this, it has been crucial the introduction of the  parameter c > e. Moreover for all t ≥ 0, the limit α→ 1 of (<ref>) gives:

    lim_α→ 11/1-α   [ - α1/4(e/c r^1-α/α)^l_α +11/e-1 + r^(1-α)/4^α(e/c)^α(l_α + 1)1/e^α-1] 
        = log r/4(e-1)(e/c)^et'+1≤log r/4(e-1)(e/c)

For qubits, r=2, and c/e=1.2, 1/4(e-1)e/c≃ 0.12. For all t > 0 the upper bound (<ref>) is exponentially decreasing in t, this is also the case in the limit α→ 1 as seen in (<ref>). 




Denoting K'-log r the upper bound in (<ref>), in agreement with (<ref>), it is:

    Δ S_α(t) ≤ l_α(t')  log r + K'

Then:

    Δ S_α(t) ≤  c   r^1/α-1  t'  log r  + K'

Recollecting that t':= 4   t max_j {H_j,j+1} =: 4   t J, we obtain, with K := 4    c

    Δ S_α(t) ≤ K   r^1/α-1  J   t  log r + K'

as in (<ref>).











§ ACCOUNTING FOR TAILS OF INTERACTIONS IN THE HAMILTONIAN FOR AN UPPER BOUND OF Α-RÉNYI ENTROPY
 








In this section we provide the details to obtain the upper bound (<ref>).
Applying (<ref>) we obtain:

    V_Λ_k+1(t) ρV^*_Λ_k+1(t) -  V_Λ_k(t) ρV^*_Λ_k(t) _1 ≤ 2 t sup_s∈[0,t]V^*_Λ_k+1(s) V_Λ_k(s) d/ds(  V_Λ_k^*(s) ) V_Λ_k+1(s) + V^*_Λ_k+1(s) d/dsV_Λ_k+1(s)

Recalling that 

    V_Λ_k(t) := e^it ( H_[-k,-b] + H_[b,k])   e^-it H_[-k,k]

it holds:

    i d/dsV_Λ_k(s) = e^it ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-it ( H_[-k,-b] + H_[b,k])V_Λ_k(s)

Comparing this equation with (<ref>), where the generator of V_Λ_k was a unitary conjugation of the same Hermitean operator, H_I, for all k, we see that in (<ref>) instead we have the operator H_[-k,k] - H_[-k,-b] - H_[b,k] that, despite decaying outside of the interval [-b,b], is supported on [-2k,2k], therefore it is k-dependent. All together we have:  

    V_Λ_k+1(t) ρV^*_Λ_k+1(t) -  V_Λ_k(t) ρV^*_Λ_k(t) _1 
       ≤ 2 t sup_s∈[0,t]  e^is ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k,-b] + H_[b,k]) +  
        - e^is ( H_[-k-1,-b] + H_[b,k+1])( H_[-k-1,k+1] - H_[-k-1,-b] - H_[b,k+1]) e^-is ( H_[-k-1,-b] + H_[b,k+1])
       ≤ 2 t sup_s∈[0,t]  e^is ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k,-b] + H_[b,k]) +  
        - e^is ( H_[-k-1,-b] + H_[b,k+1])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k-1,-b] + H_[b,k+1]) +  
        + 2 t  H_[-k-1,k+1] - H_[-k-1,-b] - H_[b,k+1]  -  ( H_[-k,k] - H_[-k,-b] - H_[b,k])

We see that the first operator norm of (<ref>) has almost the structure of a Δ_k(s) as in appendix <ref>, the difference with (<ref>) is that, in the case of a strictly local Hamiltonian the operator whose Heisenberg evolutions are computed had a fixed support, while in (<ref>) it depends, despite only with tails, on k.
We recollect that with r inside the region X, that for simplicity is assumed connected, it holds:

    1/2^|X^c|( _X^c H_r ) ⊗1_X^c - H_r ≤  J    f ( dist(r,X^c)/ξ)



Let us show that the second operator norm of (<ref>) is bounded by O(J C_ξ f(k/ξ)), with f denoting the rate of decrease of interactions, as defined in (<ref>). We start considering H_[b,k+1] - H_[b,k]. To shorten the equations we define: _[m,n]^c(A) := 1/2^|[m,n]^c|_[m,n]^c (A) ⊗1_[m,n]^c. Let us assume for now 2k < L, the case 2k ≥ L is discussed in (<ref>). 

    H_[b,k+1] - H_[b,k] :=   _([1,2(k+1)]∩[-L,L])^c∑_r=b^k+1 H_r  -   _([1,2k]∩[-L,L])^c∑_r=b^k H_r  
        =  _([1,2(k+1)]∩[-L,L])^c( 1 -  _ ([1,2k]∩[-L,L])^c ∖ ([1,2(k+1)]∩[-L,L])^c ) ∑_r=b^k H_r   +  _([1,2(k+1)]∩[-L,L])^c   H_k+1

The norm of the first term in (<ref>) is small, of order J f( k/ξ), see (<ref>), in fact being the complements of the sets evaluated with respect to [-L,L]:

    ([1,2k]∩[-L,L])^c ∖ ([1,2(k+1)]∩[-L,L])^c = [2k+1,2(k+1)]∩[-L,L]

 the Hamiltonian term of ∑_r=b^k H_r with the closest centre to [2k+1,2(k+1)]∩[-L,L] is H_k. We also observe that the normalized trace defined above is a projection, see <cit.>, namely: _[m,n]^c( _[m,n]^c(A) ) = _[m,n]^c(A), therefore, as a superoperator, it has norm equal to 1. This implies that:

    _([1,2(k+1)]∩[-L,L])^c( 1 -  _([2k+1,2(k+1)]∩[-L,L])) ∑_r=b^k H_r ≤( 1 -  _ ([2k+1,2(k+1)]∩[-L,L]) ) ∑_r=b^k H_r  
       ≤   J ∑_r=b^k f(2k+1-r/ξ)  ≤   J C_ξ( f(k/ξ) - f(2k-b/ξ) )

with C_ξ a function of ξ. If, for example, the interactions have exponential tails, it is C_ξ = ξ, in fact:

    J ∑_r=b^k f(2k+1-r/ξ)     = J ∑_r=b^k e^-2k+1-r/ξ = J ∑_l=k+1^2k+1-b e^-l/ξ = J ∑_l=0^2k+1-b e^-l/ξ - J ∑_l=0^k e^-l/ξ
        = J ( 1-e^-2k+2-b/ξ/1-e^-1/ξ -  1-e^-k+1/ξ/1-e^-1/ξ) 
     ≤ J ξ( e^-k/ξ - e^-2k-b/ξ)

In the last line we have used e^1/ξ-1 ≥1/ξ 


It is important to discuss what happens with 2k ≥ L, in this case we must recollect that the support, for example, of H_[b,k] is [1,2k] ∩ [-L,L], therefore in (<ref>) we have: 

    ( [1,2k] ∩ [-L,L] )^c ∖( [1,2(k+1)]  ∩ [-L,L] )^c  = [-L,0] ∖ [-L,0] =  ∅

The partial trace associated with the empty set is the identity, therefore the term among parenthesis in (<ref>) is vanishing. 


We mention the fact that within this formalism we are able to recover the results for nearest neighbor Hamiltonians that we obtained in the first part of this paper, and actually to generalize those to the case of a k-neighbor Hamiltonian. In fact with f compactly supported and ξ the size of its support, the RHS of (<ref>) is vanishing. More in detail, considering

    f(j/ξ) = δ( ⌊ j - 1/2/ξ⌋ , 0 )

⌊ a ⌋ denotes the largest integer such that ⌊ a ⌋≤ a. δ(·,·) is the Kronecker delta. We take ξ integer, and ξ≥ 1. From the assumption r ∈ X in (<ref>), follows that (r,X^c) ≥ 1, then j ≥ 1 and integer. According to (<ref>), with  j ≤ξ, it is f(j/ξ) = 1, with j > ξ, f(j/ξ) = 0. This also shows that the value of b, that we have introduced in (<ref>), for the case of a strictly local Hamiltonian that is the sum of ξ-neighbors terms  equals ξ.


We are now ready to upper bound the operator norm of the second term in equation (<ref>). To ease the notation all the sets are meant to be the intersection with [-L,L].

    H_[-k-1,k+1] - H_[-k-1,-b] - H_[b,k+1]  -  ( H_[-k,k] - H_[-k,-b] - H_[b,k]) 
        = ( _[-2(k+1),2(k+1)]^c( 1 -  _ [-2k,2k]^c ∖ [-2(k+1),2(k+1)]^c ) ∑_r=-k^k H_r )  +  _[-2(k+1),2(k+1)]^c( H_-(k+1) + H_k+1) 
        -( _[-2(k+1),-1]^c( 1 -  _ [-2k,-1]^c ∖ [-2(k+1),-1]^c ) ∑_r=-k^-b H_r )  -  _[-2(k+1),-1]^c   H_-(k+1) 
        -( _[1,2(k+1)]^c( 1 -  _ [1,2k]^c ∖ [1,2(k+1)]^c ) ∑_r=b^k H_r )  -  _[1,2(k+1)]^c   H_k+1

Combining together in (<ref>) the partial traces on H_k+1 and H_-(k+1), using the same procedure as for (<ref>), and employing (<ref>), we obtain that (<ref>) is upper bounded by O( J C_ξ f(k/ξ)).


More precisely from (<ref>) we have five terms to keep into account. Term I:

    ( _[-2(k+1),2(k+1)]^c( 1 -  _ [-2k,2k]^c ∖ [-2(k+1),2(k+1)]^c ) ∑_r=-k^k H_r ) ≤( 1 -  _ [-2k-2,-2k-1] ∪ [2k+1,2k+2] ) ∑_r=-k^k H_r  
       ≤( 1 -  _ [-2k-2,-2k-1] ∪ [2k+1,2k+2] ) ∑_r=0^k H_r   + ( 1 -  _ [-2k-2,-2k-1] ∪ [2k+1,2k+2] ) ∑_r=-k^0 H_r  
       ≤ 2 J ∑_r=0^k f(2k+1-r/ξ) ≤ 2 J C_ξ( f(k/ξ) - f(2k/ξ) )



Term II:

    ( _[-2(k+1),-1)]^c( 1 -  _ [-2k,-1]^c ∖ [-2(k+1),-1]^c ) ∑_r=-k^-b H_r ) ≤( 1 -  _ [-2k-2,-2k-1]  ) ∑_r=-k^-b H_r  
       ≤  J ∑_r=b^k f(2k+1-r/ξ) ≤  J C_ξ( f(k/ξ) - f(2k-b/ξ) )



Term III:

    ( _[1,2(k+1))]^c( 1 -  _ [1,2k]^c ∖ [1,2(k+1)]^c ) ∑_r=b^k H_r ) ≤   J C_ξ( f(k/ξ) - f(2k-b/ξ) )



Term IV:

    _[-2(k+1),2(k+1)]^c H_k+1 -  _ [1,2(k+1)]^c  H_k+1 = _[-2(k+1),2(k+1)]^c( 1 -  _ [1,2(k+1)]^c ∖ [-2(k+1),2(k+1)]^c )  H_k+1
       ≤( 1 -  _ [-2(k+1),0]) H_k+1≤ J f(k+1/ξ)



Term V:

    _[-2(k+1),2(k+1)]^c H_-(k+1) -  _ [-2(k+1),-1]^c  H_-(k+1)≤ J f(k+1/ξ)

Overall: 

    Term I + ... + Term V≤ 4 J C_ξ( f(k/ξ) - f(2k/ξ) ) + 2 J f(k+1/ξ)

  
  
So far with (<ref>) we have upper bounded the second term in equation (<ref>), as mentioned above the first term is in relation with the object that in appendix <ref> is called Δ_k(t). In fact, from the first term in (<ref>):

    e^is ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k,-b] + H_[b,k]) +  
        - e^is ( H_[-k-1,-b] + H_[b,k+1])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k-1,-b] + H_[b,k+1])
    
      =∫_0^s du    d/du[  e^i(s-u) ( H_[-k-1,-b] + H_[b,k+1]) 
     e^iu ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) 
        e^-iu ( H_[-k,-b] + H_[b,k])  
      e^-i(s-u) ( H_[-k-1,-b] + H_[b,k+1])] 
    ≤∫_0^s du    [ H_[-k-1,-b] + H_[b,k+1] - 
      H_[-k,-b] - H_[b,k] ,  
        e^iu ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k])
      e^-iu ( H_[-k,-b] + H_[b,k])]

H_[-k-1,-b] + H_[b,k+1] - H_[-k,-b] - H_[b,k] is “mostly” supported around ± (k+1) but with tails on [-2k,2k], whereas H_[-k,k] - H_[-k,-b] - H_[b,k] is “mostly” supported around [-b,b] with tails on [-2k,2k].
The commutator in (<ref>) has the structure of a Lieb-Robinson bound, once we have shown that these two operators  can be approximated with operators that have supports disjoint and enough far apart to ensure a small upper bound. We now perform such approximations.


Analogously to what done in (<ref>) we have that:

    H_[-k-1,-b] + H_[b,k+1] -   H_[-k,-b] - H_[b,k]≤ 2J C_ξ(f(k/ξ)-f(2k-b/ξ) ) +  H_k+1 + H_-(k+1)
       ≤ 2J C_ξ(f(k/ξ)-f(2k-b/ξ) ) +  2J ≤ O(2J)

(<ref>) bounds the norm of the first entry in the commutator of (<ref>).


To apply the theory of Lieb-Robinson bounds for local Hamiltonians with tails we need to ensure that the two operators that enter the commutator are far apart, since they are actually terms of the Hamiltonian, namely they have tails, we need to restrict their supports, provided the error in doing so is small. Let us consider H_k+1 and H_-(k+1), that have supports [1,2(k+1)] and [-2(k+1),-1]. We define:

    H_k+1 := _[ ⌊3(k+1)/4⌋, ⌊5(k+1)/4⌋ ]^c H_k+1
       H_-(k+1) := _[ ⌊-5(k+1)/4,-3(k+1)/4⌋ ]^c H_-(k+1)

⌊ a ⌋ denotes the largest integer such that ⌊ a ⌋≤ a.  With this choice the size of the support is approximately |k|/2. It is easy to see that the error made is upper bounded by  J f(k/4 ξ), in fact:

    H_k+1 -  _ [ ⌊3(k+1)/4⌋, ⌊5(k+1)/4⌋ ]^c   H_k+1 = ( _[1,2(k+1)]^c( 1 - _ [ ⌊3(k+1)/4⌋, ⌊5(k+1)/4⌋ ]^c ∖ [1,2(k+1)]^c )  H_k+1)

is such that: [ ⌊3(k+1)/4⌋, ⌊5(k+1)/4⌋ ]^c ∖ [1,2(k+1)]^c = [1,⌊3(k+1)/4⌋ -1 ]  ⋃  [ ⌊5(k+1)/4⌋ , 2(k+1) ], therefore the distance of k+1 from this set is approximately ⌊k/4⌋.


On the other hand:

    H_[-k,k] - H_[-k,-b] - H_[b,k]
        =  _[-2k,2k]^c(∑_r=-k^- ⌊k/4⌋-1 H_r  + ∑_r=-⌊k/4⌋^⌊k/4⌋ H_r   + ∑_r=⌊k/4⌋+1^k H_r  )   
       - _[-2k,-1]^c( ∑_r=-k^-⌊k/4⌋-1 H_r + ∑_r=-⌊k/4⌋^-b H_r  )   - _[1,2k]^c( ∑_r=b^⌊k/4⌋ H_r + ∑_r=⌊k/4⌋+1^k H_r )

We pair terms from (<ref>) as follows:

    ( _[-2k,2k]^c - _[-2k,-1]^c) ∑_r=-k^-⌊k/4⌋-1 H_r ≤ J C_ξ(  f ( k/4 ξ) - f ( k/ξ) ) 
       ( _[-2k,2k]^c - _[1,2k]^c) ∑_r=⌊k/4⌋ +1^k H_r ≤ J C_ξ( f ( k/4 ξ) - f ( k/ξ) )

It is left _[-2k,2k]^c∑_r=-⌊k/4⌋^⌊k/4⌋ H_r - _[-2k,-1]^c∑_r=-⌊k/4⌋^-b H_r - _[1,2k]^c∑_r=b^⌊k/4⌋ H_r. We replace this operator with the same one but with support restricted to [-⌊k/2⌋, ⌊k/2⌋], we denote this operator as:

    H_[-b,b] : = _[-⌊k/2⌋,⌊k/2⌋]^c∑_r=- ⌊k/4⌋^⌊k/4⌋ H_r - _[-⌊k/2⌋,-1]^c∑_r=-⌊k/4⌋^-b H_r - _[1,⌊k/2⌋]^c∑_r=b^⌊k/4⌋ H_r

As shown in (<ref>), it follows that: 

    _[-2k,2k]^c∑_r=-⌊k/4⌋^⌊k/4⌋ H_r - _[-2k,-1]^c∑_r=- ⌊k/4⌋^-b H_r - _[1,2k]^c∑_r=b^⌊k/4⌋ H_r -H_[-b,b]
        =  _[-2k,2k]^c( 1 - _[- ⌊k/2⌋,⌊k/2⌋]^c ∖ [-2k,2k]^c ) ∑_r=-⌊k/4⌋^⌊k/4⌋ H_r 
        - _[-2k,-1]^c( 1 - _[-⌊k/2⌋,-1]^c ∖ [-2k,-1]^c ) ∑_r=-⌊k/4⌋^-b H_r 
       - _[1,2k]^c( 1 - _[1,⌊k/2⌋]^c ∖  [1,2k]^c ) ∑_r=b^⌊k/4⌋ H_r 
       ≤ J C_ξ( 4f ( k/4 ξ) - 2f ( k/2 ξ) - 2f ( k/2 - b /ξ) )



To upper bound the norm of the commutator (Lieb-Robinson bound) in (<ref>), we also need to upper bound the norm of each entry of the commutator. The norm of the first entry is provided by (<ref>).


The norm of the second entry of the commutator in (<ref>) is bounded as follows:

    H_[-k,k] - H_[-k,-b] - H_[b,k] := 
     _[-2k,2k]^c∑_r=-k^k H_r - _[-2k,-1]^c∑_r=-k^-b H_r
     - _[1,2k]^c∑_r=b^k H_r 
       ≤ | _[-2k,2k]^c( 1 - _ [-2k,-1]^c ∖ [-2k,2k]^c ) ∑_r=-k^-b H_r  + _[-2k,2k]^c( 1 - _ [1,2k]^c ∖ [-2k,2k]^c ) ∑_r=b^k H_r  +  H_[-b+1,b-1]
       ≤ |  ( 1 - _ [-2k,-1]^c ∖ [-2k,2k]^c ) ∑_r=-k^-b H_r  + ( 1 - _ [1,2k]^c ∖ [-2k,2k]^c ) ∑_r=b^k H_r  +  (2b-1)J 
       ≤ 2J ∑_r=b^k f( r/ξ) +  (2b-1)J ≤ 2JC_ξ(f( b/ξ) - f( k/ξ) ) +  (2b-1)J ≤ O(2bJ)



Overall the norm of the commutator in (<ref>) is bounded as follows:

    [ H_[-k-1,-b] + H_[b,k+1] - 
      H_[-k,-b] - H_[b,k] ,  e^iu ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k])
      e^-iu ( H_[-k,-b] + H_[b,k])] 
       ≤[ H_k+1 + H_-(k+1) ,  e^iu ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k])
      e^-iu ( H_[-k,-b] + H_[b,k])]  + 
        + 2 ( 2J C_ξ(f(k/ξ)-f(2k-b/ξ) ) + 2 J  f(k/4ξ) ) ( 2JC_ξ(f( b/ξ) - f( k/ξ) ) +  (2b-1)J  )  
       ≤[ H_k+1 + H_-(k+1) ,  e^iu ( H_[-k,-b] + H_[b,k])H_[-b,b] 
      e^-iu ( H_[-k,-b] + H_[b,k])]  + 
        + 8 O(bJ^2) ( 2 C_ξ(f(k/ξ)-f(2k-b/ξ) ) + 2  f(k/4ξ) )  + 4 O(C_ξ J^2)   ( 6f ( k/4 ξ) - 4f ( k/2 ξ) - 2f ( k/2 - b /ξ) )  
       ≤ O(bJ^2)(e^vt-1) f( k/2ξ) + J^2 D_ξ

In the last line, (<ref>), we have introduced D_ξ≤ O (f( k/4ξ) ) to shorten the notation. We see that in comparison to the Lieb-Robinson bound of strictly local operators there is a term, D_ξ that vanishes for large distances, of O(k), among the supports of H_k+1 and H_[-b,b], as f( k/2ξ). We stress that when f has a compact support, as discussed before in (<ref>), we recover the case of strictly local Hamiltonians and D_ξ is identically vanishing. 
The norm of the commutator in (<ref>) is upper bounded by a Lieb-Robinson bound for Hamiltonians with rapidly decaying interactions <cit.>, we denote v the corresponding velocity. We notice that the supports of the operators involved in the Lieb-Robinson bound are at a distance of the order of k/2. 


We are now ready to go back to (<ref>), that we copy below, to obtain the bound on the trace distance that we are looking for:

    1/2V_Λ_k+1(t) ρV^*_Λ_k+1(t) -  V_Λ_k(t) ρV^*_Λ_k(t) _1  
       ≤  t sup_s∈[0,t]  e^is ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k,-b] + H_[b,k]) +  
        - e^is ( H_[-k-1,-b] + H_[b,k+1])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k-1,-b] + H_[b,k+1]) +  
        +  t  H_[-k-1,k+1] - H_[-k-1,-b] - H_[b,k+1]  -  ( H_[-k,k] - H_[-k,-b] - H_[b,k]) 
       ≤  t sup_s∈ [0,t]∫_0^s du  [ H_[-k-1,-b] + H_[b,k+1] - 
      H_[-k,-b] - H_[b,k] ,  
        e^iu ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k])
      e^-iu ( H_[-k,-b] + H_[b,k])]  + 
        + 4   t    J   C_ξ( f(k/ξ) - f(2k/ξ) ) + t  J   f(k+1/ξ)  
       ≤  t^2 O(bJ^2) (e^vt-1) f( k/2ξ) + t^2J^2 D_ξ   +  t    J   D'_ξ
       ≤ t   J   O (  g(b,J,t)   f ( k/4 ξ) )

In (<ref>) we have introduced 

    D'_ξ:= 4    C_ξ( f(k/ξ) - f(2k/ξ) ) + f(k+1/ξ) ≤ O (f( k/ξ) )

When f has a compact support, see equation (<ref>), D'_ξ, as D_ξ, vanishes.
In (<ref>), it is: 

    g(b,J,t) ≤ b  t  J  ( (e^vt-1) + O(1) ) + O(1)

The minimization with respect to b of the trace distance (<ref>) is straightforward, in fact being g proportional to b in (<ref>) up to corrections of O(1), the minimum is reached for the smallest b that makes the theory consistent. Since we have seen for the strictly local case, see (<ref>), that b=ξ, then we pick b=ξ also in the general case.


We are now ready to upper bound the sum in (<ref>), that we copy below (<ref>), to obtain Δ S_α(t) of equation (<ref>) in the case of an Hamiltonian with exponential decrease of interactions .
As done in the case of strictly local Hamiltonians we need to find out the value of l that minimizes the sum. This will turn out to be dependent on α, v ad ξ. We adopt a different, but equivalent, approach than the one of appendix <ref> where we had an approximate value of l from the solution of the minimization condition. Here we will pick up l=β t with a β to be determined in such a way to hold a linear increase in t for large t of Δ S_α(t), we know in fact that a “slower” increase has been already ruled out by <cit.>. 
With, for large enough t, R_k(t) ≤ξ (t J)^2 e^vte^-k/4ξ and 

    Δ S_α (t)    ≤1/1-α∑_k=l+1^Llog[ (1-R_k(t) )^α + (2^2(k+1)+1-1)^1-αR_k(t)^α] + 2l 
       ≤1/1-α∑_k=l+1^∞( -α R_k(t)  + 2^(2k +3)(1-α)R_k(t)^α) + 2β t 
       ≤1/1-α∑_n=1^∞( -α R_l+n(t)  + 2^(2(l+n) +3)(1-α)R_l+n(t)^α) + 2β t 
       ≤1/1-α∑_n=1^∞( -αξ (t J)^2 e^vte^-β t+n/4ξ  + e^ln 2(2(β t+n) +3)(1-α)(ξ (t J)^2 )^α e^α vte^-αβ t+n/4ξ) + 2β t 
       ≤1/1-α∑_n=1^∞( -αξ (t J)^2 e^vte^-β t+n/4ξ  + e^3ln 2(1-α)(ξ (t J)^2 )^α e^α vte^-λ(β t+n)) + 2β t
 
In the last step we have introduced λ:= α/4ξ - 2ln2(1-α). We now carry out two important steps. The first is to ensure that the sum over n is finite, this will give a condition on set of allowed α. The second step would be to impose that the sum of the series decreases exponentially in time, this will provide a restriction on the value of β and therefore the large t behaviour of Δ S_α (t).
From the second sum in (<ref>) we get a restriction on the value of α as follows:

    λ > 0 ⇒α/4 ξ - 2ln2(1-α) > 0 ⇒α > 1/1+1/8ξln2

With ξ≪ 1, that corresponds to the Hamiltonians' term becoming almost on site, (<ref>) reduces to α≳ 0. In the limit of extendend interactions instead ξ≫ 1, being by assumption 0 < α≤ 1, (<ref>) gives α≈ 1.


On the other hand the restriction on β to ensure that the sum in (<ref>) decreases exponentially in t provides:

    λβ - α v > 0 ⇒β > α v/λ = v/1/4ξ- (2ln2)1-α/α

It can be checked that the condition for the exponential decrease in time of the first term of the sum in (<ref>) is less restrictive than (<ref>).


Overall, making use of the conditions (<ref>) and (<ref>) we have that:

    Δ S_α (t)  ≤ -α/1-αξ (t J)^2 e^vte^-β t/4ξ1/e^1/4ξ-1 + 1/1-α e^(3ln 2)(1-α)(ξ (t J)^2 )^α e^α vte^-λβ t1/e^λ -1  + 2β t 
       lim_t→∞Δ S_α (t)/t <  2v/1/4ξ- (2ln2)1-α/α

The limit α→ 1, that gives the von Neumann entropy, is:

    Δ S_1 (t)  ≤ (3ln 2) ξ (t J)^2 e^vte^-β t/4ξ1/e^1/4ξ-1  + 2c t 
       lim_t→∞Δ S_1 (t)/t < 8vξ
















§ AN UPPER BOUND ON THE CONCAVITY OF THE Α-RÉNYI ENTROPIES, 0 < Α < 1
   

 
 
 
 
 
 
Given a convex combination of states ρ := ∑_i=1^N p_i ρ_i, with ρ_i: C^d →C^d, denoting ρ_i= ∑_j=1^n_iλ_j^i |e_j^i ⟩⟨ e_j^i | the spectral  decomposition of each ρ_i, for 0 < α < 1 it holds:

    ∑_i=1^N p_i S_α (ρ_i) ≤ S_α (ρ) ≤ H_α{p_i λ_j^i }

where H_α{p_i λ_j^i } := 1/1-αlog∑_i,j ( p_i λ_j^i)^α is the α-Rényi entropy of the probability distribution {p_i λ_j^i }. Moreover:
 
    H_α{p_i λ_j^i }≤ H_α{p_i} + max_i { S_α(ρ_i) }

Therefore if {ρ_i } are pure states, it is S_α (ρ) ≤ H_α{p_i }, with the upper bound depending only on the probability distribution { p_i }.




We now sketch a proof, for more details consult lemma 1.24 (page 18) and example 11.12 (page 178) of <cit.>.
The first inequality in (<ref>) is the concavity of the Rényi entropy with 0 < α <1, <cit.>.
 The proof of the second inequality in (<ref>) is a simple application of the theory of majorization, see chapter II of <cit.>. The spectral decomposition of ρ reads: ρ = ∑_k=1^m a_k |v_k ⟩⟨ v_k |. Since {|v_k ⟩} is an orthonormal set, it follows that m ≤ d. The set { |e_j^i ⟩}, that collects the eigenvectors of all {ρ_i }, is in general not orthonormal. In fact, for example, two states ρ_i can share an eigenvector.  It follows that, completing the matrix V with entries V_j,k^i := √(p_i λ_j^i/a_k)⟨ v_k |e_j^i ⟩ to a unitary matrix U, it holds
 
    |e_j^i ⟩ = ∑_k U_j,k^i |v_k ⟩

It follows from the definition of V, and the unitarity of U, that 

    p_i λ_j^i = ∑_k |U_j,k^i|^2 a_k

The matrix with entries |U_j,k^i|^2 is a double stochastic matrix, meaning that ∑_k |U_j,k^i|^2 = ∑_i,j |U_j,k^i|^2 = 1, this implies, see theorem II.1.10 (page 33) of <cit.>,   that the set {p_i λ_j^i } is majorized by the set  { a_k }. Each set is supposed to be ordered in a non increasing fashion. It then follows from theorem II.3.1 (page 40) of <cit.>, and the concavity of the function x^α, with 0< α <1, that:

    ∑_k a_k^α≤∑_i,j (p_i λ_j^i)^α

That implies, being the logarithm an increasing function: S_α (ρ) := H_α{a_k }≤ H_α{p_i λ_j^i }. 


We now prove equation (<ref>).
 
    H_α{p_i λ_j^i } := 1/1-αlog∑_i,j ( p_i λ_j^i)^α =
     1/1-αlog∑_i=1^N  p_i^α∑_j (λ_j^i)^α
       = 1/1-αlog∑_i=1^N  p_i^αexp( (1-α) 1/1-αlog∑_j (λ_j^i)^α) = 1/1-αlog∑_i=1^N  p_i^αexp( (1-α) S_α (ρ_i) ) 
       ≤1/1-αlog∑_i=1^N   p_i^αexp( (1-α) max_i { S_α(ρ_i) }) =  H_α{p_i} + max_i { S_α(ρ_i) }




Equation (<ref>) holds also for the von Neumann entropy, with the Rényi entropy H_α replaced by the Shannon entropy H. In fact the same proof applies with the function x^α replaced by -x log x. See equation 2.3 of <cit.>. See also Theorem 11.10 (page 518) of <cit.>, or theorem 3.7 (page 35) of <cit.> for alternative proofs.  
For the von Neumann entropy we can see that the upper bound takes a more explicit form:
 
    S(ρ) ≤ H{p_i} + ∑_i p_i S(ρ_i)

In fact:

    H {p_i λ_j^i }:= -∑_i,j p_i λ_j^i log ( p_i λ_j^i ) = -∑_i,j p_i λ_j^i ( log p_i + logλ_j^i ) 
        = -∑_i p_i log p_i  -∑_i,j p_i λ_j^i logλ_j^i = H{p_i} + ∑_i p_i S(ρ_i)

This implies that the difference S(ρ) - ∑_i p_i S(ρ_i) ≤  H{p_i} is upper bounded by a quantity only depending on the probability distribution { p_i }, but not on other quantities like the states {ρ_i } or the dimension of the Hilbert space. The bound (<ref>) has been improved in theorem 14 of <cit.>, where also appears an upper bound on the difference of Holevo quantities for different ensembles that is independent from the Hilbert space dimension.


In general this is not the case for the Rényi entropy. Let us consider the state ρ of equation (<ref>) and the maximally mixed state of C^d, 1/d1_d. It is easy to see that S_α( p ρ + (1-p) 1/d1_d ), with fixed α, p and d ≫ 1 goes like log d. This also follows from (<ref>). We see that contrarily to (<ref>), for α-Rényi entropy it is: S_α( p ρ + (1-p) 1/d1_d ) -(1-p) S_α( 1/d1_d ) ≈ p log d. 


Within the setting of the physical system described in <ref>, and the notations of Lemma <ref>, let us consider the case where max{ S_α(ρ_i) }≤ O( log L ), and N ≤ O(L), then according to equation (<ref>), we have:

    S_α(ρ) ≤1/1-αlog∑_i=1^N p_i^α + O(log L ) ≤ O(log L)

This shows that when convex combinations of few, O(L), low entangled states, S_α(ρ_i) ≤ O(log L), are considered then ρ = ∑_i p_i ρ_i is still low entangled: S_α(ρ) ≤ O(log L).  
















































 














 Paper Title: Half-sided translations broken to pieces.


[
    Bruno S. Dandogbessi^2
    August 5, 2024
==========================










§ INTRODUCTION


The half-sided translations <cit.>
is a group of unitary transformations
built from the algebraic structure of the underlying quantum field theory, that can translate operators through spacetime. 
Specifically, their definition requires a pair of von Neumann algebras ℳ and 𝒩, where 𝒩 is a subalgebra of ℳ, and when sufficiency condiitons are met, they are generated by a positive operator 𝒢 that satisfies 

    e^i 𝒢 sℳ e^-i 𝒢 s⊂ℳ,    s ≤  0  (or s ≥ 0).

Therefore, ℳ is preserved under the flow of 𝒢 for “half of the real line", and only for the remaining half, 𝒢 can translate operators outside ℳ; a property that earns half-sided translations their name. This way, the half-sided translations connect the algebraic structure of the quantum field theory to translations within the spacetime, and serve as a powerful tool for studying quantum field theories and quantum gravity. In particular, they have been used in AdS/CFT for understanding the algebras of gravitational theories, state dependence of black hole interiors, and for doing explicit bulk reconstruction in the black hole interior <cit.>.


The sufficiency conditions for their existence for a given pair ℳ, 𝒩 are (i) of existence of a common cyclic and separating state |Ω⟩ 
and (ii) thus derived modular operators[We review these notions briefly in section <ref>.]
 Δ_ℳ, Δ_𝒩 that generate automorphisms of the algebra, to satisfy the property

    Δ_ℳ^-it𝒩Δ_ℳ^it⊂𝒩,    t≤ 0  (or  t ≥ 0).

Then 𝒢 is given by

    𝒢 = 1/2π (logΔ_𝒩 - logΔ_ℳ).

The operator  K_ℳ = -logΔ_ℳ 

is called the modular Hamiltonian <cit.> for algebra ℳ, however, it acts on both ℳ and its commutant ℳ', generically in a non-factorizable way. (Likewise is true for K_𝒩). For theories with a UV cut off however, a factorization is possible, such that K_ℳ, K_𝒩 and thus 𝒢 can be written as a combination of logarithms of density operators.








To see the factorization, let Σ be a Cauchy slice corresponding to the state |ω⟩ of the quantum field.
Consider the density operator ρ_M for a spatial subregion M on Σ, whose causal diamond ♢_M defines the algebra ℳ.
Likewise, let ρ_M' be the density operator for the complement region M', whose causal diamond ♢_M' gives the commutant algebra ℳ'. 
These are well defined positive operators if the field theory has a UV cut off. 
Therefore, one can take their logarithms  H_ℳ = -logρ_M, H_ℳ' = -logρ_M'.[These operators have been studied
extensively in high energy physics and condensed 
matter physics communities 
<cit.> 
and are also often referred to as modular Hamiltonians,
but to avoid confusion,
we will call them entanglement Hamiltonians following the condensed matter community.]
If |ω⟩ is a cyclic separating state for ℳ, then <cit.> the modular operator Δ_ℳ is

    Δ_ℳ = ρ_M ⊗ρ_M'^-1,   or  
        K_ℳ = H_ℳ - H_ℳ'.





Likewise repeating for Δ_𝒩 permits us to write 
𝒢 = G + G', where 

    G = H_ℳ - H_𝒩,   
    G'= H_𝒩' - H_ℳ'

where G is made only from ℳ, 𝒩, and G' involves only their commutants. 
This observation naturally raises the question whether one could break down the action of 𝒢 into the action of its constituents G and G', 
but its answer is not straightforward, because [G, G']≠ 0, and thus action of  𝒢 does not split cleanly into those of G and G'. In fact, since a finite action is implemented through exp(i𝒢 s), an attempt at the split will require the  Zassenhaus formula, and depending on the severity of the non-commutativity, an infinite product may be needed <cit.>.

Despite this, however, causality suggests that action of 𝒢 should be indistinguishable from G  in the region where G' has no causal influence. Likewise, the action of 𝒢 should be same as that of G' in regions that G cannot influence causally. 
In our setup, these would be regions ♢_N' and ♢_M' respectively. Moreover, it would be interesting to see the deviations between actions of G and G' compared with 𝒢 in the remaining regions of spacetime, and whether highly nested commutators are needed to approximate the action of 𝒢 in these regions. This is one of the main questions we will address in this paper.

In this paper, we will study flow of local operators ϕ under G and G', and the all the nested commutators involving [G, G'], presenting a breakdown of the action of 𝒢 into the actions of G and G'. We will use the 1+1D massless scalar field in Minkowski spacetime for our work as it is the simplest example where computations can be done explicitly. Let us list our main results here:

    
  * For ℳ, 𝒩 corresponding to Rindler wedges related by translation of origin in Minkowski spacetime, we compute all the nested operators that appear in the Zassenhaus product, and show that
        
    e^is 𝒢 = e^is G e^is G'exp([G, G'] ·1-4s - e^-4s/16 s^2 )
            .

        
    
  * As one would expect, in regions where G' has no causal influence, the action of G is indistinguishable from that of 𝒢, and likewise for G' in regions where G has no causal influence. We show this, and more importantly, in the region where both G and G' together create 𝒢, 
        
    i [G, ϕ(x,t)] = (x-t) ∂_- ϕ(x,t)

        where x-t = 2a is the boundary of ♢_N. 
         This is to be compared with the known equation
        
    i [𝒢, ϕ(x,t)] = 2a ∂_- ϕ(x,t)

        The equation (<ref>) highlights, among other things, that without G', G cannot take the operator outside ♢_M.


Let us give the outline of the paper. In section 2, we give the essentials of half-sided translations, and the main content begins in section 3, where we study all the nested commutators of G and G' for the massless scalar field, including the Zassenhaus product. In section 4, we study the action of G on local operators within the regions where G has a causal influence. The action of G' can be obtained by a symmetry. We conclude in section 5, discussing how our results can be applied to the island paradigm, which originally inspired the work. Finally, in the appendix, we have given derivations of unequal time commutators we use repeatedly in the main text.













§ BASICS OF HALF-SIDED TRANSLATIONS



Let us consider a quantum field theory in flat spacetime in a state |Ω⟩ which is cyclic and separating for a von Neumann algebra ℳ of operators. Cyclicity of the state means that ℳ when acting on |Ω⟩ produces a dense subset of the full Hilbert space, and separability means that no operator in the algebra ℳ, except the trivial operator, annihilates |Ω⟩. These conditions are necessary and sufficient to define the Tomita operator S whose polar decomposition is given by


    S = J_ℳΔ_ℳ^1/2

where Δ_ℳ = exp (- K_ℳ) implements the automorphisms of the algebra ℳ, and J_ℳ is an anti-unitary operator satisfying J_ℳℳ J_ℳ=ℳ', the commutant of ℳ. Therefore, the S operator knows about ℳ'; in fact Δ_ℳ also generates the automorphisms of ℳ'. 

The one parameter group of automorphisms  defined by Δ_ℳ^it is called the modular flow of ℳ, and its generator K_ℳ is called the modular Hamiltonian.  In mathematical form,


    e^i K_ℳ tℳ e^-i K_ℳ t = ℳ, t ∈ℝ


This is a point where we can introduce the half sided modular translation. Quite similarly to the modular flow, ℳ may have an additional structure associated to it which may allow us to write



    e^i 𝒢 sℳ e^-i 𝒢 s⊂ℳ,    s ≤  0  (or s ≥ 0)
 
with 𝒢|Ω_0⟩ = 0, and 𝒢≥ 0, like K_ℳ. 

The operator G is generating a one parameter unitary group called the half sided translations, which we will denote as ℋst(ℳ)^- for s≤ 0, or ℋst(ℳ)^+ for s≥ 0.
Clearly, the difference between (<ref>) and (<ref>) is that the latter preserves the algebra only for s ≤ 0 (or s ≥ 0), and the former preserves it for all t ∈ℝ. 


Interestingly, it is impossible to have s ∈ℝ, because when combined with the properties 𝒢≥ 0 and G|Ω_0 ⟩, it would force 𝒢 = 0 <cit.>. Below we'll mostly focus on the case s ≤ 0.









 §.§ Theorems for half-sided translations


Reiterating the conditions for their existence, we need two von Neumann algebra ℳ, 𝒩 with 𝒩⊂ℳ that satisfies the following properties:



    
  * The state |Ω_0⟩ is cyclic and separating for 𝒩.
    
  * The half sided modular flow of 𝒩 with Δ_ℳ lies within 𝒩:
        
    Δ_ℳ^-it𝒩Δ_ℳ^it⊂𝒩,    t ≤ 0  (or t ≥ 0).



If satisfied, there are several theorems that guarantee the existence of the half sided translations and describe their properties <cit.>. Most notably for us, there are four theorems to highlight; for a more detailed discussion, see <cit.>.



    
  * There exists a unitary group U(s) = e^-i 𝒢 s, s ∈ℝ generated by a positive generator 𝒢, such that 
        
           
  * U(s) |Ω_0⟩  = |Ω_0⟩, s ∈ℝ, or equivalently 𝒢 |Ω_0⟩ = 0.
           
  * U^† (s) ℳ U(s) ⊂ℳ,     s≤  0 
           
  * 𝒩 = U^† (-1) ℳ U(-1).
        


  * The commutator [K_ℳ, K_𝒩] = - 2π i (K_ℳ - K_𝒩) and K_ℳ- K_𝒩 = 2π𝒢.

    
  * Given an ℳ, 𝒩 and  |Ω_0⟩ defined this way, U(s) is unique.

    
  * The von Neumann algebra ℳ needs to be type  III_1 for 𝒢 to be non-trivial.



Despite working with entanglement Hamiltonians H_ℳ, H_𝒩, whenever we refer to 𝒢,
we will mean the one in the UV limit, where the algebras are type III_1.



  §.§.§ Half sided translations in Minkowski spacetime


Let us introduce, as an example, the setup that will become the main focus of this paper.
Consider a quantum field theory in 1+1 dimensional Minkowski spacetime having coordinates (x,t). 
Let the system be in the vacuum state |0⟩ with respect to t, and let the algebras ℳ, 𝒩 correspond
to two Rindler wedges associated with different origins in Minkowski spacetime: 

    ♢_M = {x^+ > 0, x^- < 0},   ♢_N = {x^+ > 0, x^-<-2a}.

Their associated intervals are M=(0,∞) at t=0, and  N = (a, ∞)  at t=-a respectively. See figure <ref>.




There are several ways to derive H_ℳ, H_𝒩 and the corresponding K_ℳ, K_𝒩 for these algebras <cit.>. Explicitly, we note that 


    H_ℳ = ∫_0^∞ dx  x T_00(x,0),    H_ℳ' = - ∫_-∞^0 dx  x T_00(x,0),


    H_𝒩 = ∫_a^∞ dx  (x-a) T_00(x,-a),    H_𝒩'=  -∫_-∞^a dx  (x-a) T_00(x,-a).


The operators K_ℳ and K_𝒩 are the boost operators preserving (x,t) = (0,0) and (x,t)=(a,-a) respectively. One way to relate these two, which allows one to identify 𝒢 = K_ℳ - K_𝒩, is using spacetime translations like in <cit.>. Another is to use transformation properties of the stress tensor[This one is shown in the appendix for a conformal field theory.]. One finds that 

    𝒢 = 2 a P^+≡ a (H+P)

is the generator of translations in the x^- direction for the algebras (<ref>). It generates a flow in the null direction, and is an example of ℋst(ℳ)^-. [For the same ℳ, one can get a different 𝒢 = 2a P^- by considering a different algebra 𝒩, associated to the region {x^+ > 2a, x^-< 0}, of the type ℋst(ℳ)^+.]



 §.§ A non example


An arbitrary 𝒩⊂ℳ does not satisfy conditions of half-sided translations.  Regions of the type shown in figure <ref> do not satisfy the inclusion property (<ref>). This is because the modular flow mimics the action of a boost operator <cit.>, which means any local operator in the wedge associated to ℳ under modular flow will approach the upper and the lower vertices of the wedge, as t →∞ and t →  -∞ respectively. Both of the associated regions however are not covered by the algebra 𝒩, hence evolution of operators in 𝒩 under the modular flow of ℳ goes outside 𝒩 for both t≥ 0 and t≤  0. For more examples, see <cit.>.







§ COMMUTATORS FOR THE MASSLESS SCALAR FIELD


Now that we have defined our algebras (<ref>) and the entanglement Hamiltonians (<ref>), (<ref>), we can generally say that

    [G, G'] =    [H_ℳ - H_𝒩, H_𝒩' - H_ℳ'] 
    
        =    [H_ℳ, H_𝒩']

where the second line follows from the definitions (<ref>), (<ref>) and causality. See figure <ref>.
Thus by making a change of variables, we can write

    [H_ℳ, H_𝒩']  = ∫_0^∞ dx · x·∫_0^∞ dy · y· [T_00(x,0), T_00(a-y, -a)]


The y integral runs on thin line from -∞ to a, effectively, and the one on thick x axis from 0 to ∞. Now, clearly all the T_00(x,0) operators commute with T_00(a-y,a) for y<-a, and all those with x> 2a commute with all of them, so this operator becomes

    [H_ℳ, H_𝒩']  = ∫_0^2a dx · x ∫_0^2a dy · y·  [T_00(x,0), T_00(a-y,-a)]

but that is all we can say in general, without assuming a specific form of the stress tensor. 
Similarly, for studying the effect of entanglement Hamiltonians on local operators, we need a theory where computations can be done explicitly.
Here onwards, we will specialize to the massless scalar field in 1+1D Minkowski spacetime, and compute the quantities explicitly.





The massless scalar field ϕ(x,t) in 1+1D Minkowski spacetime satisfies the Klein-Gordon equation 

    ∂π/∂ t = ∂ ^2 ϕ/∂ x^2,   π≡∂_t ϕ

and the equal time commutation relations

    [ϕ(x,t), π(x', t)] = i δ(x-x')
    .

The stress tensor components are
 the Hamiltonian density ℋ = T^00 = T_00 and the momentum density 𝒫 = T^01 = -T_01,

    T_00 = 1/2 ( π^2 + (∂_x ϕ)^2),    T_01 = 1/2 (π·∂_x ϕ + ∂_x ϕ·π).



Our computations will involve studying the operators defined at different times, therefore we will need unequal time commutators. While I have discussed them in great detail in the appendix <ref>, for the clarity of the main text, I will summarize their computation here. The crucial points are to write the Heisenberg operator 

    ϕ(x,t) = exp(i H t) ϕ(x) exp(-i Ht)

using the Schroedinger operator, and then use Hadamard's lemma and the equation of motion (<ref>). That gives us two commutators we will need in next section:

    [T_00(y,0), ϕ(x,t)]    =  - i  ∂_+  ϕ(y,0) ·δ(x+t-y) - i ∂_- ϕ(y,0) ·δ(x-t-y)
    
        [T_01(y,0), ϕ(x,t)]   =  - i ∂_+ ϕ(y,0) ·δ(x+t-y) + i ∂_- ϕ(y,0) ·δ(x-t-y)


Note that the effect of evolution, because of the equation of motion, is coming from only two points: (x+t, 0) and (x-t, 0). 

Further, for computing [G, G'] and subsequent nested commutators, we will need commutators of stress tensors at different times as well. In particular, we will need

    [T_–(x), T_–(y)] = i D(x,y) (∂_y - ∂_x) δ(x-y)

where

    D(x,y) = π(x,0) π(y,0) + χ(x,0) χ(y,0) -{π(x,0),  χ(y,0)}.

To reach this expression, one has to recognize the commutator is of the form [A^2, B^2] for appropriate A,B, after which simplifications are easier. Note that in the left side the operators depend only on one variable since the theory is conformal. 



 §.§ The commutator [G, G']


Using the transformation properties of the stress tensor for the conformal field, 

    T_00(x + a, -a) = T_++(x) + T_–(x+ 2a)
 
we can write G = I_1 + I_2, where

    I_1 = 2a ∫_2a^∞ T_–(x) dx,   I_2= ∫_0^2a x · T_–(x)dx.

This derivation is given in the appendix <ref>. Further, we can convert the equation (<ref>) into



    [H_ℳ, H_𝒩']= ∫_0^2a dx · x ∫_0^2a dy ·  y  · [  T_– (x) ,  T_–(-y+2a)]
    .

Direct use of (<ref>) will yield 4 terms, in which only this [T_–,T_–] commutator will survive. Using (<ref>), we see this integral will split into two, one involving the derivative over x and another on y. One also has to carefully note that argument of second T_– is 2a-y. Let us write it as [H_ℳ, H_𝒩'] = I_x - I_y, then the integral with x derivative becomes 

    I_x = - i/4∫_0^2a du · (u-2a) ·   [D(u,u) + u ·∂_ (1) D(u,u)]

and the integral with y derivative becomes

    I_y = - i/4∫_0^2a du · u·[D(u,u)+ (u-2a) ∂_ (2)  D(u,u) ] 
    .

Here the ∂_(1), ∂_(2) denote derivatives of D with respect to first and second variable respectively. D symmetrically has same derivatives and thus when taking the difference this term gets cancelled. Ultimately we are left with
[
Following this method, one can show that

    [H_ℳ, H_𝒩] =  - 2 ia ∫_2a^∞ T_–(x) dx

giving us [H_ℳ, H_𝒩] = -i(H_ℳ-H_𝒩) +i I_2, which is in stark contrast with [K_ℳ, K_𝒩] = -i(K_ℳ - K_𝒩).
]

    [G, G'] = [H_ℳ, H_𝒩'] = i · 2a/4∫_0^2a dx ·  D(x,x)  = 2a i ∫_0^2a T_–(x) dx
    .





 §.§ The nested commutators


We shall now show that all the nested commutators are proportional to [G, G'] and find them explicitly. Before we do it, let us use C = [G, G'] notation to help readability. Proceeding, to get all the nested commutators, note that we only need to evaluate two terms: [G, C] and [G', C]. The same integrals that appeared above will make appearance here.

First we note that [G, C] =  [I_2, C], since I_1 is integrated over (2a,∞) and thus would contain operators spacelike to C. Therefore

    [G, C] = 2i a ∫_0^2a dy ∫_0^2a dx · x · [T_–(x), T_–(y) ]

Again, one can split this into two integrals involving derivatives over x and y, and like in (<ref>), (<ref>), the terms with ∂ D cancel, and we're left with

    [G, [G,G']] =-2a ∫_0^2a dx · D(x, x) = i^2 · 2a  · 4 ∫_0^2a dx · T_–(x)


    = 4 i [G, G'].

Now let us look at [G', C]. One can get G' = 𝒢- G, and from there it follows that

    [G', C] =(2ia) ∫_0^2a dx · (2a-x) ∫_0^2ady·[ T_–(x), T_–(y)]
    .

which is the same as [G, C], except for x → 2a-x in the integrand. When integrating, this just leads to a minus sign on the term that does not get cancelled, and we get

    [G', [G, G']] = -4i [G, G']
    .

Therefore, all the nested commutators are just (4i)^n(-1)^k [G, G'] for some n,k. 



 §.§ The Zassenhaus formula


The action of half sided translations formally can be written as an infinite product:

    exp(i s 𝒢)  = exp[ i s (G + G') ] = exp(is G) exp(i s G') · e^Z_2 e^Z_3 e^Z_4⋯

using the Zassenhaus formula <cit.>. The terms Z_2, Z_3, … involve nested commutators of  G and G', and the procedure of finding them is very tedious in general. Although there are some cases where a simplification can be made <cit.>, we will see that our situation doesn't satisfy their conditions, and hence we must compute the Z 's iteratively. Because all the nested commutators are simply proportional to  [G,G'] however, this is a doable task. 

To start, let us take A = i s G, B = i s G';  both  A, B are anti-hermitian, and so is [A, B]. Because all the Z's commute and are proportional to [G, G'], we will end up with the form

    exp(is 𝒢) = exp (i s G) exp(i s G') ·exp( [G, G'] f(s))

for some f(s). For this operator to be unitary, f(s) must obey f(s)= f(s). 
As noted earlier, [G, C] = -[G',C] = 4iC, thus we can say that [A, C] = α = - [B, C] for α = -4s. In this case we find using the recursive formulas <cit.> that

    Z_n = - · C α ^n-1/(n+1)! 
    .

Therefore the resultant series is easily summable, giving us

    ∑_n=2^∞ Z_n =  -C ·(e^α - 1 - α)/α^2


    e^is 𝒢 = e^is G e^is G'exp([G, G'] ·1-4s - e^-4s/16 s^2 )

where we've substituted back α = -4s, and obtained a real function f(s),

    f(s) = 1- 4s - e^-4s/16 s^2

ensuring unitarity of e^is 𝒢. 

This expression clearly reflects that if you're looking at action of 𝒢 on local operators within ♢_N, they would commute with C and G' and therefore only effect would be of G'. In the next section, we will see this point in more detail and further study the significance of this result.












§ THE FLOW OF G = H_ℳ - H_𝒩





Let us consider a local operator ϕ(x,t), where x,t belongs to any of the regions 1, 2, or 3, marked in the figure <ref>. Explicitly, the regions are

    R_1 = ♢_N, R_2 = ♢_M ∩♢_N,    R_3 = {x^+ > 0, x^- > 0}.

In this section we will study the action of the operator exp[i s G] on ϕ(x,t), where G =H_ℳ - H_𝒩, made out of the entanglement Hamiltonians (<ref>), (<ref>) associated to the algebras ℳ and 𝒩 (<ref>) respectively. 
As we mentioned, G = I_1 + I_2, where I_2 is integrated over [0, 2a], and thus has no effect on local operators in R_1. Hence, as far as the action of G on region 1 is concerned, we will ignore I_2 and focus only on I_1.

Before we proceed, we need to highlight an important result that will simplify the subsequent calculations enormously. 
Since ϕ is a scalar field, for a fixed point P with coordinates (x,t) and x^± = t± x,

    ϕ(x,t) = ϕ̂(x^+,x^-)

where we use ϕ̂ to denote the field in null coordinates. Further note, translations in x^- are generated by P^+, giving us

    ϕ(x^++a, x^-)    = e^i P^- aϕ(x^+,x^-) e^-i P^- a
       =  ϕ̂(x^+,x^-) + i a [P^-, ϕ̂(x^+,x^-)] + i^2 a^2/2 [P^-, [P^-, ϕ̂(x^+,x^-)]] + …
       =  ϕ̂(x^+,x^-) +  a ∂_+ϕ̂(x^+,x^-) + a^2/2∂_+^2 ϕ̂(x^+,x^-) + …
    .

Now, note a must not depend on x^+ for this to work. To connect with our problem, let P be (x,t),  Q be (x-t,0), and R be (x+t,0). The latter we will use in the next section. Then note that

    ϕ̂(x^+_Q,x^-_Q) = ϕ̂(x^+_P, x^-_P) + (Δ x^+) ·∂_+ϕ̂(x^+_P, x^-_P) + (Δ x^+)^2/2·∂_+^2 ϕ̂(x^+_P, x^-_P) + …

because x^-_P = x^-_Q = t-x. On the other hand, note Δ x^+ = x^+_Q - x^+_P = -2t does not change with x^-_Q, because Δ x^+ vector is orthogonal to x^-_Q variations. This is an important point because it allows for an enormous simplification:

    ∂_-ϕ̂(x^+_Q, x^-_Q)  = ∂_-ϕ̂(x^+_P, x^-_P) + ( Δ x^+ ) ·∂_- ∂_+ϕ̂(x^+_P, x^-_P) + (Δ x^+)^2/2∂_-∂_+^2 ϕ̂(x^+_P, x^-_P) + …


    = ∂_-ϕ̂(x^+_P, x^-_P)

where I've used the equation of motion in null coordinates: ∂_- ∂_+ ϕ = 0 and the commutativity of partial derivatives ∂_+, ∂_-. As a consequence, ∂_- ϕ(x-t,0) = ∂_- ϕ(x,t). The same analysis can be done in x^+ direction, and that tells us ∂_+ ϕ(x+t,0) = ∂_+ ϕ(x,t).

















 §.§ Flow in Region 1






To understand the flow in this region, let us define

    ϕ(x,t,s) = e^is I_1 ϕ(x,t) e^-is I_1

where I_1 is in (<ref>). Now note that T_–(x) = 1/2 (T_00(x) - T_01(x)) =  1/2 (ℋ(x) + 𝒫(x)). This is true for any conformal field theory with vanishing stress tensor trace. In other words,
    
    I_1 =  2 a ∫_2a^∞ dx ·   T_–(x) = 2a ∫_2a^∞ dx   1/2(ℋ(x) + 𝒫(x))

which makes it evident that if the range of integration were the whole line, integral would become 2 a P^+. As mentioned, this observation alone, in conjunction with causality, should convince the reader that the flow in region 1  generated by I_1 is same as that generated by 𝒢. 

Let us do an explicit computation now to show this, that will also help us set up the machinery for the more complicated cases of regions 2 and 3.
Using the Hadamard Lemma, we can write

    e^is I_1ϕ(x,t) e^-is I_1 = ϕ(x,t) + is [I_1, ϕ(x,t)] + (is)^2/2 [I_1, [I_1, ϕ(x,t)]] + …
    .
 
Now note, 
    
    [I_1, ϕ(x,t)] = a ∫_2a^∞ dy    [(T_0 0 - T_0 1) , ϕ(x,t)]
        .

For ϕ(x,t) to be in region 1, ♢_N = {x^+ > 0, x^-< -a}, by definition, both the delta functions in the commutators (<ref>) and (<ref>) contribute ordinarily, but because we are dealing with T_00 - T_01, the integral becomes 

    [I_1, ϕ(x,t)] = - 2ai ∫_2a^∞ dy ·∂_- ϕ(y,0) δ(x-t-y)
 

    = -2ia ∂_- ϕ(x-t,0)
    .

which proves that 
    
    ∂ϕ(x,t, s)/∂ s = 1/i [ϕ(x, t), I_1] = 2a ∂_- ϕ(x,t)
        .

where we have used (<ref>). The right side is independent of s, and therefore as long as the point x,t is in region 1, the flow is along the null direction x^-; we don't need to compute the higher order terms in the expansion of e^is I_1ϕ(x,t) e^-is I_1.








 §.§ Flow in Region 2


In region 2, R_2 = ♢_M ∩♢_N, the derivative is 

    ∂ϕ(x,t,s)/∂ s = i [I_1, ϕ(x,t)] + i [I_2, ϕ(x,t)].

As we make the transition from region 1 to 2, we're losing a part of the effect previously generated by I_1, while gaining an effect generated by a part of I_2. Below we will see this explicitly.

To begin the computation, we need to use the results (<ref>), (<ref>) to write

    [T_– (y), ϕ(x,t)] = - i∂_- ϕ(y,0) δ(x-t-y)
    .

Note that in region 2, delta function contributes at y = x-t, which must be less than 2a, and therefore I_1 contributes nothing to the derivative. Therefore

    ∂_s ϕ(x,t,s) = i [I_2, ϕ(x,t)] = i ∫_0^2a  dy · y · [T_–(y), ϕ(x,t)]


    = + ∫_0^2a  dy · y ·∂_- ϕ(y,0) ·δ(x-t-y)


    = (x-t) ·∂_- ϕ(x-t,0)  = (x-t) ∂_- ϕ(x,t)

where again we have used (<ref>).
Note that as x - t → 2a, the boundary of regions R_1, R_2, ∂_s ϕ is same as (<ref>), therefore the flow is continuous across the boundary.


This result would have been very surprising had we not known that all the non-commutativity of [G, G'], and the ensuing (infinite) Zassenhaus series can be tamed. Ultimately is shows that the effect of non-commutativity is to create an effect of a graded deceleration of the vector field, all the way to zero as one reaches the x=t line, the boundary of ♢_M. It means that the local operator localized in region ♢_M is never able to escape. Mathematically this seems very interesting,
    
    e^i s Gℳ e^-i s G⊂ℳ,    s ∈ℝ
        ,

because it is in stark contrast to (<ref>), which holds only for half of the line, and like mentioned in the discussion that followed, it is impossible to find a 𝒢 such that (<ref>) is true for s ∈ℝ, which is the distinguishing property of half sided translations from modular flows of ℳ.







 §.§ Flow in Region 3


In region 3, R_3 = {x^+ > 0, x^- > 0}, as we can again look back from [T_–, ϕ] commutators, the derivative is exactly zero. G operator, in other words, is unable to cause any effect in the region 3, despite it being a region where G has a causal influence.

Here we should mention that because of the symmetry between G and G' it is evident that same kind of results can be expected for the flow of G'. Therefore we can apply exactly the same analysis with ℳ→𝒩' and 𝒩→ℳ since ℳ' ⊂𝒩'.



















































































































































































 §.§ General evolution operator


Let us now consider the operator

    𝒪 = ∫_a^b dx ·  (f_0(x) T_00 (x,0) + f_1(x) T_01(x,0) )

where b > a >0. Given that our field is conformal, this is the most generic operator one can take for creating flows for local operators. Let us see whether this operator has the capacity to take a local operator outside the region ♢_M. In other words, let us compute [𝒪, ϕ(x,t)]. This is fairly easy now since we've already obtained the tools for it.  First let us write down

    F(x)  = f_0(x) T_00 (x,0)+ f_1(x) T_01(x,0),

then again using the results (<ref>), (<ref>), we get

    [F(x), ϕ(x,t)] = i [f_1(x)  - f_0(x)] ∂_- ϕ(y,0) ·δ(x-t-y)  - i[f_1(x)+f_0(x)] ∂_+ ϕ(y,0) ·δ(x+t-y)
    .

To see the meaning of this result,[Post integration one would also have to use (<ref>) to connect derivatives at (x± t, 0) to (x,t). ]  we will take the limit x → t so that we are on the boundary, as we want to see if the operator can take a local operator outside ♢_M. Well, if x - t = 0, then δ(-y) is left, and that means the integral better pass the origin for delta function to click and give possibility of motion along x^-. Clearly however, for any f_1, f_0, this will never happen. On the other hand, there will be generic motion along x^+, given by ∂_+ ϕ(x,t). So, no 𝒪 will be able to lead to an exit for the local operator. 
It doesn't matter what a,b are, so long as a >0. And δ(x+t-y) will click if x+t lies in (a,b), otherwise it will be zero.  

Since (<ref>) is the most general operator one could take that is made up of degrees of freedom of system contained in the  ♢_M region, or the algebra  ℳ, this result implies that any shouldn't permit the flow of local operator ϕ(x,t) outside ℳ.
The generality of this result may make one suspect whether it can be derived with just the properties of the von Neumann algebras, however, I am not aware of any such argument, and the proof here is based on the explicit computation of the commutator for the massless scalar field. 






§ CONCLUSION


In this work, we have studied the flow of local operators in a 1+1 dimensional conformal field theory, using the massless scalar field as an example. We have shown that for the regions ℳ, 𝒩 chosen in the Minkowski spacetime, the effect of the half-sided translation generator 𝒢 can be understood in terms of the entanglement Hamiltonians H_ℳ, H_𝒩, H_ℳ', H_𝒩'. The non-commutativity of [H_ℳ, H_𝒩'] creates the central challenge in splitting the effect of 𝒢 in two parts, which introduces the need for the infinite Zassenhaus product. Here we have been able to compute this product exactly.

Further, we have shown that the flow generated by the entanglement Hamiltonian difference H_ℳ - H_𝒩 is unable to take a local operator outside the region ♢_M, which is the region of spacetime accessible to the algebra ℳ. This result is in contrast to the half sided translations, which can take a local operator outside the region ♢_M. We have also shown that the flow generated by H_ℳ - H_𝒩 is continuous across the boundaries of the regions ♢_M, ♢_N, and that the flow is exactly zero outside the region ♢_M, even though causally H_ℳ - H_𝒩 is permitted to have an influence there. We have generalized this result to a generic operator 𝒪 made out of ℳ 's degrees of freedom, and have shown that it cannot take a local operator outside the region ♢_M.


 

 §.§ Application to AdS/CFT and the island formula


In the work <cit.>, the authors considered a thermal bath in equilibrium with an AdS_2 black hole, which lead to unbounded growth of entanglement entropy of outgoing radiation, leading to a Hawking like paradox. The unitarity was restored using the island formula <cit.>, which concluded that after the Page time, the bath degrees of freedom can access the black hole interior. 

The island formula asserts that before the Page time, the left and right CFTs alone are enough to reconstruct the black hole interior while after the Page time, the bath degrees of freedom are needed to the reconstruction. In <cit.>, the authors showed that a half-sided translation protocol using the left and right CFTs succeeds to do interior reconstruction before Page time, but fails to do so after the Page time. 

In the same spirit, the island formula implies that before the Page time the bath degrees of freedom are not needed, This part should follow easily by using the methods in this paper.
For instance, by considering the algebras ℳ, 𝒩 shown in the figure <ref> depicting AdS_2 in equlibrium with the left and right baths, and asking the question whether the bath operators can be brought into the AdS region is exactly the problem we've studied in this paper, because bath is a flat spacetime, and the field theory considered in the work is also conformal. 










We will try to make this above connection with island paradigm more explicit in a future work. We also suspect that several of these features may extend beyond the massless scalar, which is an interesting direction for us. 



  §.§.§ Acknowledgements


I'd like to thank Budhaditya Bhattacharjee and Adarsh Sudhakar for helpful discussions. 









































































§ APPENDIX


















































































 §.§ Unequal time commutators for massless scalar field




For our calculations, we will need the commutators


    [T_00(y, 0), ϕ(x, t)],   [T_0 0(y,0), π(x, t)]
 

which we'll compute using the equal time commutation relations of the field operators, and the equations of motion. Below we use ∂_y ϕ(y,0) ≡χ(y,0).


    [T_00(y, 0), ϕ(x,t)]  = 1/2 [ϕ̇ (y,0)^2 + ∂_y ϕ(y,0)^2, ϕ(x,t)]


    =  1/2{π (y,0),   [π(y,0), ϕ(x,t)] } + 1/2{χ (y,0),   [χ(y,0), ϕ(x,t)] }

Now, let us write

    ϕ(x,t) = exp (i H t) ϕ(x,0) exp(-i H t)
    .
 

    = ϕ(x,0) + i t [H, ϕ(x,0)] + t^2/2 [H, [H, ϕ(x,0)]] + …

using the Hadamard's lemma. Here however we note that from Heisenberg equation of motion, [H, ϕ(x,0)] = -i π(x,0), and [H, π(x,0)] = -i ∂_t^2 ϕ(x,0), and so on, which makes this simply a Taylor series expansion, so that we can write

    [π(y,0), ϕ(x,t)]  = ∑_k t ^k/k![π(y,0),  ∂_t ^kϕ(x,0)]

This we will find using the equation of motion ∂_x^2ϕ = ∂_t π = ∂_t^2 ϕ. It is easy to see that k=0 gives -iδ(x-y). Then k=1 makes it a commutator between two π' s, and we get 0. Then k=2 gives ∂_t^2ϕ which equals ∂_x^2 ϕ and we can bring ∂_x^2 outside commutator, and so on, to get

    [π(y,0), ϕ(x,t)]  
       = ∑_k=0^∞t ^2k/(2k)!∂_x^2k[π(y,0),  ϕ(x,0)]
       = -i ∑_k=0^∞t ^2k/(2k)!∂_x^2kδ(y-x) 
       = -i cosh(t ∂_x ) (δ(x-y) )

Now spending a few minutes staring at it, one may think perhaps we can write it as some δ(x + f - y) because it does look like some Taylor series. Well, kind of; we can write it as sum of two Taylor expansions:

    [π(y,0), ϕ(x,t)] = -  i/2(δ(x+t - y) + δ(x-t - y))


The opposite signs of t lead to only even terms, which we can divide by 2, to get our answer. Likewise, a similar analysis for χ term yields sinh (t∂_x) acting on the delta function, giving us

    [χ(y,0), ϕ(x,t)] = - i/2 (δ(x+t-y)  - δ(x-t-y))


which only picks up odd terms from the full expansion. Thus

    [T_00(y,0), ϕ(x,t)] =  - i  ∂_+  ϕ(y,0) ·δ(x+t-y) - i ∂_- ϕ(y,0) ·δ(x-t-y).


Likewise we can compute [T_00(y, 0), π(x,t)], if we wish, which may be needed for computing nested commutators of the form [H, [H, ϕ]]. However we will not need them. The only other commutator we need would be [T_01(y,0), ϕ(x,t)], which simplifies to 


    [T_01(y,0), ϕ(x,t)] =  ∂_y ϕ(y,0) ·   [π(y,0), ϕ(x,t)]  + π(y,0) · [∂_y ϕ(y,0), ϕ(x,t)]

and by using (<ref>) and (<ref>) can be written as

    [T_01(y,0), ϕ(x,t)]    =  - i  ∂_+  ϕ(y,0) ·δ(x+t-y) + i ∂_- ϕ(y,0) ·δ(x-t-y).

Note that compared to (<ref>), the only difference is a sign for ∂_- ϕ term.






 §.§ Simplifying H_ℳ - H_𝒩 for Minkowski spacetime




Note that

    H_𝒩 = ∫_a^∞ (x-a) T_00(x,-a)dx

since its a CFT, T_00(x, -a) = T_++(x-a) + T_–(x+a), so

    H_𝒩 = ∫_a^∞ (x-a) ( T_++(x-a) + T_–(x+a))dx


    = ∫_a^∞ (x-a)  T_++(x-a) dx + ∫_a^∞ (x-a)T_–(x+a)dx


    = ∫_0^∞ x  T_++(x) dx + ∫_2a^∞ (x-2a)T_–(x)dx


    = ∫_0^∞ x  T_++(x) dx + ∫_0^∞ (x-2a)T_–(x)dx - ∫_0^2a (x-2a)T_–(x)dx


    = ∫_0^∞ x  T_00(x) dx -2a ∫_0^∞ T_–(x)dx - ∫_0^2a (x-2a)T_–(x)dx

where we use T_00(x) = T_++(x)+ T_–(x), then note that second and third integrals combine to form

    H_𝒩= H_ℳ-2a ∫_2a^∞ T_–(x)dx - ∫_0^2a x · T_–(x)dx

so

    H_ℳ - H_𝒩 = 2a ∫_2a^∞ T_–(x) dx + ∫_0^2a x · T_–(x)dx


Similarly one can get H_𝒩' - H_ℳ' and that way verify that 𝒢 = a P^+.




JHEP
 






 Paper Title: Kramers-Kronig relations via Laplace formalism and $L^1$ integrability.






Department of Physics and Astronomy “Augusto Righi”, University of Bologna, 40127 Bologna, Italy

Department of Physics, University of Trento, 38123 Trento, Italy
TIFPA-INFN, University of Trento, 38123 Trento, Italy

[]leonardo.ricci@unitn.it
Department of Physics, University of Trento, 38123 Trento, Italy



§ ABSTRACT 
Kramers-Kronig relations link the real and imaginary part of the Fourier transform of a well-behaved causal transfer function describing a linear, time-invariant system. From the physical point of view, according to the Kramers-Kronig relations, absorption and dispersion become two sides of the same coin. Due to the simplicity of the assumptions underlying them, the relations are a cornerstone of physics. The rigorous mathematical proof was carried out by Titchmarsh in 1937 and just requires the transfer function to be square-integrable (L^2), or equivalently that the impulse response of the system at hand has a finite energy. Titchmarsh's proof is definitely not easy, thus leading to crucial steps that are often overlooked by instructors and, occasionally, prompting some authors to attempt shaky shortcuts. Here we share a rigorous mathematical proof that relies on the Laplace formalism and requires a slightly stronger assumption on the transfer function, namely its being Lebesgue-integrable (L^1). While the result is not as general as Titchmarsh's proof, its enhanced simplicity makes a deeper knowledge of the mathematical aspects of the Kramers-Kronig relations more accessible to the audience of physicists.


Kramers-Kronig relations via Laplace formalism and L^1 integrability
    Leonardo Ricci
    August 5, 2024
====================================================================





§ INTRODUCTION

Linear and time-invariant (LTI) systems are among the most basic, widespread and studied models in physics. Restricting the discussion to one-dimensional functions of time, an LTI system is characterized by a transfer function, G(t), whose convolution with an input function provides the output function. In the frequency domain, this property assumes an even simpler form: in the Fourier or Laplace formalism, the transform of the output is given by the product of the transforms of the input and the transfer function.

As if that were not enough, requiring the transfer function G(t) to be causal and have a finite energy, as it is always the case in the real physical world, produces a spectacular result <cit.>: the mutual dependency of the physical descriptions underlying absorption and dispersion, which describe how a system reacts to an input in terms of energy and phase delay, respectively. So the rainbow exists because at some wavelengths other than visible ones water is opaque, and vice versa. More specifically, the real and imaginary parts of the Fourier transform G_F(ω) of G(t), or susceptibility χ(ω), are the Hilbert transforms of one another. The result was first derived, independently, by R. de L. Kronig <cit.> and H. A. Kramers <cit.>. However, the eponymous relations got a solid and definitive mathematical justification only with the work by E. C. Titchmarsh <cit.>, who proved the Kramers-Kronig relations to hold if and only if the causal transfer function G(t) is square-integrable; i.e. it belongs to L^2:

    ∫_0^∞| G(t) |^2  t < +∞ .


The Kramers-Kronig relations, sometimes referred to as “dispersion relations” <cit.>, are a cornerstone of physics, whose implications are broadly investigated in a wide range of fields <cit.> and thus go beyond the prototypical problem of interpreting the refraction index n(ω), for which they were first devised <cit.>. Although Titchmarsh's contribution was recognized long ago <cit.>, a few decades after the formulation and proof of the relations, the awareness among the community of physicists about Titchmarsh's achievement was not unanimous. So Sharnoff in 1964 still argued <cit.>: “It is paradoxical that although the Kramers-Kronig relations are so widely used, the literature contains neither a convincing proof of their general validity nor a careful discussion of sets of conditions under which they might be expected to hold.”

The likely reason is that the Titchmarsh theorem in Fourier analysis–as the theorem is named—is definitely not straightforward to prove. Indeed most textbooks and papers citing it arrive just short of a complete proof when they typically take for granted the crucial and most difficult step: how to prove that the Fourier transform G_F(ω), where the usually real frequency ω is extended to the complex plane, is analytic not only on the open, upper half-plane Im(ω) > 0, but also on the real axis Im(ω) = 0. So, for example, in J. D. Jackson's classic book on electrodynamics <cit.> the step is justified with the words “On the real axis it is necessary to invoke the `physically reasonable' requirement that G(τ)→ 0 as τ→∞ to assure that ε(ω)/ε_0 is also analytic there.”[In Jackson's book, ε(ω)/ε_0 - 1 corresponds to the Fourier transform of G(τ). The requirement G(τ)→ 0 as τ→∞ is physically reasonable because dissipative mechanisms loom everywhere, so the impulse response of any real system must fade out some time. While this behavior translates, as a consequence of Parseval's theorem, in G_F(ω) being vanishing too as |ω| →∞, how this implies the analyticity on the real axis is less immediate.] A similar argument is used by Bechhoefer <cit.> who, in order to provide “a brief derivation of the Kramers-Kronig relations”, states: “for simplicity, we will also assume that G(ω) has no poles on the real axis”. Finally, the likewise classic book by Landau and Lifshitz on Statistical Physics <cit.> proposes a proof that is based on a previous theorem, proved by N. N. Meĭman, which derives asymptotic properties of χ(ω) as a consequence of Cauchy's argument principle <cit.>. However, exactly as in the previous cases, also this theorem implicitely takes for granted the analyticity of χ(ω) on the real axis Im(ω) = 0.

The arduousness of the proof, combined with the importance of the result, has prompted several attempts to find simpler approaches. Searching the internet is likely to provide alleged solutions from non-peer-reviewed sources and, occasionally, peer-reviewed ones <cit.>, which invariably fail to live up to the promises. A common trait of these attempts is their being based on the combination of two ingredients: the convolution theorem, and the Fourier transform of the Heaviside step function θ(t), in fact the very expression of causality. The convolution theorem states that the Fourier transform of the convolution of two functions or distributions of time is equal to the product of the Fourier transforms of the factors. Due to the symmetry of the Fourier transform and its inverse, the theorem can be read the other way round, i.e. the inverse Fourier transform of the convolution of two functions or distributions of frequency is equal to the product of the inverse Fourier transforms of the factors multiplied by 2 π. With regard to the Fourier transform of the Heaviside step function θ(t), χ(ω) is given by the sum of the two distributions πδ(ω) and i P ω^-1, where P indicates the Cauchy principal value. Consequently, starting from the expression G(t) = θ(t) G(t), which is true because of causality, and applying the “inverse” version of the convolution theorem one can derive the expression

    χ(ω) ⋆P1/ω = -i πχ(ω)  ,

which corresponds to the Kramers-Kronig relations being written as a convolution.

This result is well known (see, for example, Sec. 1.8 of Ref.Nussenzveig1972), but it definitely not easy to derive, the most difficult part being the conditions under which the convolution theorem holds. In fact, the derivation of the expression above requires the theory of distributions, and it is far from being more elementary than Titchmarsh's one. On the other hand, the alleged solutions mentioned above go straight to the final expression, disregarding essential aspects of validity, which essentially coincide with those set by the Titchmarsh theorem and whose omission leads to miscalculations. For example, referring to the attempt by Hu, described in the 1989 paper “Kramers-Kronig (relations) in two lines” <cit.>, one could try to verify whether the relations work when the function Ŷ(t) is given by a constant value, or by the sign function sgn(t) = 2·θ(t) - 1. They do not, as a direct evaluation promptly shows. To conclude, the Kramers-Kronig relations are a powerful tool that stems from linearity, causality, and energy boundedness: proving the link is arduous and admits no shortcuts.

Here we show that the Kramers-Kronig relations can alternatively be derived in a way that is simpler than Titchmarsh's one. The derivation relies on the intrinsically causal Laplace formalism and on the assumption that the transfer function is Lebesgue-integrable rather than square-integrable, i.e. belonging to L^1 (rather than L^2):

    ∫_0^∞| G(t) |  t < +∞ .

However, simplicity comes at a cost: as it will be shown later, G(t) ∈ L^1 provides a sufficient condition for the Kramers-Kronig relations to hold, rather than a necessary and sufficient one as in Titchmarsh's formulation. Therefore, though providing a solid path to the Kramers-Kronig relations, the present proof does not replace Titchmarsh's one, which remains unsurpassed.

In the following, after a brief review of the Laplace and Fourier transforms and the related properties that are functional to the proof, we introduce Laplace-transformable, Lebesgue-integrable functions, for which the Kramers-Kronig relations are thereupon proved. The limit of the present proof compared with Titchmarsh's classic one is finally discussed.



§ A REVIEW OF LAPLACE AND FOURIER TRANSFORMS


To aid the reader, and despite their being common knowledge, we summarize here the definition and some properties of the Laplace transform that are important for the discussion below: analyticity, Riemann-Lebesgue lemma, and the inversion of the transform via Bromwich, or Fourier-Mellin, integral. For the same purpose, at the end of the section we also review the definition of the Fourier transform and when a function is Fourier-transformable.

Let s be a complex variable and s^', s^'' its real and imaginary part, respectively. By definition, a function f(t) is causal if, for all t < 0, f(t) = 0, and is locally integrable if the integration on any compact subset[For a function of a real variable, compact is equivalent to closed and bounded.] of its domain is finite. The Laplace transform F_L(s) of a causal and locally integrable f(t) is defined as

    F_L(s) ≡∫_0^∞ e^-st f(t)  t .

The complex half-plane where the above integral absolutely converges (á la Lebesgue), i.e. the set of complex numbers s such that

    ∫_0^∞| e^-st  f(t) |  t = ∫_0^∞ e^-s^' t | f(t) |  t < ∞ ,

is left-bounded by the so-called abscissa of absolute convergence λ_0: λ_0 is the minimum real number such that absolute convergence occurs for any s^' > λ_0. A function f(t) that is causal, locally integrable, and has a finite λ_0 is henceforth referred to as a Laplace-transformable function.

The main consequence of the absolute convergence condition is the analyticity of the Laplace transform F_L(s) in the half-plane of absolute convergence, i.e. for s^' > λ_0. Another consequence is the Riemann–Lebesgue lemma, which follows from Lebesgue's dominated convergence theorem:

    lim_s^'→ +∞F_L(s) = 0  .


Finally it is worth stating the general expression for the inverse Laplace transform, which corresponds to the Bromwich, or Fourier-Mellin, integral:

    f(t) = 1/2π i∫_a-i∞^a +i ∞F_L(s) e^st s ,

where a is any constant real number such that a > λ_0.

The Fourier transform of a function f(t), not necessarily causal, is

    F_F(ω) ≡∫_-∞^∞ e^i ω t f(t)  t ,

where ω is a real frequency. Here “the physicists's notation” for the phasors of positive frequency, namely e^-iω t, is used.[The expression above carries out a “projection” of the original function f(t) onto the phasor corresponding to the frequency ω. In quantum mechanics, the projection of a wave-function onto another wave-function is a scalar product that requires the complex conjugation of the latter. This is the reason why, within the Fourier integral, the complex conjugate of the phasor appears.] Engineers typically use e^jω t instead. The two notations are completely equivalent, due to the invariance of the real world under complex conjugation and the freedom we have in choosing the “reference” root of x^2 = -1: there are indeed two, i and -i, or, better, i and j = -i. As a result, one can recover the engineers' notation by replacing here and henceforth i with -j and, in particular, setting s = j ω.

It is now worth mentioning that a common mistake consists of assuming square-integrability (f(t) ∈ L^2). Indeed, f(t) is Fourier-transformable if it is Lebesgue-integrable (f(t) ∈ L^1), i.e. if it satisfies Eq. (<ref>) (with f instead of G). Remarkably, f(t) ∈ L^2 does not necessarily imply f(t) ∈ L^1, and thus the Fourier-transformability of f(t). An example is provided by f(t) = θ(t-1)/t, which belongs to L^2 though not to L^1. Conversely, f(t) ∈ L^1 does not imply f(t) ∈ L^2 either: the function f(t) = θ(t)  θ(1-t) /√(t) provides a counterexample. On the other hand, it is well known that, in the case of square-integrability, Parseval's theorem holds and the inverse Fourier transform is essentially the same operator as the direct Fourier transform. The conundrum of a function f(t) that belongs to L^2 but not to L^1 can be overcome by redefining the Fourier transform as follows <cit.>. One can consider the sequence of functions

    f_n(t) = f(t) [θ(t+n) - θ(t-n) ]  ,

where n is a positive integer number. Each function f_n(t), which can be shown to belong simultaneously to L^1 and L^2, tends to f(t) as n →∞ with respect to the L^2-norm given by ‖ f‖ = ∫_ℝ|f(t)|^2 t. Defining the Fourier transform of F_F(ω) of f(t) as the limit of the sequence of Fourier transforms F_F,n(ω) when n →∞ eventually settles the problem.



§ LAPLACE-TRANSFORMABLE, LEBESGUE-INTEGRABLE FUNCTIONS


In the following discussion, besides being Laplace-transformable, the function f(t) is assumed to be Lebesgue-integrable, i.e. to belong to L^1 and thus to satisfy

    ∫_0^∞| f(t) |  t < ∞ .

Comparing this last equation with Eq. (<ref>) requires the abscissa of absolute convergence λ_0 to be negative, so F_L(s) is analytic on the closed right-half plane (RHP), namely the set of s such that s^'⩾ 0.

Due to λ_0 < 0, one can set a=0 in Eq. (<ref>), so the integration occurs on the imaginary axis. The substitution s = -iω, where ω is a real variable, yields

    f(t) = 1/2π∫_-∞^∞F_L(-iω) e^-iω t ω .


Upon noting that f(t) ∈ L^1 is the basic condition for the Fourier-transformability of f(t), it is straightforward to recognize that F_L(-iω) is equal to the Fourier transform F_F(ω) of f(t):

    F_F(ω) = F_L(-iω)  .

The inverse is true as well: f(t) being causal and Fourier–transformable implies its Fourier transform F_F(ω) to correspond to the Laplace transform F_L(s), with s^' = 0, s^'' = -iω, and to this Laplace transform having λ_0 < 0.



§ KRAMERS-KRONIG RELATIONS


We now suppose that, besides being causal, the transfer function G(t) of an LTI system is Lebesgue-integrable. Upon setting a complex number s_0 = -iω lying on the imaginary axis, we then consider the following function

    H(s,ω) ≡G_L(s)/s-s_0 = G_L(s)/s+iω .

Due to G_L(s) being analytic on the closed RHP, H(s,ω) is analytic on the closed RHP except at the point s = s_0 = -iω. By virtue of Cauchy residue theorem, an integration along the closed path shown in Fig. <ref> yields a vanishing result because no poles lie within the path:

    ∫_-iR^-iω-iε H(s,ω)  s + ∫_γ(ε) H(s,ω)  s + ∫_-iω+iε^iR H(s,ω)  s + ∫_Γ(R) H(s,ω)  s = 0  ,

where the Γ(R), γ(ε) are two semicircular paths of radii R and ε, respectively, that are connected by the two linear segments joining the points on the imaginary axis of ordinates -iR, -iω-iε, and -iω+iε, iR.



Once ε→ 0^+ and R →∞, the sum of the integrals along the linear segments can be expressed as the Cauchy principal value of a single integral:

    lim_ε→ 0^+
    R →∞(∫_-iR^-iω-iε H(s,ω)  s + ∫_-iω+iε^iR H(s,ω)  s) =
    	P ∫_-i∞^+i∞ H(s,ω)  s = P ∫_-i∞^+i∞G_L(s)/s + iω s .

The integral on γ(ε), which runs counterclockwise, is equal to iπG_L(-iω).

Now comes the crucial part of the theorem, namely to show that the integral along Γ(R) vanishes as R →∞. Upon writing s in polar coordinates as s = R e^iθ, the integral can be written as

    lim_R →∞∫_Γ(R) H(s,ω)  s = lim_R →∞∫_π/2^-π/2G_L(R e^iθ)/R e^iθ + iω i R e^iθ θ = lim_R →∞∫_π/2^-π/2 i s G_L(s)/s + iω θ .

For any θ within the interval (-π/2, π/2) the limit R →∞ implies s^'→ +∞, so G_L(s) tends to zero as a consequence of the Riemann–Lebesgue lemma expressed in Eq. (<ref>). Therefore, because given a real number ℓ > 1, one has | s/(s+iω) | ⩽ℓ as soon as R ⩾ℓ |ω| / (ℓ - 1), the whole integral vanishes as well.[By writing s = R e^iθ, one has

    | s/s+iω| ⩽ℓ⇔ω^2/R^2 + 2 ω/Rsin(θ) + 1 ⩾1/ℓ^2 .

Because, for any real number a, a sin(θ) ⩾ -|a|, it holds

    ω^2/R^2 - 2 |ω|/R + 1 ⩾1/ℓ^2⇔|ω|/R⩽ 1 - 1/ℓ .
]

The argument used here is similar to Jordan's lemma, which states that if the maximum value of G_L(s) satisfies the Riemann–Lebesgue lemma, then, for t < 0, one has

    lim_R →∞∫_Γ(R)G_L(s)   e^st s = 0  .

The main difference between Jordan's lemma and the present argument is therefore the factor e^st, which is here replaced with 1/(s+iω). In addition, while in Jordan's lemma the sign of the parameter t plays a crucial role to achieve the convergence to zero of the integral, here the parameter ω plays no role.

Setting s = -iν, ν∈ℝ, and remembering the relation between Laplace and Fourier transform expressed by Eq. (<ref>) above, the path integral of Eq. (<ref>) can then be rewritten as

    G_F(ω)=1/i π P∫_-∞^∞G_F(ν)/ν-ω ν .


Separating the real and imaginary part of the Fourier transform by writing G_F(ω)=G_F^'(ω)+iG_F^''(ω) yields

    G_F^'(ω)    = 1/πP∫_-∞^∞G_F^''(ν)/ν-ω ν,
    G_F^''(ω)    = -1/πP∫_-∞^∞G_F^'(ν)/ν-ω ν,

that corresponds to the well known Kramers–Kronig relations <cit.>, i.e. to G_F^'(ω), G_F^''(ω) being the Hilbert transforms of one another.



§ DISCUSSION

We mentioned above that the Kramers-Kronig relations were proven by Titchmarsh to be valid for any L^2, causal function G(t). One might argue that the present proof of the Kramers-Kronig relations, in which the L^2 assumption is replaced with the L^1 assumption (see diagram below in Fig. <ref>), is, due to its enhanced simplicity, superior to Titchmarsh's approach. Indeed, there are two reasons why the Titchmarsh theorem still makes up the unsurpassed way to achieve the Kramers-Kronig relations.



First, any function G_L(s) that is analytic on the closed RHP does not necessarily correspond to a causal, L^1 transfer function: as a major counterexample, a constant G_L(s) cannot be the Laplace transform of any regular function because it would violate the Riemann–Lebesgue lemma (a constant G_L(s) is, indeed, the Laplace transform of a distribution, namely a Dirac delta in the origin). For this reason, our approach to the Kramers-Kronig relations is one-way only. Conversely, the Titchmarsh theorem can be read also backwards: a function G_F(ω) belonging to L^2 and whose real and imaginary parts are Hilbert transforms of one another does correspond to a causal, L^2 transfer function G(t) = ℱ^-1[G_F(ω)]. Second, proving the Kramers-Kronig relations for L^2 functions makes the use of the theorem more handy, and this is what matters in physical applications.

However, as mentioned above, our approach provides an easier proof in all the cases in which the causal transfer function belongs to L^1 ∩ L^2.



§ CONFLICT OF INTEREST

The authors have no conflicts to disclose.





 Paper Title: Exact surface energies and boundary excitations of the Izergin-Korepin model with generic boundary fields.










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        cmmib10 scaled1Exact surface energies and boundary excitations of the Izergin-Korepin model with generic boundary fields 
        § ABSTRACT  The Izergin-Korepin model is an integrable model with the simplest twisted quantum affine algebra U_q(A_2^(2)) symmetry. Applying the t-W method, we derive the homogeneous zero roots Bethe ansatz equations and the corresponding zero root patterns of the Izergin-Korepin model with generic integrable boundaries. Based on these results, we analytically compute the surface energies and boundary excitations in different regimes of boundary parameters of the model. It is shown that in some regimes, correlation effect appears between two boundary fields.
        
        
        
        
        
         PACS: 75.10.Pq, 03.65.Vf, 71.10.Pm
        
        
         Keywords: Bethe Ansatz; Lattice Integrable Models
        
        
        
        
        
        
        
        § INTRODUCTION
        
        The Izergin-Korepin (IK) model<cit.> is a quantum integrable model <cit.> with the twisted quantum affine algebra U_q(A^(2)_2) as underying algebraic structure. It was shown that this model is closely related to the Dodd-Bullough-Mikhailov or Jiber-Mikhailov-Shabat model <cit.>, which is a low-dimensional relativistic quantum field model.  The IK model with periodic boundary condition or diagonal boundaries has been extensively studied <cit.>. 
        The open boundary case is related to the O(n) loop model <cit.> and  self-avoiding walks  <cit.>.
        The IK model with generic integrable boundaries defined by non-diagonal reflection matrices  <cit.>) is not U(1) symmetric and its exact solution was obtained in <cit.> by means of the so-called off-diagonal Bethe ansatz (ODBA).
        However, the physical properties of the model remain an unsolved problem due to the non-homogeneity of the Bethe ansatz equations (BAEs) and the unclear patterns of the Bethe roots.
        
        In <cit.>, a systematic t-W method for calculating physical properties of quantum integrable systems with or without U(1)-symmetry has been developed based on parameterizing the eigenvalues of the transfer matrix by zero points of the Bethe ansat equations. Subsequently, this method was applied to many quantum integrable models, such as the 
        integrable J_1-J_2 model with competition interactions <cit.>
         and the D_2^(1) spin chain with generic non-diagonal boundary reflections <cit.>. The algebraic structures of these models are related to untwisted affine algebras.  
        In this paper we generalize the t-W method to integrable spin chain model with twisted affine algebra symmetry. We derive the density of zero roots and surface energy of the IK model with generic boundary fields using the t-W method.
        
        The paper is organized as follows.  Section 2 presents an introduction to the IK model and its integrability. In section 3, we derive the homogeneous zero roots BAEs by using the intrinsic properties of the transfer matrix and the zero roots of its eigenvalues. The BAEs completely determine the eigenvalues and energies of the system. In section 4, we give the detailed patterns of the zero roots in the different regimes of boundary parameters. 
        Section 5 is devoted to the computation of the surface energies induced by the generic boundary fields. The boundary excitations associated with the
        boundary magnetic fields also are calculated in section 6. Concluding remarks are given in section 7.
        
        
        
        
        
        
        
        
        
        
        
        
        § HAMILTONIAN AND  INTEGRABILITY
        
        
        The Hamiltonian of the Izergin-Korepin (IK) model with generic integrable boundaries reads
        
        H=H_bulk+H_R+H_L.
        
        Here H_bulk is the bulk Hamiltonian
        
        H_bulk                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  =                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  2/sinh 5η-sinhη∑_j=1^N-1{cosh5η (E_j^11E_j+1^11+E_j^33E_j+1^33)
        .;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             .+sinh2η(sinh3η-cosh3η)
        (E_j^11E_j+1^22+E_j^22E_j+1^33)
        .;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            . +sinh2η(sinh3η+cosh3η)
        (E_j^22E_j+1^11+E_j^33E_j+1^22)
        .;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    .+2sinhηsinh2η (e^-2η
        E_j^11E_j+1^33+e^2η E_j^33E_j+1^11)+coshη
        (E_j^13E_j+1^31+ E_j^31E_j+1^13)
        .;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            . +cosh3η
        (E_j^12E_j+1^21+E_j^21E_j+1^12
        +E_j^22E_j+1^22+E_j^23E_j+1^32
        +E_j^32E_j+1^23) .;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    .  -e^-2ηsinh2η
        (E_j^12E_j+1^32+E_j^21E_j+1^23) +
        e^2ηsinh2η (E_j^23E_j+1^21+E_j^32E_j+1^12)
        },
        where η is the crossing parameter and E_j^μν(μ, ν=1,2,3) are is the Weyl matrix on jth-site with  
        E^μν=|μ⟩⟨ν |, and 
        H_L and H_R quantify the left and right boundary fields, respectively,
        
        H_L=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  2e^-ε[1+2e^-εsinhη]^-1[sinhη(E_1^11-E_1^33 )+coshη E_1^22-e^ς E_1^13-e^ς E_1^31],;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               H_R=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    2[sinh5η-sinhη)(2e^-ε^'sinh5η-4e^-ε^'sinhηcosh4η+1+2cosh2η)]^-1;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ×{[(e^2η-2e^-4η-ε^'sinhη)cosh5η+(1+2e^-ε^'sinh5η)sinh2η(sinh3η-cosh3η);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +2(e^-4η-2e^2η-ε^'sinhη)sinhηsinh2η]E_N^11+[ (e^2η-2e^-4η-ε^'sinhη)sinh2η;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       ×(sinh3η+cosh3η)+(1+2e^-ε^'sinh5η)cosh3η+(e^-2η-2e^4η-ε^'sinhη);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    ×sinh2η(sinh3η-cosh3η) ]E_N^22+[2(e^4η-2e^-2η-ε^'sinhη)sinhηsinh2η;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +(1+2e^-ε^'sinh5η)sinh2η(sinh3η+cosh3η)+(e^-2η-2e^4η-ε^'sinhη)cosh5η]E_N^33;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   -2e^-ε^'sinh6ηcoshη(e^2η+ςE_N^13+e^-2η-ςE_N^31   )};                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          -2e^-ε^'(2sinhηsinh4η-cosh5η)/2cosh2η-4e^-ε^'sinhηcosh4η+1+2e^-ε^'sinh5ηI_N,
        
        where ε, ε^', ς and ς^' are the boundary parameters. We should note that H_L and H_R are two unparallel boundary fields, which obviously break the U(1) symmetry. The hermiticity of the
        Hamiltonian (<ref>) requires that the crossing parameter η and the four boundary parameters are real.
        
        The Hamiltonian (<ref>) is constructed by using the R-matrix and the type II  generic non-diagonal K-matrices K^±(u) classified in <cit.>. The R-matrix R(u)∈ End( V⊗ V) is U_q(A_2^(2)) symmetric and has the form <cit.>
        R_12(u)=([           [ h_3(u); h_2(u); h_4(u) ] [ ; e(u)          ;      g(u)      ]               [ ; ; f(u)           ];                   [ e̅(u); g̅(u);  ]           [ h_2(u); h_1(u); h_2(u) ]      [ ; g(u)          ; e(u)      ];                        [ f̅(u); ;  ]                   [ g̅(u); e̅(u);  ]           [ h_4(u); h_2(u); h_3(u) ] ]),
        where                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             h_1(u)=sinh(u-3η)
        -sinh 5η+sinh 3η+sinhη,  
        h_2(u)=sinh(u-3η)
        +sinh3η,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    h_3(u)=sinh(u-5η)+sinhη,   h_4(u)=sinh(u-η)+sinhη,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    e(u)=-2e^-u/2sinh2ηcosh(u/2-3η),  e̅(u)=-2e^u/2sinh2ηcosh(u/2-3η),;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                f(u)=-2 e^-u+2ηsinhηsinh2η-e^-ηsinh4η,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  f̅(u)=2 e^u-2ηsinhηsinh2η-e^ηsinh4η,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               g(u)=2e^-u/2+2ηsinhu/2sinh 2η,  g̅(u)=-2e^u/2-2ηsinhu/2sinh 2η.
         
        It satisfies the quantum Yang-Baxter equation (QYBE) <cit.>,
        
        R_12(u_1-u_2)R_13(u_1-u_3)R_23(u_2-u_3)
        =R_23(u_2-u_3)R_13(u_1-u_3)R_12(u_1-u_2).
        The type II generic non-diagonal K-matrices K^±(u) is given by <cit.>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              K^-(u)=(
                  [   1+2e^-u-εsinhη                0    2e^-ε+ςsinh u;                0 1-2e^-εsinh(u-η)                0;    2e^-ε-ςsinh u                0    1+2e^u-εsinhη;                  ]),;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     K^+(u)= MK^-(-u+6η+iπ)|_(ε,ς)→(ε^',ς^'),
        where M is the diagonal constant matrix, M=diag(e^2η,1,e^-2η). K^-(u) satisfies reflection equation (RE)
        
        R_12(u-μ)K_1^-(u)R_21(u+μ)K_2^-(μ)=K_2^-(μ)R_12(u+μ)K_1^-(u)R_21(u-μ),
        
        and K^+(u) satisfies the dual RE
        
        R_12                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (μ-u)K_1^+(u) M_1^-1R_21(-u-μ+12η) M_1K_2^+(μ);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 =K_2^+(μ) M_2^-1R_12(-u-μ+12η) M_2K_1^+(u)R_21(μ-u).
        
        
        We now define the one-row monodromy matrices
        T(u) and T̂(u), which are two 3× 3 matrices with operator-valued elements acting on quantum space V^⊗ N,
        
        T_0(u)=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               T̂_0(u)=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            R_10(u+θ_1)R_20(u+θ_2)⋯ R_N0(u+θ_N),
        where V_0 is a 3× 3 auxiliary space.
        The corresponding transfer matrix t(u) is given by
        
        t(u)= tr_0{K^+_0(u)T_0(u)K^-_0(u)T̂_0(u)},
        The Hamiltonian (<ref>) then can be expressed in terms of the transfer matrix t(u) as
        
        H=-∂ ln t(u)/∂ u|_u=0,{θ_j=0}. 
        
        The QYBE (<ref>), the RE (<ref>) and its dual (<ref>) give rise to the commutation relation [t(u),t(v)]=0, which ensures the integrability of the Hamiltonian (<ref>).
        
        
        
        § HOMOGENEOUS BAES
        
        
        Using the properties of the R-matrix (<ref>) and the K-matrices (<ref>)-(<ref>), the following operator identities can be easily proved                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   t(±θ_j)t(±θ_j+6η+iπ)=δ_1(u)× id/φ_1(2u)|_u=±θ_j,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                t(±θ_j)t(±θ_j+4η)=δ_2(u)× t(u+2η+iπ)/φ_2(-2u+8η)|_u=±θ_j,
        where j=1,⋯,N and the functions included in above operator identities are as follows
        δ_1(u)=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         -4(1-2e^-εsinh(u-η))(1+2e^-εsinh(u+η))(1-2e^-ε^'sinh(u-η));                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    ×(1+2e^-ε^'sinh(u+η))sinh(u+6η)cosh(u+η)sinh(u-6η);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ×cosh(u-η)∏_l=1^Nφ_1(u-θ_l)φ_1(u+θ_l),;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            δ_2(u)=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 -4(1-2e^-εsinh(u-η))(1-2e^-ε^'sinh(u-η))sinh(u+4η);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    ×sinh(u-6η)cosh^2(u-η)∏_l=1^Nφ_3(u-θ_l)φ_3(u+θ_l),;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            φ_1(u)=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                -4sinh(u/2-2η)sinh(u/2+2η)cosh(u/2-3η)cosh(u/2+3η),;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            φ_2(u)=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      -4cosh(u/2-5η)cosh(u/2-η)sinhu/2sinh(u/2-6η),;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            φ_3(u)=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 -2sinh(u/2+2η)cosh(u/2-3η).
        
        In addition, the transfer matrix of the model possesses the crossing symmetry and asymptotic behavior                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  t(u)=t(-u+6η+iπ),;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 lim_u→±∞t(u)=(1/2)^2Ne^± 2(N+1)(u-3η)-ε-ε^'[1+2cosh(ς^'-ς+2η)]× id+⋯.
        
        
        Let Λ(u) denote the eigenvalue of the transfer matrix t(u). Then it follows from the above results on t(u) that the eigenvalue Λ(u) satisfies                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  Λ(u)=Λ(-u+6η+iπ),;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Λ(±θ_j)Λ(±θ_j+6η+iπ)=δ_1(u)/φ_1(2u)|_u=±θ_j,   j=1,⋯,N,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Λ(±θ_j)Λ(±θ_j+4η)=δ_2(u)×Λ(u+2η+iπ)/φ_2(-2u+8η)|_u=±θ_j,   j=1,⋯,N,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     lim_u→±∞Λ(u)=(1/2)^2Ne^± 2(N+1)(u-3η)-ε-ε^'[1+2cosh(ς^'-ς+2η)]+⋯,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Λ(0)=Λ(6η+iπ)=(1+2e^-εsinhη)tr{K^+(0)}∏_l=1^Nφ_1(-θ_l),;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Λ(iπ)=Λ(6η)=(1-2e^-εsinhη)tr{K^+(iπ)}∏_l=1^Nφ_1(iπ-θ_l).
        From these relations and the definition (<ref>) of t(u), we see that Λ(u) is a polynomial of u with degree 4N+4 and can be parameterized by its zero points as
        Λ(u)=Λ_0∏_j=1^2N+2sinh(u/2-z_j/2-3η/2)sinh(u/2+z_j/2-3η/2+iπ/2),
        where Λ_0 is a coefficient and {z_j|j=1,⋯,2N+2} are the zero points of the polynomial. Putting the parameterization (<ref>) into (<ref>)-(<ref>), we can deduce the homogeneous BAEs                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Λ_0^2∏_l=1^2N+2sinh(±θ_j/2-z_l/2-3η/2)sinh(±θ_j/2+z_l/2-3η/2+iπ/2)
        sinh(±θ_j/2-z_l/2+3η/2+iπ/2);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       ×sinh(±θ_j/2+z_l/2+3η/2)=δ_1(±θ_j)/φ_1(± 2θ_j),;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          Λ_0∏_l=1^2N+2sinh(±θ_j/2-z_l/2-3η/2)sinh(±θ_j/2+z_l/2-3η/2+iπ/2)
        sinh(±θ_j/2+z_l/2+η/2+iπ/2);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              ×sinh(±θ_j/2-z_l/2+η/2)=δ_2(±θ_j)∏_l=1^2N+2sinh(±θ_j/2-z_l/2+3η/2)sinh(±θ_j/2+z_l/2+3η/2)/φ_2(∓ 2θ_j+8η),
        where j=1,⋯,N. The BAEs and Eqs.(<ref>)-(<ref>) can determine the 2N+3 unknowns Λ_0 and {z_j|j=1,⋯,2N+2} completely.
        Moreover, the energy spectrum of the system (<ref>) can be expressed as
        
        E=1/2∑_j=1^2N+2[(z_j/2+3η/2)-tanh(z_j/2-3η/2)].
        
        
        § PATTERNS OF ZERO ROOTS
        
        
        Now let us consider the patterns of the zero roots {z_j} for the ground state. For convenience, we
        put {θ_j≡ iθ̅_j} with real {θ̅_j}, and let {z̅_j≡ iz_j}. In addition, we set χ_+= arcsinh(e^ε/2) and χ_-= arcsinh(e^ε^'/2).
        
        
        
        Summarizing the numerical solutions of Eqs.(<ref>)-(<ref>)
        with finite system size N, the patterns of the z̅-roots for the ground state include: (1) a set of bulk pairs as {z̅_j∼z̃_j-5iη, (z̃_j-π)+5iη} with real {z̃_j}; (2) free open boundary pairs ±π/2± 2iη; (3) boundary pairs determined by the boundary parameters; (4) additional roots with imaginary parts greater than 5η. According to the  number and type of the boundary pairs and additional roots, the distribution of the z̅-roots can be divided into six different regimes in the χ_+-χ_- plane, as shown in Fig.<ref>. 
        
        
        
        1) In the regime I, where χ_+≥ 3η, χ_-≥ 3η, as shown in Fig.<ref>, there only exist four additional roots ±α_1+ iβ_1, ±(π-α_1)- iβ_1 with β_1≥5η;   
        
        2) In the regime II, where χ_+≥ 3η, η≤χ_-< 3η or  η≤χ_+< 3η, χ_-≥ 3η, as shown in Fig.<ref>, there exist two additional roots iβ_1, π-iβ_1 with β_1≥5η, and one boundary pair π-i(2η+χ_-), i(2η+χ_-); 
        
        3) In the regime III, where χ_+≥ 3η, χ_-< η or  χ_+< η, χ_-≥ 3η, as shown in Fig.<ref>, there exist four additional roots iβ_1, -iβ_2, π-iβ_1, π+iβ_2 with {β_1,β_2}≥5η, and two 
        boundary pairs π-i(2η+χ_-), i(2η+χ_-), π+i(4η+χ_-), -i(4η+χ_-); 
        
        4) In the regime IV, where η≤χ_+< 3η, η≤χ_-< 3η, as shown in Fig.<ref>, there exist four additional roots ±α_1-iβ_1, ±(π-α_1)+iβ_1 with β_1≥5η, and two 
        boundary pairs π-i(2η+χ_-), 
         i(2η+χ_-), π-i(2η+χ_+), 
         i(2η+χ_+); 
        
        5) In the regime V, where η≤χ_+< 3η, χ_-< η or  χ_+< η, η≤χ_-< 3η, as shown in Fig.<ref>, there exist two additional roots -iβ_1, π+iβ_1 with β_1≥5η, and three boundary pairs π-i(2η+χ_-), 
         i(2η+χ_-), π-i(2η+χ_+), i(2η+χ_+), π+i(4η+χ_-), -i(4η+χ_-); 
        
        6) In the regime VI, where χ_+<η, χ_-<η, as shown in Fig.<ref>, there only exist four boundary pairs π-i(2η+χ_-), i(2η+χ_-), π-i(2η+χ_+), i(2η+χ_+), π+i(4η+χ_-), -i(4η+χ_-) and π+i(4η+χ_+), -i(4η+χ_+).
        
        We find that the patterns of zero roots for the IK model are different from the ones for models related to untwisted A-type algebras <cit.>. 
        In Tab.<ref>, we list the differences between the zero roots of spin-1 model (the 19-vertex model related to untwisted A^(1)_1) and the IK model. We observe that the zero roots of the spin-1 model form  conjugate pairs and the bulk pairs are a set of four strings. However, the zero roots in the IK model form non-conjugated pairs and the bulk pairs locate only on the two lines with imaginary parts close to ± 5η.
        
        
        
        
        
        § SURFACE ENERGY
        
        
        We now study the surface energy induced by the generic boundary fields. The surface energy is defined by E_b=E_g-E_p, where E_g is the ground state energy of Hamiltonian (<ref>)
        and E_p is the ground state energy of periodic IK model. In the thermodynamic limit, the distribution of z̃-roots can be  described by the continuum density ρ(z̃). Furthermore, we assume that the inhomogeneity parameters also has the continuum density
        σ(θ̅)=1/[2N(θ̅_j-θ̅_j-1 )].
        
        In regime I, taking the logarithm of the BAEs (<ref>) or (<ref>) and making the difference of θ_j-1 and θ_j in the thermodynamic limit, we readily have                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2N∫_-∞^+∞[b_2(u-z̅)+b_8(u-z̅+π)]ρ(z̅)dz̅-π/2)+b_5(u+π/2)+b_5(u-π/2);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +b_β_1/η+3(u-α_1)+b_β_1/η-3(u+π-α_1)+b_β_1/η-3(u-π+α_1)+b_β_1/η+3(u+α_1);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  =                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             b_|1+χ_+/η|(u)+b_|1-χ_+/η|(u+π)+b_|1+χ_-/η|(u)+b_|1-χ_-/η|(u+π)+b_6(u)+b_6(u+π)-b_2(u);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             -b_2(u+π)-b_3(u-π/2)-b_3(u+π/2)
        +2N∫_-∞^+∞[b_4(u-θ̅)+b_6(u-θ̅+π)]σ(θ̅)dθ̅,
        where the function b_n(u)=1/2πsin(u)/cosh(nη)-cos u. Solving the Eq.(<ref>) by the Fourier transformation, we obtain
        ρ̃(k)=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     [2N(b̃_4+(-1)^kb̃_6)σ̃(k)+b̃_|1+χ_+/η|+b̃_|1+χ_-/η|+(-1)^k·(b̃_|1-α/η|+b̃_|1-α^'/η|+b̃_6-b̃_2);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +b̃_6-b̃_2-2(b̃_3+b̃_5 ) cos(π k/2)-2(b̃_β_1/η+3+b̃_β_1/η-3)cos(α_1 k)]/[2N(b̃_2+(-1)^kb̃_8)],
        where the Fourier spectrum k takes integer values and b̃_n(k)=sign(k)ie^-η|nk|. In the homogeneous limit σ(θ̅)=δ(θ̅), the ground state energy E_g1 of the
        Hamiltonian (<ref>) in regime I can be expressed as
        
        E_g1=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           N∑_k=-∞^∞[ã_8(k)e^-iπ k-ã_2(k)]ρ̃(k)+1/2[(3η+iα_1-β_1/2)+(3η-iα_1-β_1/2);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +tanh(3η-iα_1+β_1/2)+tanh(3η+iα_1+β_1/2)]+tanhη+tanh(5η),
        where ã_n(k)=e^-η|nk| is the Fourier transformation of a_n(u)=1/2πsinh(nη)/cosh(nη)-cos u. The ground state energy of the periodic IK model is
        
        E_p=2N∑_k=1^∞[(-1)^k e^-η|6k|-1]e^-η|4k|+(-1)^k e^-η|6k|/1+(-1)^ke^-η|6k|.
        Subtracting Eq.(<ref>) from Eq.(<ref>), we obtain the surface energy in the regime I                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       E_b1=e_b(χ_+)+e_b(χ_-)+e_b0,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               e_b(χ)=∑_k=1^∞[(-1)^k e^-η|6k|-1]e^-η|(1+χ/η)k|+(-1)^k e^-η|(1-χ/η)k|/1+(-1)^ke^-η|6k|,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           e_b0=∑_k=1^∞[(-1)^k e^-η|6k|-1](e^-η|6k|-e^-η|2k|)(1+(-1)^k)-2(e^-η|5k|+e^-η|3k|)cosπ k/2/1+(-1)^k e^-η|6k|;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +tanhη+tanh(5η),
        
        where e_b(χ) indicates that the contribution of one boundary field and e_b0 is the surface energy induced by the free open boundary. The absence of α_1 and β_1 in Eq.(<ref>) is due to that the bare contribution of the additional roots to the energy is exactly offset by that of the change of the continuous root density                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       1/2[(3η+iα_1-β_1/2)+(3η-iα_1-β_1/2)+tanh(3η-iα_1+β_1/2)+tanh(3η+iα_1+β_1/2)];                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              -∑_k=-∞^∞[(-1)^k e^-η|6k|-1](e^-|(β_1-3η)k|+e^-|(β_1+3η)k|)cos(α_1 k)/1+(-1)^k e^-η|6k|=0.
        
        Using a similar procedure with regime I in regimes II and IV, we can find that the boundary pairs and additional roots will also contribute nothing to the surface energy. The surface energy E_b2 and E_b4 take exactly
        the same form of (<ref>). 
        
        In regime III, the imaginary part of the inner boundary pair will satisfies {χ_+,χ_-}_min+2η<3η. In this case, the inner boundary pair indeed contributes a nonzero value to energy and the surface energy reads
        
        E_b3=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ∑_k=1^∞[1-(-1)^k e^-η|6k|](-1)^k e^-η|(1-χ_1/η)k|+e^-η|(5+χ_1/η)k|/1+(-1)^k e^-η|6k|+E_b1;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +1/2(5η/2+χ_1/2)+1/2tanh(η/2-χ_1/2),
        
        where χ_1={χ_+,χ_-}_min. The contributions of the two boundaries to the surface energy are no longer additive, which implies that a correlational effect appears between the two boundary fields in this case.
        
        In regimes V and VI, there will exist another inner boundary pair that satisfies {χ_+,χ_-}_max+2η<3η. The new inner boundary pair also contributes a nonzero value to energy and the surface energy reads
        
        E_b5=E_b6=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ∑_k=1^∞[1-(-1)^k e^-η|6k|](-1)^k e^-η|(1-χ_2/η)k|+e^-η|(5+χ_2/η)k|/1+(-1)^k e^-η|6k|+E_b3;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +1/2(5η/2+χ_2/2)+1/2tanh(η/2-χ_2/2),
        
        where χ_2={χ_+,χ_-}_max. In the two regimes, a correlation effect also appears between the two boundary fields.
        
        We note that the boundary parameters ς and ς^' do not appear in the surface energies of all regimes, implying
        that they contribute nothing to the surface energy in the leading order.
        The surface energies with different boundary parameters ε and ε^' are shown in Fig.<ref> below. From this figure, we observe that the surface energies with the increasing or decreasing of ε tend to fixed values. In addition, the surface energy behaves as a monotonic function of ε or ε^' when χ_+<η or χ_-<η.
        
        
        
        
        
        
        
        
        
        
        
        § BOUNDARY EXCITATIONS
        
        Besides the usual bulk elementary excitations, the system also exist the boundary excitations associated with the boundary magnetic fields (<ref>). By comparing the zero roots distributions of the ground state and the excited states, we find that the boundary excitations can exist in the regimes II-VI, where the boundary parameter  χ_+< 3η or  χ_-< 3η. The typical boundary excitation is putting the boundary pairs from π-i(2η+χ_±), i(2η+χ_±) to π-i(4η-χ_±), i(4η-χ_±), and from π+i(4η+χ_±), -i(4η+χ_±) (or the bulk pair) to π+i(2η-χ_±), -i(2η-χ_±). These four new boundary pairs indeed satisfy the BAEs (<ref>)-(<ref>). 
        
        
        
        As an example in Fig.<ref>, we show the pattern of zero roots at the ground state (blue asterisks) and that at the 32nd excited state (red squares) with two boundary pairs π+i(2η-χ_+), -i(2η-χ_+), π-i(4η-χ_+), i(4η-χ_+) in regime II with N=8. In the excitation process, we observe that four roots at the bulk pair parts of the ground state jump to form ±α_1+ iβ_1, ±(π-α_1)- iβ_1 with β_1>5η. From the Eq.(<ref>), we have known that the change of the zero roots contribute nothing to the energy. Without losing generality, we omit the zero roots in the subsequent discussions. In the thermodynamic limit, the resulted density change between ground and excited states reads
        δρ̃_χ_+(k)=e^-η|(5+χ_+/η)k|-e^-η|(7-χ_+/η)k|-e^-η|(1+χ_+/η)k|-(-1)^k e^-η|(5-χ_+/η)k|/2N(e^-η|2k|+(-1)^k e^-η|8k|).
        
        The corresponding excited energy is
        δ_e_χ_+=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             ∑_k=1^∞[1-(-1)^k e^-η|6k|]e^-η|(5+χ_+/η)k|-e^-η|(7-χ_+/η)k|-e^-η|(1+χ_+/η)k|-(-1)^k e^-η|(5-χ_+/η)k|/1+(-1)^k e^-η|6k|;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                +1/2(7η/2-χ_+/2)+1/2(η/2+χ_+/2)+1/2tanh(5η/2-χ_+/2); -1/2(5η/2+χ_+/2)-tanh(η/2-χ_+/2).
        The excited energies with fixed values ε of versus are shown in Fig.<ref>.
        
        We have also calculated the boundary excitations in the remaining regimes and found that the excited energies are given by expressions similar to (<ref>). Note that when computing the boundary excitation in the regime with χ_-< 3η,  χ_+ in Eq.(<ref>) should be replaced by the χ_-.
        
        
        
        
        
        
        
        
        
        § CONCLUSIONS
        
        In this paper, we have extended the t-W method to the IK model with generic integrable boundaries. By parameterizing the eigenvalues of the transfer matrix using its zero points, we derive two sets of homogeneous zero roots BAEs. The patterns of zero roots distributions in different regime are obtained by solving the BAEs. Unlike untwisted A-type models, the zero roots of the present system no longer form conjugate pairs. Based on the patterns, we have obtained the exact surface energies and boundary excitations in different regimes of the boundary parameters. The results indicate that a correlation effect appears between the two boundary fields in some regimes. The method and process presented in this paper can be generalized to other models with twisted affine algebra symmetries. 
        Explicit R-matrices for models related to quantum twisted affine algebras can be constructed by the general procedure developed in <cit.>. Work on the quantum integrable
        D_2^(2) spin chain with generic boundary fields <cit.> is in progress and results will be presented elsewhere. 
        
        
        
        § ACKNOWLEDGMENTS
        
        
        We thank Professor Yupeng Wang for valuable discussions. We acknowledge the financial support from Australian Research Council Discovery Project DP190101529, Future Fellowship FT180100099, National Key R&D Program of China (Grant No.2021YFA1402104),
        China Postdoctoral Science Foundation Fellowship 2020M680724, National Natural Science Foundation of China (Grant Nos. 12074410, 12247103, 12247179, 11934015 and 11975183), the Major Basic Research Program of Natural Science of Shaanxi Province (Grant Nos. 2021JCW-19 and 2017ZDJC-32), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB33000000).
        
        
        
        
        
        
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 Paper Title: Rotational influence on fermions within negative curvature wormholes.




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Rotational influence on fermions within negative curvature wormholes
     Abdullah Guvendi^ 1[ E-mail: abdullah.guvendi@erzurum.edu.tr  ] ,  Semra Gurtas Dogan^ 2[ E-mail: semragurtasdogan@hakkari.edu.tr (Corr. Auth.) ] ,  R. L. L. Vitória^ 3[ E-mail: ricardo-luis91@hotmail.com ]   

	^ 1 Department of Basic Sciences, Erzurum Technical University, 25050, Erzurum, Türkiye

	^ 2 Department of Medical Imaging Techniques, Hakkari University, 30000, Hakkari, Türkiye

	^ 3 Faculdade de Física, Universidade Federal do Pará, Av. Augusto Corrêa, Guamá, Belém, PA 66075-110, Brazil
    
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§ ABSTRACT 

In this research, we examine relativistic fermions within the rotating frame of negative curvature wormholes. Initially, as is typical in our context, we introduce the wormholes by embedding a curved surface into a higher dimensional flat Minkowski spacetime. Subsequently, we derive the spacetime metric that characterizes the rotating frame of these wormholes. We then investigate analytical solutions of the generalized Dirac equation within this framework. Through exploring a second-order non-perturbative wave equation, we seek exact solutions for fermions within the rotating frame of hyperbolic and elliptic wormholes, also known as negative curvature wormholes. Our analysis provides closed-form energy expressions, and we generalize our findings to Weyl fermions. By considering the impact of the rotating frame and curvature radius of wormholes, we discuss how these factors affect the evolution of fermionic fields, offering valuable insights into their behavior.








Keywords: Fermions; Non-inertial effects; Wormhole; Rotating reference frame; Graphene; Weyl fermions	
















§ INTRODUCTION



Landau and Lifshitz introduced rotation to Minkowski spacetime, analyzing two specific effects: the impact on clocks within a rotating body and the singular behavior at significant distances within a uniformly rotating frame <cit.>. Notably, the latter effect imposes a limitation on the radial coordinate. Within the domain of relativistic quantum theory, investigating the impact of a rotating frame involves understanding the principles of special relativity, where the quantized nature of fields and particle behaviors are crucial considerations. This exploration unveils intriguing and unconventional effects absent in classical Sagnac-type descriptions, representing the coupling between angular momentum and rotation angular frequency of a uniformly rotating frame, emphasized across various references <cit.>. In the non-relativistic context, this effect is known as the Page-Werner et al. term <cit.>. The incorporation of relativistic corrections within this framework underscores the potential for phase shifts induced by relative motion between an observer and the rotating frame. Analyzing relativistic quantum systems within such a rotating reference frame unveils a realm of interesting results. Additionally, for quantum fields within curved spaces, the modified effective gravitational field, influenced by the centrifugal force, alters the energy levels of quantum entities, especially in confined or bound states <cit.>. In this context, several studies explore the impact of nontrivial rotating spacetimes on quantum mechanical systems. These investigations encompass various scenarios such as scalar bosons within cosmic string spacetime <cit.>, a scalar field within a Kaluza-Klein theory <cit.>, and a scalar field within time-dislocation spacetime <cit.>. Predicting the effects of a rotating frame on relativistic quantum particles is theoretically plausible; however, practically observing these minute alterations demands precision, control, and advanced techniques in quantum optics, atomic physics, or condensed matter physics, alongside cutting-edge technology <cit.>. Directly observing these effects necessitates trapping and controlling particles within the rotating frame, potentially detectable through variations in energy profiles, trajectories, or alterations in total angular momentum or spin. While extensive theoretical studies have scrutinized the effects of rotating reference frames on physical systems within non-relativistic and relativistic frameworks <cit.>, there is a noticeable gap in announced results regarding the evolution of relativistic fermions within the rotating frame of graphene or a graphene wormhole characterized by a two-dimensional curved surface of constant negative curvature. This unexplored domain requires further investigation to comprehend the intricate interplay between relativistic fermionic fields and specific curved geometries such as graphene structures.

In astrophysics, wormholes emerge from solutions in general relativity, serving as connections between two distinct regions <cit.>. These phenomena may manifest in both astrophysical contexts and condensed matter systems (by embedding curved surfaces into higher-dimensional spacetime (see also <cit.>)). However, the latter depiction differs because it involves two-dimensional curved surfaces, with time remaining unaffected by the background curvature <cit.>. Wormholes are actively researched to understand the fundamental properties of spacetime and whether they can exist within our current understanding of physics. Current research on wormholes spans from astrophysics to condensed matter contexts <cit.>. Research examining quantum systems evolving within curved spaces has garnered considerable interest <cit.>, exemplified by investigations into Weyl particles in optical backgrounds <cit.>, fermion-antifermion pairs in magnetized elliptic wormholes <cit.>, and more <cit.>. No results were found on fermionic fields in the rotating frame of negative curvature wormholes. This research addresses a gap in the literature by offering non-perturbative insights into fermions within rotating graphene and wormhole-like graphene structures characterized by a rotating surface with negative Gaussian curvature. Pursuing exact solutions holds promise in enabling the control or observation of compelling physical properties, potentially fostering novel phenomena, validating fundamental theories, and facilitating technological advancements. Our focus is on investigating relativistic fermion evolution within the rotating framework of negative curvature wormholes by seeking exact solutions to the corresponding Dirac equation.

Our examination of Dirac particles within the rotating frame of negative curvature wormholes is structured as follows: In section <ref>, we unveil a general and non-perturbative wave equation, exploring the intricate realm of relativistic fermions in the rotating frame of negative curvature wormholes. Through a thorough analysis, we present the main wave equation governing their behavior in these environments. In section <ref>, we uncover precise and exact outcomes concerning the evolution of relativistic fermions within two distinct rotating frames: the hyperbolic wormhole and the elliptic wormhole. By investigating these specific scenarios, we reveal compelling insights into the behavior and dynamics of particles under these conditions. Finally, in section <ref>, we summarize our findings, encapsulating the essence of our results. Additionally, we provide a comprehensive discussion that explores the implications of our results across various physically plausible scenarios and limits. This reflective section aims to highlight the significance and broader ramifications of our work within the realm of theoretical physics.



§ WAVE EQUATION
 



In this section, we formulate a radial equation system for relativistic fermions in the rotating frame of negative curvature wormholes. This type of wormhole is characterized geometrically by points on its surface, which are expressed parametrically as shown below (also referenced in <cit.>):

    r⃗(u,v)=x(u,v)î+y(u,v)ĵ+zk̂,

where x(u,v)=χ( u )cos(v), y(u,v)=χ( u )sin(v), and z(u)=∫√(1-χ_,u^2) du, with _,u indicating the derivative with respect to x^u. The shape of the wormhole is generally described by the function χ(u). The constraints on z arise from the relationship:

    ds^2=dx^2+dy^2+dz^2=du^2+χ^2( u )dv^2,

which results in the Hilbert horizon at χ_,u=1 <cit.>. Embedding a three-dimensional wormhole into a (3+1)-dimensional spacetime generates effective gravity or curvature within the altered spacetime, leading to the experience of spacetime curvature by any particle or quasi-particle present in this modified background spacetime. Now, let us define the (2+1)-dimensional wormhole spacetime coordinates as dx^μ= { cdT, du, dv } and the (3+1)-dimensional Minkowski spacetime coordinates as dx^μ^'= { cdT, dx, dy, dz }, where Greek indices denote the coordinates within the curved space. The transformation between these coordinates can be accomplished via the following matrix:

    ∂ x^μ^'/∂ x^ν=([           1           0           0           0;           0  χ_,ucos(v)  χ_,usin(v) √(1-χ_,u^2);           0 -χ_,usin(v)  χ_,ucos(v)           0 ]).

Consequently, the metric of the considered wormhole spacetime takes the following form (with a negative signature) <cit.>:

ds^2=c^2dT^2-du^2-χ^2( u )dv^2.  

A wormhole of this nature is geometrically depicted as shown in Fig. <ref>. In Eq. (<ref>), c symbolizes the speed of light, and the function χ(u) takes the form χ( u )=a cosh( u/r_0) for a hyperbolic wormhole and χ( u )=b sinh( u/r_0) for an elliptic wormhole. Here, a (b) denotes the radius of the associated wormhole at the midpoint (u=0) between two ends, while r_0 represents the radius of curvature of the wormhole surface û. It is noteworthy that the Gaussian curvature for the spacetime background as expressed in Eq. (<ref>) is K =-χ_,uu/χ. Furthermore, Eq. (<ref>) describes a curved surface exhibiting negative Gaussian curvature, making such curved surfaces suitable for describing a monolayer graphene sheet and graphene wormholes. These surfaces can also be derived using the optical metric of the Banados-Teitelboim-Zanelli (BTZ) black hole <cit.>, referred to as negative curvature wormholes (generally) or hyperbolic/elliptic wormholes depending on the shape of the radius function χ( u ) (for further details, see <cit.>). In this scenario, we examine the rotating frame of spacetime given by Eq. (<ref>). This can be achieved through transformations provided by T=t, u=ρ, and v=ϕ+ω_rf T, where ω_rf denotes the angular frequency of the rotating frame. These transformations allow us to derive the spacetime interval describing a uniformly rotating frame of negative curvature wormholes (also see <cit.>).

ds^2=(c^2-ω_rf^2 χ^2)dt^2-2ω_rf χ^2  dt dϕ-dρ^2-χ^2 dϕ^2,

for which the Gaussian curvature remains unchanged but for u →ρ. In this context, it is crucial to observe that the coordinate ρ needs to adhere to a condition derived from the relationship χ(ρ)< c/ω_rf for the fermions to reside within the light-cone. This introduces an additional requirement, indicating that the corresponding wave function must vanish at the point determined by the condition χ(ρ)= c/ω_rf. Referring to the line element Eq. (<ref>), one can derive the covariant (g_μν) and contravariant (g^μν) metric tensors

   g_μν=([ c^2-ω_rf^2 χ^2              0      -ω_rf χ^2;              0             -1              0;      -ω_rf χ^2              0          - χ^2 ]), 

   g^μν=([            1/c^2                0        -ω_rf/c^2;                0               -1                0;        -ω_rf/c^2                0 ω_rf^2/c^2- χ^-2 ]).

Now, let us express the Dirac equation in its generalized form within this specific background <cit.>

   [γ^μ𝒟_μ+im̃ℐ_2]Ψ(x^μ)=0,   μ=t,ρ,ϕ,

   𝒟_μ=∂_μ-Γ_μ,  m̃=mc/ħ.

In this context, m denotes the fermion's rest mass, ħ represents the conventional Planck constant, c stands for the speed of light, ℐ_2 signifies the 2-dimensional identity matrix, Ψ(x^μ) symbolizes the Dirac field dependent on the spacetime position vector x^μ, and γ^μ denotes the space-dependent Dirac matrices, derived from the relation: γ^μ=e^μ_kγ^k, where e^μ_k are the inverse tetrad fields (k=0,1,2.) and γ^k denote the free Dirac matrices. The space-independent (free) Dirac matrices are represented in terms of the Pauli spin matrices (σ^x,σ^y,σ^z) as follows: γ^0=σ^z, γ^1=iσ^x, γ^2=iσ^y, in accordance with the signature of the line element <cit.>. In Equation (<ref>), Γ_μ refers to the spinorial affine connections for the Dirac field, attainable through the relation: Γ_λ=1/4[e^k_ν_,λe^τ_k-Γ_νλ^τ]𝒮^μν, where _,λ indicates the derivative with respect to x^λ. Here, Γ_νλ^τ represents the Christoffel symbols Γ_νλ^τ=1/2g^τϵ[∂_ν g_λτ+∂_λ g_ϵν-∂_ϵ g_νλ], e^τ_k denotes the inverse tetrad fields, and 𝒮^μν symbolizes the spin operators, defined as 𝒮^μν=1/2[γ^μ,γ^ν] <cit.>. Now, the tetrads (and their inverses) can be obtained as follows (also refer to <cit.>):

e^k_μ=([      c      0      0;      0      1      0; ω_rf χ      0      χ ]), e^μ_k=([     1/c       0       0;       0       1       0; -ω_rf/c       0    χ^-1 ])

since g_μν=e^k_μe^l_νη_kl and e^μ_k=g^μνe^l_νη_kl where η_kl is the Minkowski tensor, η_kl=diag(1,-1,-1). Accordingly, we have γ^t= σ^z/c, γ^ρ= i σ^x, γ^ϕ=-ω_rf/c σ^z+ i χ^-1σ^y <cit.>. Also, non-vanishing components of the the spinorial affine connections are Γ_t=i/2ω_rfχ_,ρσ^z and Γ_ϕ=i/2χ_,ρσ^z. Hence, one can verify that <cit.>

γ^μΓ_μ=-i/2χ_,ρ/χ σ^x.

The metric Eq. (<ref>) allows us to factorize the Dirac field as the following Ψ(x^μ)=e^-iω te^is ϕ(ψ_1(ρ) ψ_2(ρ))^T where ω is the relativistic frequency, s is the spin and T means transpose of the ρ-dependent spinor. Finally, a system of coupled equations can be derived

   [ϖ-m̃]ψ_1-[∂_ρ+1/2χ_,ρ/χ+s/χ]ψ_2=0,

   [ϖ+m̃]ψ_2+[∂_ρ+1/2χ_,ρ/χ-s/χ]ψ_1=0.

in which, ϖ=ω/c+s  ω_rf/c, describing fermions within the rotating frame of negatively curved wormholes (see also <cit.>). In the subsequent section, we aim to uncover analytical solutions to this wave equation within two specific contexts: (i) hyperbolic wormholes and (ii) elliptic wormholes.



§ EXACT RESULTS
 

By decoupling the equations presented in Eq. (<ref>), a non-perturbative second-order wave equation for the component ψ_2 is obtained in the subsequent format

[∂_ρ^2+χ_,ρ/χ∂_ρ
 +Q̂ +(ϖ^2-m̃^2)]ψ_2(ρ)=0,

where

Q̂=1/2χ_,ρρ /χ-s^2/χ^2-sχ_,ρ/χ^2 -1/4χ_,ρ^2/χ^2.

Now, we have the ability to explore the precise solution of this wave equation under two distinct scenarios.



 §.§ Relativistic fermions in the rotating frame of hyperbolic wormhole


If we examine a fermionic field within the relativistic framework of a rotating hyperbolic wormhole, the equation (<ref>) transforms into the following form

    ψ_2_,ρρ+tanh( ρ/r_0)/r_0 ψ_2_,ρ+[λ̃+Q̂_1] ψ_2=0,
       Q̂_1=1/2r_0^2-s^2/a^2/cosh^2( ρ/r_0)-s/a/r_0 tanh( ρ/r_0)/cosh( ρ/r_0)-tanh^2( ρ/r_0)/4 r^2_0,

where λ̃=ϖ^2-m̃^2. To transform this unfamiliar equation into a more recognizable form, we can employ an assumed function, ψ_2(ρ)=(sinh(ρ/r_0)+i)^p(sinh(ρ/r_0)-i)^p^*ψ(ρ), where p=3/2-isr_0/2a and ^* denotes the complex conjugate. Then, by introducing a new variable, ξ=1/2[1-i sinh(ρ/r_0)], we arrive at the following wave equation

   ξ(ξ-1)ψ_,ξξ-[γ-(α+β+1)ξ]ψ_,ξ+αβψ=0,

   α=1-ir_0√(λ̃),   β=α^*,   γ=3/2+isr_0/a, 

in which α, β and γ are constants. This equation is the renowned Gauss' hypergeometric equation, and its solution in the vicinity of the origin can be expressed as ψ(ξ)=𝒩  _2F_1(α, β ; γ; ξ) <cit.>. Here, 𝒩 denotes an arbitrary constant. The polynomial condition associated with this function is α=-n <cit.>, where n signifies the principal quantum number (n=0,1,2..). This criterion leads to the quantization principle governing the formation of the system under consideration, consequently yielding the subsequent expression

ℰ_ns=-s  ħω_rf ±ħc√(m^2c^2/ħ^2-(n+1)^2/r_0^2), 

for the relativistic energy of the system under scrutiny. In this context, it becomes evident that the energy of such a system relies on the properties of particles, including spin (assuming ω_rf≠0), particle's rest mass, and the curvature radius of the hyperbolic wormhole, in addition to fundamental constants like ħ and c. The background under consideration is a spacetime with constant negative Gaussian curvature, characteristic of structures like monolayer graphene sheets or graphene wormholes since K=-1/r_0^2 (refer to <cit.>). Consequently, there is potential utility in adjusting the outcomes for Weyl fermions by setting m=0 and substituting c with ν_f, where ν_f denotes the Fermi velocity, approximately equal to c/300. Notably, equation (<ref>) transforms accordingly

ℰ_ns=-s ħω_rf ±iħν_f (n+1)/r_0,

if m=0. The Eq. (<ref>) indicates that the energy of this system possesses a complex nature. Consequently, these states cannot remain stable, and the associated modes either decay or grow over time since Ψ∝e^-iℰ_ns/ħt. Additionally, the real oscillation of a Weyl particle is solely influenced by the coupling between the particle's spin and the rotational frequency of the frame. Our findings suggest that the stability of the background spacetime is compromised under perturbations caused by massless particles, while it remains stable under perturbations from Weyl antiparticles (also see <cit.>) when ω_rf=0. Notably, the coupling s ω_rf might induce symmetry breaking around the Dirac point when m≠ 0. Furthermore, it appears conceivable to measure, in principle, only the energy stemming from the sω_rf coupling if the Dirac particle possesses a critical (or effective) mass value, m_c=ħ (n+1)/c r_0. Such a scenario could arise particularly in condensed matter systems, especially when r_0 is on the order of a few nanometers (also see <cit.>).



 §.§ Relativistic fermions in the rotating frame of elliptic wormhole


In this scenario, we derive precise outcomes for relativistic fermions within the rotating frame of an elliptical wormhole. Consequently, the wave equation presented in Eq. (<ref>) manifests as follows:

    ψ_2_,ρρ+( ρ/r_0)/r_0 ψ_2_,ρ+[λ+Q̂_2] ψ_2=0,
       Q̂_2=1/2r_0^2-s^2/b^2/sinh^2( ρ/r_0)-s/b/r_0 ( ρ/r_0)/sinh( ρ/r_0)-^2( ρ/r_0)/4 r^2_0.

Let us attempt to reformulate this equation using a new variable substitution, z=1/2[1+cosh(ρ/r_0)], aiming to eliminate the hyperbolic functions. By expressing Eq. (<ref>) in terms of the variable z, we can render it free of hyperbolic functions, facilitating an exploration of its asymptotic behaviors. Consequently, employing an ansatz function, ψ_2(z)=z^2sr_0-b/4b(z-1)^-2sr_0+b/4bψ(z), the resulting equation can be transformed into the following wave equation

   z(z-1)ψ_,zz-[γ̃-(α̃+β̃+1)z]ψ_,z+α̃β̃ψ=0,  , 

   α̃=-ir_0√(λ̃),   β̃=α̃^*,   γ̃=1/2+sr_0/b. 
In this context, α̃, β̃, and γ̃ represent fundamental constants within Gauss hypergeometric wave equation, as extensively documented in references <cit.>. The regular solution of this equation in the vicinity of the origin can be elegantly expressed as ψ(z )=𝒞 _2F_1(α̃, β̃ ; γ̃; z ), where 𝒞 stands for an arbitrary constant. Remarkably, when the parameter α̃ assumes the value -n, with n representing the radial quantum number, the solution function ψ(z) transforms into a polynomial of degree n concerning the variable z. This establishes a link between the polynomial nature of the solution function and the quantum characteristics of the system, offering valuable insights into the system's energy behavior at different quantum states. This condition establishes an intriguing connection to the system's energy spectrum, unveiling the following set of energy levels

    ℰ_ns=-s  ħω_rf±ħ c√(m^2c^2/ħ^2-n^2/r_0^2).

While bearing resemblance to the expression presented in Eq. (<ref>), the relativistic energy pertaining to a fermion precisely equals the rest mass energy ± mc^2 under specific conditions where ω_rf=0 and n=0. This distinction stands as the fundamental dissimilarity between the outcomes delineated in Eqs. (<ref>) and (<ref>). Our investigations substantiate that within the rotating frame of an elliptic wormhole, the energy of a relativistic fermion manifests an intriguing independence from the curvature radius of the wormhole as the system converges toward its ground state. Notably, our observations assert that analogous deliberations, explicated in sec. <ref>, remain viable and generalizable when transitioning solely through n+1⟶ n. The inference arises from the energy profiles derived in both situations, suggesting that the baseline energy level for relativistic fermions in the rotating frame of a hyperbolic wormhole appears akin to the first excited state for those in the rotating frame of an elliptic wormhole. Moreover, this correlation remains intact even when ω_rf=0.



§ SUMMARY AND DISCUSSIONS


In this study, we analyse the influence of non-inertial effects arising from the uniformly rotating reference frame of negative curvature wormholes, such as the hyperbolic wormhole and elliptic wormhole, on relativistic fermions. Our approach involves the investigation of analytically permissible solutions of the associated Dirac equation. Initially, we derive a comprehensive non-perturbative second-order wave equation. Subsequently, we proceed to ascertain the exact solutions of this wave equation within two distinct scenarios. In the first scenario, we focus on a background that characterizes the hyperbolic wormhole. Within this context, we determine the solution function expressed in terms of Gauss hypergeometric function. Consequently, we derive a non-perturbative energy spectrum in a closed mathematical form. The expression for this energy spectrum is provided by equation (<ref>).

The findings reveal an intriguing interdependence between the energy states of relativistic fermions, the curvature radius of a wormhole, and the angular frequency of the uniformly rotating frame. Alongside the intrinsic quantum attributes of the fermionic field, our investigation underscores the direct influence of the particle's spin on the energy manifestation. Notably, this interaction may induce symmetry breaking centered around the Dirac point (zero energy), with energy levels diminishing until they reach a critical curvature radius of the wormhole, precisely at r_0=ħ(n+1)/mc, where n signifies the principle quantum number. Importantly, the system's stability is contingent upon the relationship between r_0 and ħ(n+1)/mc; instability ensues if r_0 surpasses ħ(n+1)/mc, irrespective of whether the angular frequency of the rotating frame is null or not. Our investigation further reveals intriguing insights into Weyl fermions (m=0), suggesting that states associated with these fermions may not exhibit stability due to potential damped modes characterized by a decay time τ_n=ħ/|ℰ_ns_Im|, where ℰ_ns_Im denotes the imaginary component of the energy. In an alternative scenario involving an elliptic wormhole background in spacetime, our analysis of the primary equation yields precise solutions for the relativistic energy within the system under consideration. Surprisingly, the ground state energy identified in the earlier scenario appears to correspond to the first excited state of relativistic fermions within the rotating frame of an elliptic wormhole. Intriguingly, when the angular velocity of the rotating frame reaches zero, our results demonstrate that the relativistic energy of the system aligns with the rest mass energy of the fermion upon reaching the ground state (n=0). This alignment reflects an inevitable outcome given our understanding that the system will ultimately settle into its ground state. Furthermore, our outcomes suggest the possibility of measuring energy solely derived from the interplay between the particle's spin and the angular frequency of the uniformly rotating frame. This measurement becomes feasible if the particle possesses a critical (or effective) mass value represented by m_c=ħ (n+1)/c r_0 and m_c=ħ n/c r_0 for the first and second scenarios, respectively. Particularly intriguing is the potential manifestation of this specialized case in condensed matter systems, where c can be equated to ν_f (Fermi velocity). Notably, this scenario appears plausible when the curvature radius of the wormhole approximates a few nanometers, assuming the particle's mass is the typical electron mass.

Recently, there has been a notable focus on performing thermodynamic evaluations using energy levels from quantum systems in complex environments. Several studies in the field have investigated this, setting the stage for future exploration of thermodynamic properties <cit.>. Our precise findings could be valuable for analyzing the thermal properties of graphene structures, presenting an intriguing avenue for future research.







§ DATA AVAILABILITY

This manuscript does not contain any associated data.



§ CONFLICTS OF INTEREST STATEMENT


The authors have disclosed no conflicts of interest.



§ FUNDING


This research has not received any funding.

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 Paper Title: 2D Skyrmion topological charge of spin textures with arbitrary boundary conditions: a two-component spinorial BEC as a case study.





romero@fisica.unam.mx
Instituto de Física, Universidad
Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de México, Mexico. 


§ ABSTRACT 

We derive the most general expression for the Skyrmion topological charge for a two-dimensional spin texture, valid for any type of boundary conditions or for any arbitrary spatial region within the texture. It reduces to the usual one Q = 1/4π∬f⃗·(∂_x f⃗×∂_y f⃗) for the appropriate boundary conditions, with f⃗ the spin texture. The general expression resembles the Gauss-Bonet theorem for the Euler-Poincaré characteristic of a 2D surface, but it has definite differences, responsible for the assignment of the proper signs of the Skyrmion charges. Additionally, we show that the charge of a single Skyrmion is the product of the value of the  normal component of the spin texture at the singularity times the Index or winding number of the transverse texture, a generalization of a Poincaré theorem. We illustrate our general results analyzing in detail a two-component spinor Bose-Einstein condensate in 2D in the presence of an external magnetic field, via the Gross-Pitaevskii equation. The condensate spin textures present Skyrmions singularities at the spatial locations where the transverse magnetic field vanishes. We show that the ensuing superfluid vortices and Skyrmions have the same value for their corresponding topological charges, in turn due to the structure of the magnetic field.

2D Skyrmion topological charge of spin textures with arbitrary boundary conditions: a two-component spinorial BEC as a case study
    S. Sánchez-Reséndiz, E. Neri, S. González-Hernández, and V. Romero-Rochín
    
=================================================================================================================================

Keywords: Skyrmion charge, Vortices, BEC, Ultracold Quantum Matter





§ INTRODUCTION


Skyrmions are topological defects or excitations that emerge in continuous fields and spin textures due to intrinsic chiral interactions  or to the presence of external inhomogeneous magnetic or gauge fields. These excitations arise in diverse materials or fields as in nuclear physics <cit.>, particle physics <cit.>, notably in magnetic systems in condensed matter, such as in 2D electron gas <cit.>, magnetic metals <cit.>, transition metal films <cit.>, frustrated magnets <cit.> and in a host of other magnetic situations <cit.>, liquid crystals <cit.> and, the context of the present paper, in Bose-Einstein condensates, both 2D <cit.> and 3D  <cit.>. Ever since Skyrme seminal work <cit.>, its analysis has been pursued in field theoretical models such as the non-linear σ-model in high energy physics <cit.>, or in terms of the Dzyaloshinskii-Moriya theory in magnetic systems <cit.>, or within the Gross-Pitaevskii (GP) equation in Bose-Einstein condensates, as in essentially all of the theoretical works already referenced. In this paper we deal with Skyrmions generated by an external gauge field in a two-component spinorial Bose-Einstein condensate in two spatial dimensions, described by two coupled GP equations. This stylized model allows for a simple and clear description of  spin textures that can show a rich variety of Skyrmions, with the allowance for the determination of all its  inherent and topological properties. In most of the experimental Skyrmion realizations, BEC studies have been performed in real alkaline weakly interacting Bose clouds with spin F=1  <cit.> and  F = 2 <cit.>, however, the topological properties of a given spin structure should be independent of their physical origin.




An outstanding feature of Skyrmions is their topological character ingrained and described by its topological charge. In most of the studies on 2D Skyrmions  the topological charge has been considered to be given by the expression <cit.>

    Q = 1/4π∬f⃗·( ∂f⃗/∂ x×∂f⃗/∂ y) dxdy ,

where f⃗(x,y) is the 3D vector spin texture or continuous field of the problem at hand, obeying f⃗·f⃗ = 1 everywhere in the XY-plane. It is expected that the charge Q is an integer. This expression, however, requires particular boundary conditions. Skyrme <cit.> and others afterwards <cit.>  considered that f⃗ = constant as |r⃗| →∞ and, in such a case, Q is a positive or negative multiple integer, with the interpretation that Q is the number of times the mapping of f⃗ covers the unit sphere. Additionally, if the spin texture lies on the plane at the boundary, then Q = ± 1/2, a defect called a half Skyrmion <cit.>. These boundary conditions, however, need not appear or exist in a given physical problem <cit.> or in a multiple Skyrmion configuration, where one may need to evaluate the local charge of the Skyrmions present. In these cases expression (<ref>) cannot account for the calculation of the whole charge that, from topological consideration, must be an integer. As an example, in figure <ref> we present a spin texture arising in the 2D spin 1/2 spinor Bose-Einstein condensate (BEC) here studied, the details of such a solution given further below. We observe a configuration of three Skyrmions, one pointing on the positive z-direction and the other two in the negative one. Each defect should have its own topological charge, properly defined, and the total one of the full configuration should be the sum of the individual ones. The main purpose of this article is to show that a topological charge Q_S of the whole or part of the spin texture configuration can be defined within an arbitrary closed curve, which may or may not be the true boundary of the spin texture. Parametrizing the curve by r⃗_c(t), with t: 0 → T and  r⃗_c(0)= r⃗_c(T), we will find that such a topological charge is given by  

    2π Q_S    =   ∬_A f⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy + ∮_r⃗_cf⃗·( d f⃗/dt×d^2f⃗/dt^2) /|d f⃗/dt|^2 dt  
           + ∑_i=1^s sin^-1f⃗_i ·( . d f⃗/dt|_i^ +×. d f⃗/dt|_i^ -)/| d f⃗/dt|_i^ +| d f⃗/dt|_i^ -,

with Q_S ∈ℤ, a positive or negative integer. The first term is the usual one, except for a factor of 2, with A the area enclosed  by the curve r⃗_c. The second is a boundary term with the integral along the curve r⃗_c, and the third one takes into account the possible fact that the vector t⃗ = (d f⃗/dt)/|d f⃗/dt|, tangent to the parametric 3D curve f⃗(r⃗_c(t)), 
may be discontinuous at a finite number of s points along the curve. All these aspects should become clearer in the derivation given below. The above expression indeed reduces to the usual one Q_S = 2 Q for the boundary conditions described in the previous paragraph. Figure <ref> shows the same spin texture as in figure <ref> but with several boundary curves r⃗_c. The small circular ones each enclose the local Skyrmion singularities, yielding Q_S = +1 for the  two of them pointing down and Q_S = -1 for the one pointing up, using (<ref>). The large circle and the square curve enclose the three Skyrmions and yield Q_S = +1, which is the sum of the local ones. We remark that, for the square curve, the last term of (<ref>) must be taken into account since it gives rise to discontinuities on the tangent vector mentioned above.




While we claim that the above expression is a general one for any spin texture, since the derivation that we provide makes no reference to the physical origin of the spin texture, for purposes of illustrations and for a better understanding of the Skyrmion charge, we will use the spin textures arising from  a spinorial BEC in the presence of an external magnetic field, as a case study. In addition, an important and interesting point, is that such a system shows another well known type of topological defects, namely, quantum vortices <cit.>. We will explicitly show the profound relationship between the quantum vortices and the Skyrmions, namely, that their corresponding local charges have exactly the same values. The present stylized model allows for finding that, for the present problem, the external magnetic or gauge field ultimately defines the topological structure of the vortices and the Skyrmions.


The article is organized as follows. In Sections II and III we present a detailed analysis, both in a simplified analytic and in a full numerical fashion, of stationary solutions to the Gross-Pitaevskii equations of a 2D 1/2 spinorial Bose gas in the presence of an external inhomogeneous magnetic field. We show how the quantum vortices and Skyrmions can be predicted from the knowledge of the magnetic field. In Section IV, the main contribution of this article, we provide a general derivation of the topological charge of Skyrmions present in any given spin texture, for arbitrary boundary curves. Section V is devoted to a discussion between the similarities and discrepancies between the Skyrmions topological charge and the Euler-Poincaré characteristic of a surface in terms of the Gauss-Bonet theorem <cit.>. 
In an Appendix we illustrate our main result with a variety of numerical solutions of different spin textures generated by different magnetic fields, and for different types of arbitrary boundary curves. Final remarks are given in Section VI.



§ 1/2 SPINOR BEC IN THE PRESENCE OF AN EXTERNAL MAGNETIC FIELD


We consider stationary states of a two-component spinorial BEC in 2D, with a Zeeman coupling to an external magnetic field, as described by the coupled Gross-Pitaevskii (GP) equations

    μψ_α(r⃗) = [-ħ^2/2m∇^2 + 1/2 mω^2 r⃗^ 2
    + g (|ψ_+|^2 + |ψ_-|^2) ]ψ_α(r⃗) - B⃗(r⃗)·μ⃗^αβψ_β(r⃗) ,

where r⃗ = (x,y) and α = +,- denotes the two components of the spinor. Summing over repeated indices is assumed. μ is the chemical potential and the magnetic dipole operator is given by

    μ⃗^αβ = μ_0 σ⃗^αβ,

where σ⃗= x̂σ_x + ŷσ_y + ẑσ_z  are Pauli matrices and  μ_0 > 0 the magnitude of the magnetic dipole moment. The magnetic field is 

    B⃗(r⃗) = x̂ B_x(r⃗) + ŷ B_y(r⃗) + ẑ B_z .

We denote B⃗_T(r⃗) = B_x(r⃗) x̂ + B_y(r⃗) ŷ as the transverse magnetic field and, for simplicity, we will consider B_z(r⃗) = constant . We choose  B⃗_T(r⃗)  such that it vanishes at certain points r⃗_1, r⃗_2, …, r⃗_s. We shall see that at those points there appear quantum vortices and the origins of Skyrmions <cit.>.  The normalization condition is

    ∫ d^2 r  (|ψ_1|^2 + |ψ_2|^2) = 1 .


Writing the solution as ψ_α = √(ρ_α) e^i Θ_α, the superfluid density ρ_α and  the superfluid velocity v⃗_α, in each component, are given by

    ρ_α   =    |ψ_α|^2 ,
    v⃗_α    =    ħ/m∇Θ_α.


The circulation C of the velocity field in an arbitrary closed circuit C is defined by

    C =  ∮_ Cv⃗_α· dr⃗.

As we will explicitly calculate, if the circuit does not enclose a quantum vortex the circulation vanishes, but when a vortex or vortices are enclosed, the circulation yields the vortex topological charge Q_V within the circuit <cit.>

    C =  2 πħ/m Q_V

with Q_V ∈ℤ, a positive or negative integer. 


The condensate spin texture f⃗(x,y) is an observable, independent of the spin basis, and defined at any spatial point within the condensate,

    f⃗ = 1/ρ_+ + ρ_-(ψ_+^* ψ_-^*) [x̂σ_x + ŷσ_y + ẑσ_z]
    (
    [ ψ_+; ψ_- ]) .

It can be verified that the spin texture has magnitude one f⃗·f⃗ = 1 everywhere.  


Writing f⃗ = x̂ f_x + ŷ f_y + ẑ f_z, the components explicitly are

    f_x    =   ψ_+^* ψ_- + ψ_-^* ψ_+/ρ_+ + ρ_-
       =    2 √(ρ_+ρ_-)/ρ_+ + ρ_-cos(Θ_- - Θ_+) 
    
    f_y    =   - iψ_+^* ψ_- - ψ_-^* ψ_+ /ρ_+ + ρ_-
       =    2√(ρ_+ρ_-)/ρ_+ + ρ_-sin(Θ_- - Θ_+)
    
    f_z    =   ρ_+ - ρ_- /ρ_+ + ρ_-.




§ VORTICES AND SKYRMIONS IN THE VICINITY OF POINTS OF VANISHING TRANSVERSE MAGNETIC FIELD


While we will consider a variety of magnetic fields, all have in common that they have spatial points r⃗_1, r⃗_2, …, r⃗_s where the transverse magnetic field vanishes, B⃗_T (r⃗_i) = 0. This property imprints both the appearance of vortices and of Skyrmions at those points. To understand how this occurs, with no loss of generality, we consider a transverse magnetic field that vanishes at the origin r⃗ = 0 and analyze the behavior of the spinor wavefunctions in its vicinity. For this we explicitly write both GP equations for the two spinor components, assuming that B_z = 0,

    μψ_+    =   [-ħ^2/2m∇^2 + 1/2 mω^2 r^2
    + g ρ]ψ_+ - μ_0 (B_x - i B_y) ψ_- 
    μψ_-    =   [-ħ^2/2m∇^2 + 1/2 mω^2 r^2
    + g ρ]ψ_- - μ_0 (B_x + i B_y) ψ_+ ,

where ρ = |ψ_+|^2 +  |ψ_+|^2 and r= √(x^2+y^2). In the vicinity of the origin the magnetic field generically vanishes as <cit.>

    B_x - i B_y ≈ B_0  r^n  e^i n φ

where B_0 is the field strength, not necessarily real, tanφ = y/x and n a positive integer.  We note that if B_0 is real, the resulting field obeys ∇·B⃗ = 0 and ∇×B⃗ = 0, namely, it would be a true magnetic field. However, one can consider arbitrary “artificial” gauge fields <cit.> that do not necessarily satisfy Maxwell equations but that allow for different sorts of fields and which in turn give rise to different spin textures, as we show in the rest of the paper. In this way, generically, there are four different types of fields in the vicinity of a vanishing location, with B_0 real, positive or negative,

    I)B_x - i B_y    ≈    B_0  r^n  e^i n φ II) B_x - i B_y ≈ i  B_0  r^n  e^i n φ
     III) B_x - i B_y    ≈    B_0  r^n  e^-i n φ IV) B_x - i B_y ≈ - i  B_0  r^n  e^-i n φ.

The first two obey Maxwell equations, the last two do not; as a matter of fact, case (I) with n=1 is commonly produced in a Ioffe-Pritchard trap, and n=2 is a quadrupole trap <cit.>.  Figure (<ref>) shows the structure of the magnetic fields for the case n = 1. Case (III) is the hedgehog configuration usually considered in Skyrmion discussions <cit.>. 

As we will see, the local structure of the transverse magnetic field at its vanishing points plays a fundamental role in the final signs of the topological charges of the vortices and Skyrmions. As discussed in Section V below, such a structure can be associated to a topological quantity, namely, the Index of the field at the vanishing point. <cit.>


Let us consider case (I) for the moment. We will summarize all cases below.  For this case there are two possible solutions,

    i)ψ_+(r⃗)    =    F_+(r) e^-inφ andψ_-(r⃗) = F_-(r)  
     ii)ψ_+(r⃗)    =    F_ -(r)   andψ_-(r⃗) = F_+(r) e^inφ.

Substitution of these solution into GP equations (<ref>) shows that, very near r = 0, one obtains

    F_+(r)    ≈    A  r^n 
    
    F_-(r)    ≈   μ/g,

where A is a positive constant that can be numerically determined. The important features are that F_+(r) vanishes as r^n while  F_-(r) yields a constant as r → 0. Therefore, case (i) in (<ref>) is a vortex in the ψ_+ component and a density “spike” in the ψ_- one (see below), and the opposite in (ii), namely a spike in ψ_+ and a vortex in ψ_-. As we see now, in the former case the vortex charge is Q_V = - n, while in the latter Q_V = + n. This can be concluded right away from the velocity fields,

    i)v⃗_+(r⃗)    =    - n ħ/mφ̂/r andv⃗_-(r⃗) = 0  
     ii)v⃗_+(r⃗)    =    0   andv⃗_-(r⃗) = + n ħ/mφ̂/r,

where φ̂= - x̂sinφ + ŷcosφ. Indeed, the velocity fields circulate in the clockwise (negative) direction in the ψ_+ component for case (i), and in the anticlockwise (positive) direction for the ψ_- one, in case (ii). The circulations are, using a circular contour of radius r and dr⃗ = φ̂  r dφ,

    C_±   =   ∮_ Cv⃗_±· dr⃗
        =    ∓  n ħ/m∮ dφ
        =    ∓ 2π nħ/m,

finding the values of the topological charges already quoted. Clearly, if the vortex is not enclosed by the contour, the circulation vanishes, C_± = 0. Figure <ref> shows the solution to the GP equations (<ref>), as well as the velocity fields, corresponding to figures (<ref>) and (<ref>), including the magnetic field that generates such a configuration; see the Appendix for details of the calculation. One can observe that, in the vicinity of the vortices, the top one is of type (II) while the lower ones are of type (I) in (<ref>). In panels (A) and (C) of <ref> we can also see what we mean by a “density spike”: while the vortex in one component occurs, accompanied by a density depletion, in the other component there is a rapid increase of the density at the same spatial location, namely a spike, identified in the figure by a bright spot in the density profile. This increment of the density in one spinor component almost cancels the depletion at the vortex in the other component <cit.>.  
From expressions (<ref>) of the fields, the vortex structure of case (II) is the same as case (I) just discussed, however in cases (III) and (IV) the solution is as in (I) given by (<ref>), but with the velocity circulation reversed and, therefore, the charges are now Q_V = +n for solution (i) and Q_V = -n for (ii). As we will see below, the Skyrmion charges will have exactly the same values as Q_V, with their own spin texture structure.




We now turn our attention to the spin texture and, for the sake of argument, we also study case (I). Again, in the vicinity of zero field, one finds the spin texture from (<ref>), for both solutions (i) and (ii) in (<ref>)

    f_x    =   F_+ F_-/F_+^2 + F_-^2cos n φ
    
    f_y    =    -  F_+ F_-/F_+^2 + F_-^2sin n φ
    
    f_z     =    ±F_+^2 - F_-^2/F_+^2 + F_-^2.

where the ± signs in front of f_z refer to solutions (i) and (ii), respectively, see (<ref>). First, we note that the transverse spin texture f⃗_T = f_x x̂ + f_y ŷ  is parallel and in the same direction to the transverse magnetic field B⃗_T, as physically expected, see (I) in (<ref>), thus showing that both have the same spatial pattern. This is true in general and it is exemplified in the gallery of figures in the Appendix.
Evidently, as r → 0, the components f_x and f_y vanish since either F_+ or F_- vanish as well, and the spin texture becomes parallel to the ẑ vector, pointing upwards or downwards, f⃗ = ±ẑ. This latter direction is determined by whether the solution is (i) or (ii) as above in (<ref>): for solution (i) |ψ_+| = F_+(0) = 0, |ψ_-| = F_-(0)  0 and, therefore, the spin texture points downwards, f⃗(0) = - ẑ; for solution (ii) |ψ_-| = F_+(0) = 0, |ψ_+| = F_-(0)  0 and the spin texture now points upwards, f⃗(0) = + ẑ. For the other fields (II), (III) and (IV) the same rule applies for the peaks: if the vortex appears in the ψ_+ component, the texture points downwards, but if the vortex is in ψ_- then it points upwards. The ensuing structure of the spin texture is what we call the Skyrmion and, for the sake of describing it, we shall call the location r⃗_i where f⃗(r⃗_i) = ±ẑ, as the origin of the Skyrmion. 





§ THE TOPOLOGICAL CHARGE OF THE SKYRMIONS


As mentioned in the Introduction, the topological charge of the Skyrmions is usually given by (<ref>) but, as also argued, such an expression requires particular boundary conditions on the spin texture f⃗. As also already mentioned, in general those boundary conditions need not occur, or a spin texture can have a multiplicity of coexisting local Skyrmions, whose boundaries among them can also be quite arbitrary. Moreover, in all cases one should be able to asses the individual charges of the local Skyrmions in order to validate that the charges add as any topological index. The spin textures here presented are clear examples of the just two described situations. There is another fundamental aspect to be considered as well. In descriptions of common spin textures and their charge given by (<ref>), it is affirmed that the topological charge is the number of times that the unit sphere is covered by the spin texture. This follows, on the one hand,  by assuming that f⃗(x,y), being a unit vector, is by itself a parametrization or mapping of the unit sphere and, on the other, again, by the mentioned usual boundary conditions. While such a parametrization can indeed be assumed, there is subtle aspect that we believe has not been addressed before, namely, that different spin textures give rise to mappings to the unit sphere but with different orientations. That is, the normal vector to the sphere, generated by the parametrization f⃗(x,y), can point outwards or inwards to the sphere, depending on the given spin texture. And, for configurations with several Skyrmions, this can occur within the same spin texture. This is a very interesting and important point to consider in the elucidation of a general expression for the identification of the Skyrmion topological charge. In this section we present a derivation of the Skyrmion topological charge, in an as possible straightforward fashion, by a procedure similar to the derivation of the Gauss-Bonet (GB) theorem <cit.>. We will  leave the discussion of the mentioned delicate issues to the following section, where we also argue about the similarities or, rather, dissimilarities of the Skyrmion charge with the Euler-Poincaré characteristic χ, arising from the GB theorem.


Let us assume that a spin texture f⃗(x,y) is given in all or part of the XY-plane, such that it has a single location r⃗_0 where either f⃗(r⃗_0) = +z or f⃗(r⃗_0) = -z. We call this point the origin of the Skyrmion. Consider now an area A that encloses this point. The area 
has a boundary ∂ A described by a parametrized closed curve

    r⃗_c(t) = x̂ x_c(t) + ŷ y_c(t) t : 0 → T r⃗_c(0) = r⃗_c(T) .

 Notice that although the curve is continuous, its tangent may be discontinuous at a finite number of points r⃗_c(t_1), r⃗_c(t_2), …, r⃗_c(t_s). Further, and very important,  the curve is assumed, one, to run or be oriented in the positive counterclockwise direction and, two,  the curve does not cross itself, say the red curve in figure (<ref>).
 


Evidently, since the texture f⃗(r⃗) is a unit vector, it can be considered to be the outward normal vector r̂(θ,ϕ)  to the unit sphere S^2, with (θ,ϕ) the usual azimuthal and polar sphere angles, (please note that we use ϕ for the polar angle in 3D and φ in 2D)

    cosθ   =    f_z 
    tanϕ    =    f_y/f_x.

At the same time one could assume that f⃗(r⃗) is by itself a parametrization of the unit sphere. As discussed in detail below why, we will not consider this point of view. As we show below, perhaps surprisingly at  first sight, the two mentioned assumptions are not necessarily consistent with each other. Nevertheless, any value of the spin texture f⃗(r⃗) can be identified with a single point (θ,ϕ) on the sphere, although the opposite is not true in general.  That is, depending on the order of the Skyrmion, parts of the sphere can be covered more than once. Figure (<ref>) shows a simple example of these observations, where we can see how the spin texture within and on the square are mapped to unique points on the sphere. And, with the identification of the unit normal and the spherical angles, the usual orthogonal sphere unit vectors (r̂, θ̂, ϕ̂) along the directions (r,θ,ϕ), can also be expressed in terms of the spin texture,

    r̂    =    x̂ f_x + ŷ f_y + ẑ f_z 
    θ̂    =    x̂f_z f_x/√(f_x^2+f_y^2)+ ŷf_z f_y/√(f_x^2+f_y^2) - ẑ√(f_x^2+f_y^2)
    ϕ̂    =     -x̂f_y/√(f_x^2+f_y^2)+ ŷf_x/√(f_x^2+f_y^2).

 

In an analogous way, we can observe that the curve r⃗_c(t) on the plane gets mapped to a 3D curve f⃗(r⃗_c(t)), denoted as C, that lies on the surface of the unit sphere. The arclength of such a curve can be found from

    ds = |df⃗(r⃗_c(t))/dt| dt ,

and the tangent t⃗(s) to the curve is given by, see again figure (<ref>) as an example, 

    t⃗(s) = df⃗(s)/ds.

This is a unit vector, t⃗·t⃗ = 1. Since it is also tangent to the sphere, then t⃗·r̂ = 0 and, therefore, t⃗ lies on the θ̂-ϕ̂ plane. We define  α as the positive angle that t̂ makes with θ̂ and, thus, we can write

    t⃗ = θ̂cosα + ϕ̂sinα.

α can take any positive value, see figure <ref>. Note again that for every texture vector f⃗ on the curve C there is an angle α = α(f⃗), but the inverse is not necessarily true.


With the above preliminaries we can now start the derivation of the expression for the topological charge Q_S of a single Skyrmion. Consider the following integral along the curve C,

    I = ∮_ Cf⃗(s) ·(t⃗(s) ×dt⃗(s)/ds) ds.

Before we proceed with the evaluation of the integral, we note two points. First, while the integrand may seem to be the geodesic curvature of C on the sphere, we will see that it is not in general, differing by a minus sign when it is not. And second, although the curve r⃗_c(t) on the plane has been considered to run in the positive counterclockwise direction in the XY-plane, the curve C on the sphere, in general, is not necessarily oriented in the same direction. These apparent discrepancies, to be discussed in the Section V, have their origin in the previous observation that f⃗ is not being formally considered to be a parametrization of the sphere.


 Substituting the tangent vector (<ref>) into the integral (<ref>) yields,

    ∮_ Cf⃗(s) ·(t⃗(s) ×dt⃗(s)/ds) ds = ∮_ C dα + ∮_ Ccosθ∇ϕ· ds⃗.

Let us analyze the first integral over the α angle. Here it is important to notice that, because of the assumptions on the boundary curve r⃗_c(t), the tangent t⃗ to the curve C is also discontinuous at the points (f⃗(r⃗_1), f⃗(r⃗_2), ⋯, f⃗(r⃗_s)). This, in turn, implies that the angle α is discontinuous at those  points: Let t⃗_i^ - be the incoming tangent of the curve at the point f⃗(r⃗_i) and  t⃗_i^ + the outgoing one. We call the corresponding angles α_i^- and  α_i^+. Hence, 

    ∮_ C dα   =   ∫_α(0)^α_1^- dα + ∫_α_1^+^α_2^- dα + ⋯ + ∫_α_s^+^α(T) dα
        =    α(T) - α(0) - ∑_i=1^s ε_i,

where α(0) = α(f⃗(0)) and α(T) = α(f⃗(T)) with f⃗(0) = f⃗(T). We have assumed that such a point is not on any discontinuity. ε_i = (α_i^+ - α_i^-) is the angle between the tangents t⃗_i^ + and t⃗_i^ - at the discontinuity, given by

    sinε_i = f⃗_i ·(t⃗_i^ -×t⃗_i^ +) ,

see figure <ref>. 
Below we return to the analysis of the values of α(T) - α(0). 


The second integral in (<ref>), over the angle ϕ, can be rewritten in a way that allows for the determination of another contribution of the Skyrmion charge, using Green theorem. For this, we first recall that the origin r⃗_0 of the Skyrmion occurs where the spin texture becomes f⃗(r⃗_0) = ±ẑ, that is, where the azimuthal angle equals θ = 0 or π, respectively. Since the contour C encloses either of those points, the integrand of the second integral in (<ref>) is not analytic at θ = 0 or at π, preventing the direct use of Green theorem. However, adding and subtracting an appropriate quantity one obtains 

    ∮_ Ccosθ∇ϕ· ds⃗ = - ∮_ C(± 1 - cosθ) ∇ϕ· ds⃗±∮_ C∇ϕ· ds⃗,

where the + sign is used when the singularity is at θ = 0, namely f⃗(r⃗_0) = + ẑ, and the - sign when θ = π or f⃗(r⃗_0) = - ẑ. In this way, the integrand of the first integral on the right side of (<ref>) is analytic at the respective points and the integral can be performed using Green Theorem. This yields, up to a sign, the solid angle subtended by the contour C around the North or the South Pole,

    ∮_ C(± 1 - cosθ) ∇ϕ· ds⃗  = ±∬_ C dΩ.

The sign on the right side depends on the positive or negative orientation of the contour C, which in turn depends on the structure of the Skyrmion in the vicinity of the singularity. The above term can also be seen as a Berry phase type contribution <cit.>. However, the previous integral can be written in a manner that is more appropriate for our purposes, namely, by rewriting it in terms of the spin texture itself:

    ∮_ C(± 1 - cosθ) ∇ϕ· ds⃗    =    ∮_r⃗_c(± 1 - f_z )  ∇(arctanf_y/f_x) · d r⃗
        =    ∬_A{∂/∂ x[ f_z ∂arctanf_y/fx/∂ y] -
    ∂/∂ y[ f_z ∂arctanf_y/fx/∂ x] } dxdy 
        =    ∬_A f⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy .

The first equality follows by writing the angles θ and ϕ in terms of the f⃗ components, see (<ref>), hence allowing for expressing the line integral along the original contour r⃗_c(t). The second equality is obtained by means of Green Theorem, the surface integral being performed over the whole area A enclosed by the curve r⃗_c(t). The derivation of the last equality is a laborious exercise that requires the use of f⃗·f⃗ = 1. Note that in the expression of the last line one does not need to worry about the sign of the integral, as this will emerge naturally from the current spin texture.  The important issue of these signs will be further addressed below. 


Collecting the previous results (<ref>), (<ref>) and (<ref>) into (<ref>), we partially obtain

    α(T) - α(0) ±∮_ C∇ϕ· ds⃗   =   ∬_A f⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy + ∮_ Cf⃗(s) ·(t⃗(s) ×dt⃗(s)/ds)ds 
           + ∑_i=1^N arcsinf⃗_i ·(t⃗_i^ +×t⃗_i^ -)  .

First of all, it is simple to conclude that the left-hand-side of this expression must be 

    2π Q_S = α(T) - α(0) +  ∮_ C∇ϕ· ds⃗

 with Q_S ∈ℤ. This integer will be carefully  identified below as the topological charge of the Skyrmion enclosed by the curve r⃗_c. Quite generally, it must be an integer since, first, f⃗(0) = f⃗(T), then sinα(f⃗(0)) = sinα(f⃗(T)) and, hence, α(T) - α(0) = 2 π N with N an integer. Similarly, since

    ±∮_ C∇ϕ· ds⃗   =   ±∮_ C d ϕ
        =     2 π M,

with M a positive or negative integer, since the contour C encloses either the North or the South Pole of the unit sphere. Therefore, Q_S =  N +  M is an integer. However, it remains to understand how this integer number relates to the given spin texture f⃗(x,y) and to its behavior in the neighborhood of the singularity f⃗(r⃗_0) = ±ẑ. 



Let us address the role of the angles α and ϕ in the determination of the integer value of the Skyrmion charge Q_S in (<ref>). First, one can conclude that α(0) = α(T), namely N = 0. 
This follows from expression (<ref>) for α and from the geometrical facts, one, that 
θ̂ always points in the direction of increasing θ, two, that the tangent vector t⃗ has a definite orientation, counter or clockwise for any given curve C and, three, that either the North or South Pole are enclosed by C.
These three facts imply that the angle α retraces itself around C without completing a single turn. Thus, this leaves the value of Q_S to depend solely on the integral of ϕ in (<ref>) and (<ref>). This has an intrinsic topological solution. For this, we write the integral over ϕ in terms of an integral of the spin texture over the original contour r⃗_c(t),

    ±∮_ C_i dϕ = ±∫_r⃗_c^(i)∇(arctanf_y/f_x) · d r⃗_c^(i).

The interesting and relevant observation if that the integral on the right-hand-side is precisely the Index I(f⃗_T(r⃗_0)) of the transverse spin texture field f⃗_T(r⃗), times 2π, around r⃗_0 <cit.>. Such an Index is I = ± n with n a positive integer that counts the number of turns the angle ϕ does around the corresponding Pole. 
Based on our findings that the  transverse spin texture f⃗_T(r⃗) follows the transverse magnetic field B⃗_T(r⃗), we can extrapolate and assert that any spin texture, in the vicinity of the singularities f⃗_T(r⃗_0) = 0, behaves just as the magnetic field given in (<ref>). Hence, for cases (I) and (II) the index is I = -n while for (III) and (IV) one obtains I = +n. Therefore, in a very general way, the topological charge of a single Skyrmion can be expressed as,

    Q_S = f_z(r⃗_0)  I(f⃗_T(r⃗_0))  ,

demonstrating that the topological charge is a positive or a negative integer, its sign being determined by both, the value of f_z and the Index of f⃗_T, at the singularity. We believe that this last identification of the Skyrmion charge has not been pointed out before. Figure <ref> shows two cases of  Skyrmions with singularities at f⃗ = +ẑ and f⃗ = +ẑ, produced by the same transverse magnetic field but with opposite B_z component; all the thermodynamic properties are the same, except that the Skymion charges have opposite values.




Returning to the identification of  Q_S in (<ref>), and rewriting the integrals in terms of the spin texture f⃗ and the original boundary curve r⃗_c, one arrives at the sought for expression for the topological Skyrmion charge given in (<ref>) in the Introduction, but for a single Skyrmion. The extension to the total topological charge of a spin texture that holds an arbitrary number N of Skyrmions, whose origins are at r⃗_1, r⃗_2, …, r⃗_N, is straightforward. This is achieved with the 
reasonable assumption that, while in the vicinity of the origins the spin texture behaves as in the case of a single Skyrmion, the “local” textures can be  joined smoothly in a continuous and differentiable fashion. This assumption is verified by the numerical examples shown in the Appendix.
Very importantly, the boundaries between those local cases can be quite arbitrary and complicated, such as the boundary of our example of the three Skyrmions in figure (<ref>). This corroborates the importance of having obtained the local charge of a Skyrmion for an arbitrary boundary curve r⃗_c. Figure <ref> shows a 
sketch of the union of local boundary curves r⃗_c^(i) of several Skyrmions that yields a total curve r⃗_c, inside of which all Skyrmions are enclosed. Writing the different contributions of a local Skyrmion charge as Q_S^(i) = Q_2^(i) + Q_1^(i) +Q_0^(i), where

    Q_2^(i)   =   ∬_A_if⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy 
    
    Q_1^(i)    =    ∮_r⃗_c^(i)f⃗·( d f⃗/dt×d^2f⃗/dt^2) /|d f⃗/dt|^2 dt  
    
    Q_0^(i)    =    ∑_j=1^s^(i)sin^-1f⃗_j ·( . d f⃗/dt|_j^ -×. d f⃗/dt|_j^ +)/| d f⃗/dt|_j^ -| d f⃗/dt|_j^ +,

it is evident, since all boundary curves r⃗_c^(i) run with same positive orientation, that 

    Q_S = ∑_i=1^N Q_S^(i)

with an analogous expression for all contributions Q_2, Q_1 and Q_0. The obtained expression for the total Skyrmion charge Q_S = Q_2 + Q_1 + Q_0 is given by (<ref>) of the Introduction, with r⃗_c = ⋃_i=1^N r⃗_c^(i) the external boundary curve, A =  ⋃_i=1^N A_i, the total area, and the sum over the angles is only on those on the external boundary r⃗_c since all the internal angles cancel out, see figure <ref>. It is also of interest to conclude that the total Skyrmion charge can be written as,

    Q_S = ∑_i=1^N  f_z(r⃗_i)  I(f⃗_T(r⃗_i))  .

As we will argue in the following section, this last expression appears as a generalization of Poincaré theorem <cit.>. 



In the Appendix we illustrate the results of this section with explicit numerical calculations of several spin textures obtained by solving GP equations (<ref>) for our case study of a spinor BEC in the presence of an external field. 




§ DISSIMILARITIES BETWEEN THE SKYRMION CHARGE Q_S AND THE EULER-POINCARÉ CHARACTERISTIC Χ



The expression (<ref>) for the Skyrmion topological charge Q_S has a deceivingly similar appearance to the Euler-Poincaré characteristic χ of the Gauss-Bonet theorem of a given 2D surface embedded in 3D <cit.>. As a matter of fact, the derivation here presented for Q_S is similar, but not equal, to the derivation of the Gauss-Bonet theorem presented by do Carmo <cit.>. However, as we have been alluding to in the text, there are notable differences between them: A technical difference being that the spin texture vector f⃗ cannot always be both the normal vector and the parametrization to the sphere at the same time. A fundamental difference being that in the Gauss-Bonet theorem the main role is played by the surface itself, while in the Skyrmion case the protagonist is the spin texture f⃗. 


To be precise, given an orientable surface S, the GB theorem states that the Euler-Poincaré characteristic χ of a region R of the surface S, with a given closed boundary C and with M possible exterior angles θ_i is, <cit.>

    2πχ = ∬_R K  dA + ∮_C k_g ds + ∑_i=1^M θ_i

where K is the gaussian curvature at a point on the surface, k_g the geodesic curvature at a point of the curve C, the boundary of R. The curve C must be oriented in the positive direction. These are intrinsic properties of the surface and of curves drawn on it. If the region R is a simple region of the surface S, then χ = +1 always.  If the surface is compact with no boundaries, then χ = 2 - 2g where g is the genus or number of handles of the surface. And there are surfaces with χ = -1 such as the pseudo sphere and the so-called pair of pants. The above expression for χ is written in an invariant form, independent of any parametrization of the surface. For sure, one can write it in terms of a parametrization, but this one must be consistent with an assumed orientation of the surface and with the positive orientation of the curve C. To see this, let X⃗(σ_1,σ_2) be such a parametrization, then, the normal to the surface is

    n̂ = ∂_1 X⃗×∂_2 X⃗/|∂_1 X⃗×∂_2 X⃗|,

where ∂_α is the derivative with respect to σ_α. This normal must have the same orientation on the whole surface. For instance, if the surface is a sphere, then n̂ must point outwards or inwards over the whole sphere. The gaussian curvature can be expressed as,

    K = n̂·(∂_1 n̂×∂_2 n̂) .

Along the same lines, the tangent to the curve C is

    t⃗ = ∂X⃗(s)/∂ s,

and, hence, the geodesic curvature of C is

    k_g = n̂·(t⃗×dt⃗/ds) .

The exterior angles θ_i are, in an obvious notation,

    sinθ_i = n̂·( t⃗_i^-×t⃗_i^+) .


Therefore, if we compare χ, from (<ref>)-(<ref>), with Q_S from (<ref>), they look quite similar if we identify the parametrization X⃗ with f⃗, the surface thus being a sphere, and the normal n̂ with f⃗. However, these two identifications are incompatible in general. This can be readily seen if we calculate the normal to the sphere from the parametrization X⃗ = f⃗, 

    n̂ = ∂_x f⃗×∂_y f⃗/|∂_x f⃗×∂_y f⃗|.

One finds that, depending on the spin texture, n̂ = f⃗ or n̂ = -f⃗. And this can occur within the same spin texture, such as in the example of figure (<ref>). Therefore, if f⃗ is a parametrization, then f⃗ cannot be the normal always, and viceversa. This would result in severe problems since the normal would discontinuously change on the sphere. Hence, we have opted to consider 
that f⃗ is the outward normal to the sphere, at the expense of abandoning f⃗  as a parametrization. It is then not a surprise that we do not obtain GB theorem and, therefore, Q_S is not χ, since the sign of the former depends on the spin texture f⃗ and the latter is a topological intrinsic property of the given surface. Indeed, the fact that we can find positive or negative Skyrmion charges, simply by inverting the value of the origin of the Skyrmion f⃗(r⃗_0) is a consequence of not having considered f⃗ as a parametrization of the sphere. In this way, the integrands of Q_2 and Q_1 are not necessarily the gaussian or the geodesic curvatures of the sphere or of the curves drawn on it.
It is also of interest to point out that there is a theorem, due to Poincaré <cit.>, that states that χ = ∑_i I_i with I_i the Indices of a differentiable vector field on a compact surface with isolated singular points. That is, given the surface, the singularities must exist and their Indices add to χ. In the present case, see (<ref>), the singularities can be quite arbitrarily given by the the spin texture and, as a consequence, their topological charges depend on an additional sign given by f_z(r⃗_0) at the singularity. 


The previous observations are not definite, however, as we are aware that topology can yield results that may be difficult to visualize geometrically. That is, although we have pointed out evident differences between the Skyrmion topological charge and the Euler-Poincaré characteristic, we also concede that their respective expressions appear strikingly similar. Hence, we leave open the question of whether they may truly be the same topological object or not, perhaps for not so obvious or evident surfaces such as the unit sphere. This should be the subject of further scrutiny. Along the same lines, an approach in terms of homotopy theory <cit.> should complement our approach, basically rooted in differential geometry.




§ FINAL REMARKS


In this article we have provided a derivation of the Skyrmion topological charge Q_S of an arbitrary spatial region A ⊂ℝ^2 of a spin texture f⃗(r⃗), given by (<ref>). If the spin texture is continuous and differentiable everywhere, except perhaps at the origins of the Skyrmions, where f⃗ = ±ẑ, one can find the local topological charges of the present Skyrmions and, simply by joining the local regions, one can calculate the topological charge of several Skyrmion regions and, eventually, of the whole area where the spin texture is defined. While we can calculate the topological charge as the sum of the three main contributions Q_S = Q_2 + Q_1 + Q_0, one can also demonstrate that the local Skyrmion charge is the product Q_S = f_z(r⃗_0) I(f⃗_T(r⃗_0)), with I the Index of the transverse spin texture at the singularity r⃗_0. The total Skyrmion charge of a region, or of the whole spin texture, is the addition of the local Skyrmion charges. It may be of formal interest to highlight that the 2D contribution Q_2, the 1D Q_1 and the 0D Q_0 are all expressed in terms of a triple scalar product involving the spin texture f⃗ times the relevant cross product of the contribution at hand. We also find that, depending on the value of the spin texture at the boundary curve or curves, one recovers the usual expression for the Skyrmion charge, given solely in terms of the contribution Q_2. For instance, if  the texture completely covers the sphere, f⃗ = ±ẑ, or f_z = 0 at the boundary, the last two terms in (<ref>) or (<ref>) vanish. Notice also that we have defined the Skyrmion charge with a factor 1/2π in (<ref>), instead of 1/4π as in (<ref>), because of the similarities  with the Gauss-Bonet representation of the Euler-Poincare characteristic. Thus, while usually the Skyrmion charge for a closed sphere is Q = ± 1, here it is Q_S = ± 2, and as the charge of a half Skyrmion is considered Q = ± 1/2, here it is a Q_S = ± 1. As we indicated in the Introduction, the derived expression is quite general and independent of the origin of the spin texture.


In order to produce spin texture within a physically sensible system, we have studied a 1/2 spinor BE condensate in the presence of an external magnetic field. This system, in addition to generate interesting and non trivial spin textures, with which we can make a detailed and systematic verification of our main result, presents quantum vortices, defects of topological nature as well. As we have indicated, one can trace the appearance of the vortices and of the Skyrmions to the behavior of the external magnetic field in the vicinity of the locations r⃗_0 where the transverse magnetic field vanishes. Indeed, by looking a the emergence of quantum vortices, say, in (<ref>), one can conclude that the vortex local topological charge is Q_V = ± I(B⃗_T(r⃗_0)), where the sign ± is if the vortex appears in the ψ_+ or ψ_- spinor component, respectively, and I(B⃗_T(r⃗_0)) is the Index of the transverse magnetic field in the vicinity of its vanishing location r⃗_0. However, since the transverse spin texture f⃗_T follows exactly the same pattern as the transverse magnetic field B⃗_T, see Appendix, then the local quantum vortex and Skyrmion topological charges are exactly the same. That is, for the present case study, the external magnetic field determines the topology of all ensuing structures. These results can definitely be tested with the currently achieved experimental capabilities in the study of spinor Bose-Einstein condensates. For other physical spin textures one can perhaps always find the underlying field or interaction responsible for the emergence of the corresponding topological structures. 


While we have left out several additional observations of this problem, related to its intrinsic differential geometry and topology, we mention one that should be of importance for further analysis. This regards the issue of what we call a “Skyrmion” structure in a spin texture. That is, the emphasis is always in the role of the location where one finds the origin of the Skyrmion f_z(r⃗_0) = ±ẑ and, therefore, one may believe that the determination of the topological charge needs of a boundary curve that encloses such a point or points. However, this is not completely true or necessary. For this we point out expression (<ref>) where we identified the Skyrmion charge in terms of the angle difference α(T)-α(0) and the contour integral over ϕ. We recall that we derived our results by assuming a contour that enclosed the singularity yielding  α(T)-α(0) = 0 and the topological charge was contained in the ϕ contour integral. However, if we choose a contour that does not enclose the singularity, then the Index of the transverse spin texture vanishes, but now α(T)-α(0)  0. And, for sure, the topological charge is now accounted for by the angle α, namely 2π Q_S = α(T)-α(0). This indicates that if we take a fully arbitrary boundary curve r⃗_c, yet closed and positively oriented, that does not necessarily enclose singularities, one still finds the topological charge contained within the encircled spin texture. In other words, the topological content of the Skyrmion is within the whole spin texture, as Topology may have indicated us in advance.


Acknowledegment. This work was partially funded by grant PAPIIT-UNAM IN117623. VRR acknowledges fruitful discussions with Leonardo Patiño at the early stages of this work.




§ APPENDIX: A GALLERY OF VORTICES AND SKYRMIONS, SOLUTIONS TO GP EQUATIONS (<REF>).


The figures throughout the text and in this Appendix are found from numerical solutions of the two coupled Gross-Pitaevskii equations (<ref>). We use a standard method described in Refs. <cit.>, in a GPU processor, within a 256 × 256 grid. We consider arbitrary dimensionless units such that the solutions are numerically stationary and the desired configuration is clearly observed. We use ħ = m = 1, the interatomic coupling g = 15000 and, in all cases, the computational box is a square of side L_0 = 56.72. The values of the isotropic frequency trap ω, the magnetic fields and the corresponding chemical potential μ are given in the figures captions. 



 Recalling that the full magnetic field is B⃗ = B⃗_T + B_z ẑ, an interesting point to highlight is that both solutions (i) and (ii) in (<ref>) are possible in each position where the transverse magnetic field vanishes. However, if B_z = 0, we find numerically that these solutions appear at random, each 50% of the time, because our numerical methods requires a random guess of the initial wavefunctions <cit.>. However, if we turn on the constant B_z field, we find, as expected, that the sign of this field component forces one of the solutions, namely, if B_z > 0, then solution (i) occurs, while (ii) ensues for B_z < 0. 



The solutions of figures <ref> and <ref>, as well as those below <ref> and <ref>, have been obtained with transverse magnetic fields that become zero at appropriate locations on the XY-plane within the BE condensate. These fields are in turn tailored by superposing magnetic fields of a set of long (infinite) wires at appropriate locations, carrying also an appropriate current I. We recall that the field of a long straight wire in the ẑ-direction, located at  (x_0,y_0), with current I, is

    B⃗_ wire(r⃗) = 2I/c-(y-y_0) x̂ + (x-x_0) ŷ/(x-x_0)^2 + (y-y_0)^2.

In the figures captions we indicate the values of the currents and the locations of the wires. The wires are placed outside of the numerical calculation box.


In the following gallery of figures we show the condensate superfluid density ρ_α and velocity v⃗_α, for component α = +, panels (A) and (B); and for component α = -, panels (C) and (D), respectively. In panel (E), we show the Skyrmion spin texture and in panel (F) the transverse spin texture field lines f⃗_T = f_x x̂ + f_yŷ. In panel (G) we show the transverse magnetic field lines B⃗_T =  B_x x̂ +  B_y x̂ that produces the corresponding condensate. Note that the transverse spin texture streamlines follow those of the transverse magnetic field. Details of the corresponding cases are given in the figures captions. 













§ REFERENCES





 Paper Title: Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability.



Grothendieck Shenanigans]Grothendieck Shenanigans:

Permutons from pipe dreams via integrable probability





[
    
	A. H. Morales,
	G. Panova,
	L. Petrov,
	D. Yeliussizov
    
============================================================



§ ABSTRACT 
	We study random permutations arising from reduced pipe dreams. Our
	main model is motivated by  Grothendieck
	polynomials with parameter β=1 arising in
	K-theory of the flag variety. The probability weight of a
	permutation is proportional to the principal specialization
	(setting all variables to 1)
	of the
	corresponding
	Grothendieck polynomial. By mapping
	this random permutation to a version of TASEP (Totally
	Asymmetric Simple Exclusion Process), we describe the
	limiting permuton and fluctuations around it as the order
	n of the permutation grows to infinity. The fluctuations
	are of order n^1/3 and have the Tracy–Widom GUE distribution,
	which places this algebraic (K-theoretic) model
	into the Kardar–Parisi–Zhang universality class.

	We also investigate non-reduced pipe dreams and make
	progress on a recent open problem on the asymptotic number
	of inversions of the resulting permutation. Inspired by
	Stanley's question for the maximal value of principal
	specializations of Schubert polynomials, we resolve the
	analogous question for β=1 Grothendieck polynomials,
	and provide bounds for general β.




§ INTRODUCTION





 §.§ Overview



Algebraic Combinatorics established itself as a field that uses combinatorial methods to understand algebraic behavior in problems ranging from Group Theory to Algebraic Geometry. It started with stark exact formulas, like the celebrated hook-length formula for the dimension of irreducible modules of the symmetric group S_n; beautiful interpretations, like the Littlewood–Richardson rule for the structure constants of representations of the general linear group GL_n; powerful and intricate bijections, like the Robinson–Schensted–Knuth correspondence. However, exact answers only go so far, leaving room for questions like  “approximately how many”, “what are the typical objects”,  and “what is the typical behavior”. These questions lead us into the realm of Asymptotic Algebraic Combinatorics, which aims to answer them with the help of tools originating outside of Combinatorics. In the present work, we employ Integrable Probability, a rapidly evolving field focused on developing and analyzing interacting particle systems and random growth models possessing a certain degree of structure or symmetry.

Many significant questions concerning exact formulas and combinatorial interpretations in cohomology and K-theory of the flag variety remain open. Understanding their asymptotic behavior is thus even more natural. Stanley <cit.> asked the most basic question on the  principal specializations of Schubert polynomials 𝔖_w
(representing cohomology classes of the flag variety): does the following limit exist

    lim_n→∞1/n^2log_2 max_w ∈ S_n𝔖_w(
    	1,…,1 _n),

and if so, what is it and for which permutations w is this achieved.
This question (including the existence of the limit) is still open.
In <cit.>, a lower bound of about 0.29 was established for layered permutations. An upper bound of about 0.37 comes from a remarkable connection with Alternating Sign Matrices and the six-vertex model (see <Ref>).

A natural one-parameter generalization of Schubert
polynomials are the Grothendieck polynomials
𝔊_w^β, which represent K-theoretic
classes of the flag variety. Extending Stanley's questions, we
would like to understand the asymptotic behavior of maximal
 principal specializations max_w∈ S_n𝔊_w^β(1^n)
of Grothendieck polynomials. This
question was first touched on in
<cit.>, <cit.>. Establishing a family of
such maximal permutations would shed light on the
Schubert questions as β→ 0.

Thanks to a combinatorial model for Grothendieck and
Schubert polynomials, these questions have very natural
statistical mechanics interpretations.
Namely, both polynomials are partition functions of tilings
into crosses and elbows of a size n triangle (staircase) shape,
which result in a configuration of n “pipes”. Such pipe configurations
are often called pipe dreams.
In the Schubert case, the only valid
tilings are the ones where no two pipes cross more than
once. This is a global (long-range interaction) condition,
which makes the model much less tractable.
In the Grothendieck case
for β=1, all tilings are allowed, but the pipes must be
resolved (reduced) to obtain a permutation. This model can be mapped to a
colored stochastic six-vertex model
(and furter to TASEP, the Totally Asymmetric Simple Exclusion Process).
These interacting particle systems have only local (short-range) interactions, and are
amenable to techniques from Integrable Probability.



In this work,
we investigate the asymptotics of
β=1 Grothendieck random permutations
and characterize their limit shape which is described by a permuton.
In principle, the typical
permutation (sampled from the limiting permuton) should give the
asymptotically maximal principal specialization
𝔊_w^β=1(1^n), but we do not address this question here.
We also study fluctuations of Grothendieck random permutations around the limiting permuton.
They
are of order n^1/3 and asymptotically have the Tracy–Widom GUE distribution.
This distribution was
first observed in the fluctuations of
the largest eigenvalue of Gaussian
random matrices
with unitary symmetry. By now, having Tracy–Widom fluctuations is an indication that a
model is within the
Kardar–Parisi–Zhang (KPZ) universality class <cit.>, which includes a wide range of
random growth models and interacting particle systems.



We also study the natural, yet not as algebraically
motivated, model, where the staircase is tiled with crosses
with probability p and elbows with probability 1-p, and
the pipes follow the tiles without any reductions or
reassignments. Using simple random walks, we
show that the resulting random permutation is close to the identity, and, moreover,
that the total
displacement
∑_i=1^n|i-w_i|
and length (number of inversions)
of this permutation
are of order n^3/2. This partially resolves a recent open problem
posed by
Colin Defant.



Returning to the original question, we also investigate the asymptotics
of max_w ∈ S_n𝔊_w^β(1^n).
When
β=1, this maximum is known to be O(2^n2). Using the
correspondence with
2-enumerated Alternating Sign Matrices
(equivalently, the six-vertex model with domain wall boundary conditions
and free-fermion weights; or the model of uniform domino tilings of the Aztec diamond),
we show that a large family of layered
permutations also achieve this asymptotic maximum. For
general β, we establish certain bounds for the maximal
 principal specialization.



 §.§ Grothendieck polynomials



The nonsymmetric Grothendieck polynomials generalize the Schubert and the Schur polynomials,
and capture the K-theory of the flag variety.
They can be defined or interpreted in a number of ways,
including via
divided difference operators <cit.>,
pipe dreams <cit.>, bumpless pipe dreams
<cit.>,
<cit.>,
and solvable lattice models
<cit.>, <cit.>.

We denote by S_n the set permutations of
{1,2,…,n} that we write in the one-line notation
w=w_1w_2⋯ w_n unless indicated otherwise.
We also denote the image of i under w by w(i), and use the notation
w_i=w(i) interchangeably when this does not lead to confusion.
The longest
permutation n n-1… 2 1 is denoted by w_0=w_0(n).
For a permutation w of length ℓ, we denote by R(w)
the set of reduced words of w, that is, tuples
(r_1,…,r_ℓ) such that s_r_1⋯ s_r_ℓ
is a reduced decomposition of w, where s_i=(i, i+1) are the
simple transpositions.

The (single) Grothendieck polynomials were introduced
by Lascoux <cit.> to study the K-theory of
flag varieties. Their original recursive definition is as
follows. Let π_i:ℤ[β][ x]→ℤ[β][ x] denote the isobaric
divided difference operator:

    π_i f(1+β x_i+1) f-(1+β x_i) s_i·  f/x_i-x_i+1,

where s_i acts on a polynomial f by permuting x_i and x_i+1.
The Grothendieck polynomials 𝔊_w^β are recursively
determined by the following conditions:

	
  * For the longest permutation, we have
		𝔊_w_0(n)^β=(1+β)^n2
		x_1^n-1x_2^n-2⋯ x_n-1.
	
  * For all w∈ S_n and i=1,…, n-1
		such that
		ℓ(w s_i)=ℓ(w)+1,
		we have
		𝔊_w^β =
		π_i 𝔊_ws_i^β.

Setting β=0 in 𝔊_w^β, we obtain a Schubert
polynomial 𝔖_w which represents cohomology
classes of Schubert cycles in the flag variety <cit.>.



 §.§ Pipe dreams and permutations



Grothendieck polynomials can also be defined combinatorially
via pipe dreams (equivalently, rc-graphs).

A pipe dream
of order n
is a tiling of the
staircase shape
(having n-1 boxes in the first row,
n-2 boxes in the second row, and so on,
with boxes left-justified)
by tiles of two types:
bumps
-2pt
    < g r a p h i c s >

and
crossings
-2pt
    < g r a p h i c s >
.
The n-th diagonal below the staircase is
equipped with half bumps
-2pt
    < g r a p h i c s >

(whose boundary we do not draw).
Each of the boxes can have a tile of any type, so there
are 2^n2 pipe dreams of order n.
See <Ref>, left, for an example of a pipe dream of order 6.



A pipe dream (a tiling of the staircase shape) forms
a collection of strands (or pipes) labeled 1 to n from
the row where they start.
A pipe dream is called reduced if
any two pipes cross through each other at most once.


	
	Given a pipe dream D
	that is not necessarily reduced, the reduction of D
	is a unique reduced pipe dream D' obtained as follows:
	starting at the bottom left tile traverse the pipe dream upwards along columns and to the right. For each encountered crossing,
	replace it with a bump if the pipes have already crossed in the already traversed squares.
	The labeling of pipes together with a reduction
	is indicated by colored paths in
	<Ref>, right.



	
	One can associate a permutation w(D)∈ S_n to a pipe dream
	of order n as
	follows. If D is reduced then w(D)^-1_j is the column where
	the pipe j ends up in.[Throughout
	the paper, the column coordinate j increases from left to right,
	and the row coordinate i increases from top to bottom.]
	Equivalently, the column j contains the exiting pipe of color
	w(D)_j. Note that to capture the column number for the pipe j,
	we need the inverse permutation w(D)^-1.

	If D is not reduced, then w(D)
	is the permutation associated to the reduction D' of D.



	Alternatively, the permutation w(D)
	from a non-reduced pipe dream
	can be defined using the Demazure product. We
	refer to <cit.> for details.


Let (n) and (n) be the sets of pipe
dreams and reduced pipe dreams of size n.
For each w∈ S_n,
let (w) and
(w) be, respectively, the sets of pipe dreams and reduced pipe dreams D such that w(D)=w. Note that
#(n)=2^n2, whereas there is no simple
known formula for #(n)
<cit.>. Given
a pipe dream D, let (D) be the set of
coordinates (i,j)
of the cross tiles.
The weight of D is the monomial
(D)∏_(i,j)∈(D) x_i.
For example, for the pipe dream in <Ref>, left, we have
(D)=x_1^3 x_2^2 x_3^2 x_4 x_5.
The following
result is due to
<cit.>, <cit.>, and <cit.>.

	
	For any w∈ S_n, we have

    𝔊_w^β(x_1,…,x_n )  = ∑_D ∈(w)β^#(D)-ℓ(w)(D).

In particular, setting β=0 forbids non-reduced
pipe dreams, so the Schubert polynomial is

    𝔖_w( x) = ∑_D∈(w)(D).





 §.§ Grothendieck random permutations from reduced pipe dreams



Fix p∈[0,1].
Equip the set of all pipe dreams
with a probability measure by independently
placing the tiles in each box:

    -6pt
        < g r a p h i c s >
      with probability p,
    	    -6pt
        < g r a p h i c s >
      with probability 1-p.

In particular, for p=1/2, we have the uniform measure on the set of pipe dreams.

By reducing this random pipe dream as in <Ref>,
we obtain a
random permutation 𝐰∈ S_n
which we call the Grothendieck random permutation (of order n and with parameter p;
we suppress the dependence on n and p in the notation).
The name is justified by a connection with the polynomials 𝔊_w^β=1.
Indeed,
we have for any w∈ S_n:

    (𝐰=w)
    	=
    	∑_D ∈(w) p^(D)(1-p)^elbow(D)
    	=
    	(1-p)^n2𝔊_w^β=1( p/1-p,…, p/1-p),

where the first equality is simply the definition of the measure (<ref>),
and the second immediately follows from <Ref> with β=1.
In <Ref> below, we explain how the
same distribution on permutations arises from
the six-vertex model with domain wall boundary conditions and free-fermion weights (corresponding
to 2-enumeration of Alternating Sign Matrices).

When p=0 or p=1, we almost surely have
𝐰=id or 𝐰=(n,n-1,…,1 ), respectively.
These cases are trivial, and in the rest of the paper we assume that p∈(0,1).
Individual samples of
Grothendieck random permutations with p=4/5 and p=1/2
and a plot constructed from averaging over many samples
are given in
<Ref>.





 §.§ Asymptotics of Grothendieck random permutations



For w∈ S_n,
define its height function by

    H(x,y)#(
    		{ w^-1(x),w^-1(x+1),…,w^-1(n)  }∩{ y,y+1,…,n  }),

where 1≤ x,y≤ n. In terms of the pipe dream as in <Ref>, right,
H(x,y) is the number of pipes of color ≥ x which exit through the positions
≥ y at the top. In the example in <Ref>, right,
we have H(4,3)=2. See also <Ref>
for an interpretation in terms of the permutation matrix of w,
where H(x,y) gives the number of 1s in the rectangle with lower left corner at (x,y).

Let now 𝐰∈ S_n be the Grothendieck random permutation with parameter p∈(0,1)
which we assume fixed.
This makes the height function H(x,y) random, too.
We show that H(x,y) satisfies the law of large numbers,
and characterize its asymptotic fluctuations:


	
	
		
  * There exists a limiting height function 𝗁^∘ such that
			
    lim_n→∞ n^-1 H(⌊ n𝗑⌋,
    				⌊ n𝗒⌋)=𝗁^∘(𝗑,𝗒),
    				    
    				(𝗑,𝗒)∈[0,1]^2,

			where the convergence is in probability.
			The function 𝗁^∘(𝗑,𝗒)
			is
			explicit, see
			(<ref>).
			It is continuous and depends only on p.
			The graph of 𝗁^∘
			is given in
			<Ref>, right.
		
  * The fluctuations of H(x,y) around 𝗁^∘
			belong to the KPZ universality class:
			
    lim_n→∞( H(⌊ n𝗑⌋,
    				⌊ n𝗒⌋)-n𝗁^∘(𝗑,𝗒)/𝗏(𝗑,𝗒) n^1/3≤ r )=F_2(r),     r∈ℝ,

			where F_2
			is the cumulative distribution function of the Tracy–Widom GUE distribution,
			and 𝗏(𝗑,𝗒) is given in (<ref>).
	

<Ref>
justifies
the simulations in
<Ref>. We prove it in
<Ref>
by realizing H(x,y) as an observable of a
discrete time
TASEP with parallel geometrically distributed jumps started from the densely packed
(step) initial configuration.
Here we omit the technical details of this coupling, and refer to
<Ref> in the text.
By employing known techniques
from Integrable Probability and passing back to the
Grothendieck random permutation, we obtain the limit shape and fluctuations.



The law of large numbers in the first part of
<Ref>
means that the Grothendieck random permutations 𝐰∈ S_n
converge to a deterministic permuton.
Recall that permutons are Borel probability measures μ
on [0,1]^2 with uniform marginals, that
is,
μ( [0,1]× [a,b] )=
μ( [a,b]× [0,1] )=b-a.
For details on permutons
and their connection to pattern frequencies in permutations, we refer to
<cit.>,
<cit.>, and the survey
<cit.>.
Our limiting Grothendieck permuton is completely determined by the limiting height function
via

    μ( [𝗒,1]× [𝗑,1] )
    	=𝗁^∘(𝗑,𝗒).

As a corollary of the permuton convergence,
one can obtain law of large numbers for
arbitrary pattern counts in Grothendieck random permutations.
The simplest example is the number of inversions:


	
	Let
	𝐰=𝐰(n)∈ S_n be the Grothendieck random permutations with
	p=1/2.
	We have
	
    lim_n→∞( 𝐰(n) )/ n2=1-π/4.


The limit in (<ref>)
exists for all p∈(0,1) and is expressed through certain integrals
of algebraic functions.
We are able to evaluate it in a closed form only for p=1/2.
We refer to <Ref> and its proof for more
discussion.


	
	In <Ref>, we connect
	Grothendieck random permutations
	to
	2-enumerated Alternating Sign Matrices
	(bijectively corresponding to the six-vertex model with
	domain wall boundary conditions and free-fermion weights, and also to
	domino tilings of the Aztec diamond).
	The connection between Grothendieck polynomials and the
	six-vertex model goes back to
	<cit.>, see also <cit.>.
	Equivalences between 2-enumerated Alternating Sign Matrices,
	six-vertex models, and domino tilings of the Aztec diamond
	are well-known in statistical mechanics and integrable probability,
	see, for example,
	<cit.>, <cit.>, and
	<cit.>, <cit.>, <cit.>.

	Via this connection,
	our asymptotic results (<Ref>)
	can be recast as results about a random permutation coming from the free-fermion
	six-vertex model
	with domain wall boundary conditions.
	Moreover, there is a vast amount of asymptotic results
	about domino tilings and free-fermion six-vertex models,
	and in <Ref>
	we show how some of them lead to further asymptotic properties
	of Grothendieck random permutations.




 §.§ A deformation and random non-reduced pipe dreams



Let us discuss a deformation of the Grothendieck random permutation
by means of a new parameter
q∈[0,1].
Take a pipe dream of order n obtained
by randomly placing the tiles in the same way as in (<ref>)
(depending on p∈ (0,1)). However, instead of reduction (<Ref>),
let us apply a modified (q-randomized) procedure:

	
	Assume that D is a pipe dream
	that is not necessarily reduced.
	For each crossing tile, consider the incoming pipes.
	There are two cases:
	
		
  * If the lower colored pipe enters from the left (and the higher colored one from the bottom),
			then the pipes cross through each other with probability 1.
		
  * Otherwise, if the lower colored pipe enters from the bottom (and the higher colored one from the left),
			then the pipes cross through each other with probability q.
			That is, with probability 1-q we replace the crossing with a bump.
	
	The random q-reduction steps are done independently at each crossing tile
	when
	reading the staircase diagonal by diagonal starting from the bottom left.
	See <Ref> below for a detailed definition
	of the Markov process for q=0, which has the same order of reduction steps.
	Denote the random permutation obtained by the q-reduction of a random pipe dream
	with the measure (<ref>) by 𝐰^(q)∈ S_n.
	See
	<Ref> for simulations.



	
	For q=0, the q-reduction procedure becomes the usual reduction from <Ref>.
	For q>0, the q-reduction no longer tracks whether the pipes
	have crossed before. Instead, it simply applies the local q-reduction rule at each
	crossing tile,
	based on the relative colors of the incoming pipes.


The asymptotic analysis of 𝐰^(q) for q∈ (0,1]
may be performed similarly to the case q=0.
The underlying particle system is a certain
q-deformation of TASEP which allows both left and right jumps (governed by different rules).
Setting q=1 removes the asymmetry, which changes the
normalization in the law of large numbers and fluctuations (as well as
the fluctuation distribution).
This behavior
parallels the different
scales of laws of large numbers
in the
KPZ and the
Edwards–Wilkinson (EW) universality classes.
The KPZ class asymptotics arise in TASEP and its asymmetric deformations,
while symmetric versions of TASEP fall into the EW class.
We refer to
<cit.> for a detailed discussion of these two universality classes.

In this paper, we mainly focus on the case q=0, and obtain
results outlined in <Ref>.  In
<Ref>, we also treat the case q=1.  In
the latter model, the q-reduction does not change the pipe
dream at all. Probabilistically, this is the most natural
way to associate a permutation to a random pipe dream with
distribution (<ref>). On the other
hand, the q=1 model apparently lacks a rich algebraic
underpinning.

Finding the asymptotic number of inversions of
𝐰^(1)=𝐰^(1)(n)
∈ S_n as n→∞ was posed as an open
problem by Colin
Defant at the Richard Stanley's 80th birthday
conference.[Conference “The Many Combinatorial Legacies of Richard P. Stanley: Immense Birthday Glory of the Epic Catalonian Rascal,” <https://www.math.harvard.edu/event/math-conference-honoring-richard-p-stanley/>. Accessed: 07/23/2024.]
We show
the following
asymptotics:

	
	For p∈[0,1) and any ε>0,
	the expected number of inversions satisfies for all sufficiently large n:
	
    2/3√(π) (1-ε)  n^3/2√(p/1-p)≤𝔼[(𝐰^(1)(n))] ≤4/3√(π) (1+ε)  n^3/2√(p/1-p).

	Moreover,
	the random permutations 𝐰^(1)(n) converge in distribution
	to the identity permuton supported
	on the diagonal of [0,1]^2.


Based on simulations (see <Ref> for examples)
and supporting numerics, we make
further conjectures about the behavior of 𝐰^(q).
(<ref>):

	
	For any p∈[0,1), we have convergence in probability:
	
    n^-3/2(𝐰^(1))→ϰ√(p/1-p),     n→∞,

	where ϰ is a constant independent of p whose value is near 0.53.

<Ref> implies that
ϰ (if it exists) must satisfy
2/(3√(π))≤ϰ≤ 4/(3√(π)), and the approximate value 0.53
is conjectured based on numerical data.

We also expect that as p=p(n)→1 with n, the random
permutations 𝐰^(1)(n) converge in
distribution to a nontrivial deterministic permuton (which
depends on the speed and the parameters in p(n)→1).



In the intermediate cases q∈(0,1), one has to make sure that pipes with larger numbers
do not get stuck close to the diagonal boundary. This leads to a
conjectural restriction pq≤1/2.
Let H^(q)(x,y) be the height function of 𝐰^(q),
defined in the same way as in <Ref> above.

	
	For any fixed p,q∈(0,1) with pq≤1/2, the random permutations
	𝐰^(q)(n)∈ S_n converge to a permuton
	determined by a height function
	𝗁^(q)(𝗑,𝗒) on [0,1]^2.
	That is, we have the limit in
	probability:
	
    lim_n→∞ n^-1 H^(q)(⌊ n𝗑⌋,
    		⌊ n𝗒⌋)=𝗁^(q)(𝗑,𝗒),
    		    
    		(𝗑,𝗒)∈[0,1]^2.

	When pq>1/2, the random
	permutations 𝐰^(q)(n) converge to the identity permuton.

When pq≤1/2, fluctuations of H^(q)(x,y) around 𝗁^(q) should
be governed by one of the laws appearing in the
KPZ universality class in the presence of a boundary
(Tracy–Widom GUE distribution,
its GOE/GSE counterparts and crossovers,
or Gaussian fluctuations on scale n^1/2).
We refer to <cit.>, <cit.> for further discussion of
fluctuations in half-space models from the KPZ universality class.






 §.§ Asymptotics and maxima of Grothendieck principal specializations



We are interested in the following principal specializations of the
Grothendieck polynomials:

    Υ_w(β) 𝔊_w^β (1,1,…,1_n),
    	     w∈ S_n.

In particular, Υ_w(0) is the principal specialization of the Schubert polynomial.
For β=1, these quantities are the (unnormalized) probability weights
of the Grothendieck random permutation 𝐰 with p=1/2,
see (<ref>).

Set

    v_n(β)∑_w ∈ S_nΥ_w(β),    
    	u_n(β)max_w∈ S_nΥ_w(β).

From <Ref>, we have:

    v_n(1)=2^n2,      u_n(1)=2^n2-o(n^2),      n→∞.

The first equality is exact, while the second one is asymptotic.
This asymptotic behavior
follows from
the cardinality
n!∼ e^nlog n+O(n)≪ 2^n2 of the set over which we take the maximum.

For u ∈ S_k, w ∈ S_m, we denote

    u × w  (u(1), …, u(k), w(1)+k, …, w(m) + k)∈ S_k+m.

For a composition b = (b_ℓ, …, b_1) of n = b_1 + ⋯ + b_ℓ, the layered permutation w(b) ∈ S_n is defined as follows:

    w(b) = w_0(b_ℓ) ×⋯× w_0(b_1),

where w_0(k) is the full reversal permutation of order k.
Denote by L_n⊂ S_n the subset of layered permutations,
and let

    u_n'(1)max_w∈ L_nΥ_w(1).





In <Ref>, we prove that
on layered permutations, the
β=1 Grothendieck polynomials
attain their asymptotic maximum:


	
There are sequences of layered permutations w(b^(n))∈ S_n so that

    lim_n →∞1/n^2log_2
    Υ_w(b^(n)) (1)
    = 1/2.


<Ref> implies that in leading order,
u_n'(1) asymptotically behaves in the same way as
v_n(1) and u_n(1) (<ref>).

Explicit constructions of such sequences of layered
permutations are given in <Ref>. In particular, we
can take the parts of the compositions b^(n) = (…, b_2, b_1)
to be
geometric b_i ∼
(1- α)α^i-1 n for any α∈ [1/√(2), 1).
See
<Ref> for an illustration.
Note, however that we do not know in the limit what
compositions b of size n yield the global maximum of
Υ_w(b) over all layered permutations. For
Schubert polynomials, the analogous question was settled in
<cit.>. We present numerics
for principal specializations of the β=1 Grothendieck polynomials in
<Ref>, and explain the difference between the (numerically)
optimal layered permutations and the ones constructed in the proof of <Ref>.

Regarding Stanley's problem on permutations achieving the
maximal Schubert specialization u_n(0) (the case
β=0), the Merzon–Smirnov conjecture
<cit.> states that the maximum
is attained on layered permutations. On one hand, our
<Ref>
establishes that an asymptotic analog of this conjecture holds for
the β=1
Grothendieck polynomials.
On the other hand, the typical shape of Grothendieck
random permutations
(<Ref>, center and right; see also
<Ref>)
suggests that the
global maximum u_n(1)
may be achieved on permutations whose shape is far from layered.
In <Ref>, we obtain new bounds on the quantities
v_n(β) and u_n(β) for general values of β,
but do not consider the relation
of maximal principal specializations to layered permutations.



 §.§ Outline


In the Introduction, we defined the model and formulated our main results.
In <Ref>, we explain how the Grothendieck random permutation
𝐰 from <Ref>
is sampled by running a Markov chain, more precisely, the colored stochastic six-vertex model.
In <Ref>, we connect the vertex model
to a discrete time TASEP with parallel update and geometrically distributed jumps.
This process is well-studied within Integrable Probability,
which leads to law of large numbers and Tracy–Widom fluctuation
results.
In <Ref>, we recast the asymptotic results about
TASEP into
limiting properties of
Grothendieck random permutations.
In <Ref>, we consider a variant of our
model coming from non-reduced pipe dreams (described in <Ref>).
In <Ref>, we
address the question of asymptotics of  principal specializations
of β=1 Grothendieck polynomials, and show that they are asymptotically the largest
on layered permutations.
We also obtain new bounds on the principal specializations for general β,
based on an interpretation of Grothendieck polynomials in terms of
Alternating Sign Matrices.
In <Ref>, we discuss further directions and a few
open problems.


In <Ref>, we provide tables of maximal  principal specializations of β=1 Grothendieck polynomials on layered permutations.
In <Ref>, we provide the steepest descent computations
leading to the asymptotics in TASEP. While these computations are standard by
now, we include them to justify the exact values of the constants
in TASEP limit shape and fluctuations.




 §.§ Acknowledgments


We are very grateful to Jehanne Dousse for her valuable insights in the development of this project and doing experiments in Sage. We thank Igor Pak for bringing us together and initiating our collaboration.
We thank Daoji Huang, Mark Shimozono and Tianyi Yu for sharing an early version of <cit.>. We would also like to thank Dave Anderson, Jacopo Borga, Filippo Colomo, Colin Defant, Philippe DiFrancesco, Rick Kenyon, Matthew Nicoletti, Mikhail Tikhonov, and Anna Weigandt for helpful discussions.

The project was begun at the SQuaRE “Young Tableaux Asymptotics”
at the American Institute of Mathematics. The authors thank
AIM for providing a supportive and mathematically rich
environment.
Part of this research was performed in Spring 2024 while
some of the authors were
visiting the
program
“Geometry, Statistical Mechanics, and Integrability”
at the Institute for Pure and Applied Mathematics
(IPAM), which is supported by the NSF grant DMS-1925919.

AM was partially supported by the NSF grant DMS-2154019, the NSERC Discovery Grant
RGPIN-2024-06246, and the FRQNT Team grant 341288. GP was partially supported by the NSF grant CCF-2302174. LP was partially supported by the NSF grant DMS-2153869 and the Simons Collaboration Grant for Mathematicians 709055. DY was partially supported by the  grant.



§ FROM PIPE DREAMS TO THE COLORED STOCHASTIC SIX-VERTEX MODEL





 §.§ Colored stochastic six-vertex model



Introduce the vertex weights

    w_p(a,b;c,d)
    =
    w_p
    (
    	[baseline=-3,scale=.7,very thick]
    			[fill] (0,0) circle [radius=0.025];
    			 (0.5,0) – (0.05,0);
    			 (-0.5,0) – (-0.05,0);
    			 (0,0.05) – (0, 0.5);
    			 (0,-0.05) – (0,-0.5);
    			 at (.25,-.4) a;
    			 at (-.7,0) b;
    			 at (-.25,.4) c;
    			 at (.7,0) d;
    ),
 where
a,b,c,d∈{0, 1,…,n  }.
Here 0 represents
the absence of a pipe, and
positive numbers indicate the
pipes' colors.
We view (a,b) and (c,d) as incoming and outgoing pipes, respectively.
The weights are
defined as follows:

    w_p(a,a;a,a)    =1;
    		
    
    		w_p(b,a;b,a)   =p,       w_p(b,a;a,b)=1-p;
    		
    
    		w_p(a,b;a,b)   =0,       w_p(a,b;b,a)=1,

where 0≤ a<b≤ n.
The weights of all other configurations not listed in (<ref>)
are zero. Let us indicate a few crucial properties of the weights:

	
  * The weights conserve the pipes:
		w_p(a,b;c,d)=0 unless {a,b}={c,d} as sets.
	
  * The weights are stochastic:
		
    w_p(a,b;c,d)≥0,    ∑_c,d=0^n w_p(a,b;c,d)=1   for all a,b.




	The weights w_p come from the
	fundamental stochastic R-matrix
	for the quantum group U_q(𝔰𝔩_n+1)
	<cit.>, see also <cit.>
	for a brief overview.
	For the application to random pipe dreams, we specialize
	the quantum parameter q to zero
	(equivalently, to infinity in the normalization of <cit.> — this choice
	depends only on
	the order in which colors correspond to basis vectors of
	the fundamental representation of 𝔰𝔩_n+1).
	The remaining parameter p is related to the spectral parameter
	in an integrable vertex model.




 §.§ Matching random pipe dreams to the stochastic vertex model



Attach coordinates
(i,j)∈ℤ_≥1 ^2
to boxes of the
staircase shape, with i increasing down,
and j increasing to the right.
The staircase shape is then
_n{ (i,j) i+j≤ n }.

Place a stochastic vertex with the weight w_p at each box (i,j)∈_n.
Let the initial condition along the left boundary of _n be
the rainbow one, with colors 1 to n from top to bottom.
Then we can sample a random configuration
of pipes as in <Ref>, right,
by running a discrete time Markov chain with time τ=j-i, where
-(n-1)≤τ≤ n-1. At each step τ→τ+1,
the configuration with j-i≤τ
is already sampled, which determines the incoming colors at all boxes
(i,j)∈_n with j-i=τ+1. The next
step τ→τ+1 consists in an independent update of the outgoing colors
at all boxes (i,j)∈_n with j-i=τ+1,
using the stochastic vertex weights
w_p(a,b;·,·)
(<ref>), 1≤ a,b≤ n.
Here we view each w_p(a,b;·,·) as a probability distribution on possible outputs,
(a,b) or (b,a).
Reading off the outgoing colors at the top boundary of _n,
we arrive at a random permutation 𝐰∈ S_n.


	
	The random permutation 𝐰∈ S_n obtained from the colored stochastic six-vertex model
	as described above has the same distribution as the Grothendieck random permutation
	defined in <Ref>.


	The Markov evolution of the stochastic vertex model
	is equivalent to a simultaneous
	exploration of a random pipe dream (that is, determining the state
	-2pt
    < g r a p h i c s >

	or
	-2pt
    < g r a p h i c s >

	at each box),
	and its reduction by following the pipes.
	Indeed, in
	the evolution of the stochastic vertex model,
	in each box (i,j), two strands of different colors meet as incoming pipes.
	There are two possibilities:
	
		
  * If the pipe of the lower numbered color
			is below, then these pipes have already met and crossed through each other
			(which is allowed only once in a reduced pipe dream).
			Thus, regardless of the state
			-2pt
    < g r a p h i c s >

			or
			-2pt
    < g r a p h i c s >

			at (i,j), the pipes must bump off each other.
			In
			(<ref>),
			this corresponds to the vertex weight w_p(a,b;b,a)=1, where 1≤ a<b≤ n.
		
  * If the pipe of the lower numbered color is to the left, then the pipes have not yet crossed
			through each other (but they may have bumped off at an elbow).
			Then, we place the tile
			-2pt
    < g r a p h i c s >

			or
			-2pt
    < g r a p h i c s >

			at (i,j) with probability p or 1-p, respectively.
			After placing the tile,
			the strands deterministically follow the paths drawn on it.
			The choice of the tile is equivalent
			to using the stochastic vertex weights
			w_p(b,a;b,a)=p or w_p(b,a;a,b)=1-p, where 1≤ a<b≤ n.
	
	We see by induction that the vertex model
	(<ref>)
	produces the same random permutation
	as reducing the random pipe dream.




 §.§ Height function and color forgetting



Let x,y∈{ 1,…,n  }.
For a fixed pipe dream D and the corresponding permutation w=w(D)
(see <Ref>),
define the permutation height function as

    H(x,y)
    		   #{pipes of colors ≥ x
    		which exit through positions j≥ y at the top}
    =   #(
    			{ w^-1(x),w^-1(x+1),…,w^-1(n)  }∩{ y,y+1,…,n  }).

Observe that H(x,y) depends only on the permutation
w and not on the structure of crossings and bumps in the pipe dream.
For a fixed w, the function H(x,y) decreases in x and y.
When the pipe dream and the permutation 𝐰
are random, the height function H(x,y) becomes a random variable.

In the permutation matrix of w∈ S_n,
H(x,y) is the number of entries in the rectangle
[y,n]×[x,n]. Recall that the pipe of color i exits at w^-1(i), so the
rectangle is transposed.
See <Ref> for an illustration.
This interpretation of H(x,y) implies that

    H(x,y)≤min(n-x,n-y).


In colored stochastic six-vertex models, quantities like H(x,y) are
referred to as colored height functions, cf. <cit.>.




	
	The color-position symmetry <cit.>
	of the colored stochastic six-vertex model
	implies that the distribution of H(x,y) corresponding to the
	Grothendieck random permutation 𝐰
	is symmetric in x,y.
	Indeed, by the color-position symmetry, 𝐰 and 𝐰^-1
	have the same distribution. Therefore,
	
    H(x,y)
    		=
    		∑_i=x^n∑_j=y^n1_𝐰^-1(i)=jd=∑_i=x^n∑_j=y^n1_𝐰(i)=j
    		= H(y,x).

	However, we can access the distribution of H(x,y) directly
	using color forgetting (<Ref> below), and do not
	rely on
	color-position symmetry.


Consider an uncolored (color-blind) stochastic
vertex model
with all pipes of the same color.
We indicate pipes and empty edges by 0 and 1, respectively.
The weights of the color-blind model are defined as

    w_p^∙(0,0;0,0)   =1,     w_p^∙(1,1;1,1)=1,
    		
    
    		w_p^∙(1,0;1,0)   =p,    
    		w_p^∙(1,0;0,1)=1-p,
    		
    
    		w_p^∙(0,1;0,1)   =0,    
    		w_p^∙(0,1;1,0)=1.



	
	Fix 1≤ x≤ n. In the colored stochastic vertex model of
	<Ref>,
	erase all pipes of colors <x,
	and identify all the remaining pipes for colors ≥ x.
	The resulting random configuration of
	uncolored pipes evolves
	according to
	the color-blind stochastic
	vertex model w_p^∙ (<ref>).



	This is a standard property of the colored stochastic six-vertex model,
	see <cit.> and references therein.
	In interacting particle systems,
	this property is known as the basic coupling, cf.
	<cit.>, <cit.>.
	Let us provide an idea of the proof.

	The crucial property is
	that the weights w_p (<ref>)
	depend only on the relative order of colors of the incoming pipes, and not the actual colors.
	Therefore, when we identify all colors ≥ x, the resolution of the pipe
	crossings does not affect the behavior of the color-blind system.
	In detail, one can consider four cases
	for the incoming pipes (a,b),
	1≤ a,b≤ n,
	depending on whether a≥ x and/or b≥ x.
	The resulting probability distribution on possible outputs,
	(a,b) or (b,a), depends only on these cases, and not on
	the actual values of a,b.
	These four cases correspond to the color-blind incoming pipes
	in w_p^∙ (<ref>).


By <Ref>,
the permutation height function H(x,y)
can be identified with an observable of the model
w_p^∙ with initial occupied configuration
{ x,x+1,…,n   } along the left boundary
(that is, the sites {1,…,x-1} are empty).
Indeed, in this color-blind model, H(x,y) is simply
the
number of pipes exiting through the top boundary at positions ≥ y.
We see that under the color forgetting, the color parameter x in H(x,y) became a parameter of the initial configuration.


	One can similarly forget the colors >x, and the resulting
	color-blind system evolves according to different
	weights (obtained from w_p^∙ by swapping 0↔ 1).
	It is more convenient for us to work with the model w_p^∙,
	which we directly relate
	to an interacting particle system in the next <Ref>.




§ FROM VERTEX MODELS TO TASEP





 §.§ TASEP with moving exit boundary




	
	Let k≥ 1.
	Let
	ξ(t)(ξ_1(t)>…>ξ_k(t) )⊂ℤ
	be the
	k-particle
	discrete time TASEP (Totally Asymmetric Simple Exclusion Process)
	having
	parallel updates
	and geometrically distributed jumps.
	In detail, ξ(t), t∈ℤ_≥0, is a discrete time Markov chain
	on particle configurations in ℤ which at each time step t→ t+1 evolves as follows:
	
    ξ_i(t+1)=ξ_i(t)+min( G_i(t+1), ξ_i-1(t)-ξ_i(t)-1 ),    
    		1≤ i≤ k,

	where G_i(t+1) are independent geometric random variables with parameter p∈(0,1),
	that is,
	
    ( G_i(t+1)=m )=(1-p) p^m,     m∈ℤ_≥0.

	The update (<ref>) occurs in parallel for all
	particles 1≤ i≤ k, that is,
	the new positions ξ_i(t+1) depend only on the configuration
	ξ(t) at the previous time step, and new independent random variables.
	By agreement, we have ξ_0(t)≡ +∞,
	so that the first particle ξ_1(t) performs an independent random walk
	with geometrically distributed jumps.
	See <Ref> for an illustration.




Start the TASEP from
the densely packed (also called step)
initial configuration
{ 1,2,…,k  }, that is,
ξ_i(0)=k+1-i, 1≤ i≤ k.
Fix n≥ k, and introduce a moving exit boundary
which starts at n+1/2, and deterministically
moves to the left with speed 1.
The exit time of a particle ξ_i(t) is defined by

    T_exit(i)min{ tξ_i(t)≥ n+1-t }.

For example, if n=k, then the first particle starting at
n exits at time t=1 (but have not exited at the initial
time t=0). The exit times are almost surely ordered:

    1≤ T_exit(1)<T_exit(2)<…<T_exit(k)≤ n.

Note that tor any n≥ k,
all particles have exited the system by the time t=n.
See <Ref> for an illustration and simulations.







Integrability of TASEP with parallel updates, geometric jumps,
and densely packed initial configuration
can be traced back to
<cit.>, see also
<cit.>,
<cit.>,
and <cit.>.
We recall the necessary
integrability
results in <Ref> below
after connecting TASEP to vertex models and
Grothendieck random permutations
in the next <Ref>.



 §.§ Identification with vertex models



Fix n and 1≤ x,y≤ n. Recall the
observables H(x,y)
defined by (<ref>).
That is, H(x,y) are random variables which are functions of the Grothendieck
random permutation (equivalently, of the colored stochastic six-vertex model).
In <Ref>,
we identified H(x,y) with observables of a color-blind vertex model with weights
w_p^∙ (<ref>).


	
	For any 1≤ x,y≤ n and 0≤ h≤ n-x+1, we have
	
    _𝐰(H(x,y)≤ h)
    			   =
    			_TASEP(T_exit(n-x+1-h)≤ y-1)
    			
       =
    			_TASEP(
    				ξ_n-x+1-h(y-1)≥ n-y+2
    			).

	Here
	_𝐰
	corresponds to the Grothendieck random permutation of order n,
	and
	_TASEP
	is the probability distribution of the TASEP
	with k=n-x+1 particles and moving exit boundary.
	In (<ref>)
	we have, by agreement,
	T_exit(m)= 0 and
	ξ_m(t)= +∞ for all m≤ 0.


	The second identity in (<ref>)
	immediately follows from the definition of the exit times
	(<ref>). Let us focus on the first identity.

	By <Ref>, let us consider the color-blind vertex model
	with k=n-x+1 identical pipes entering the left boundary
	of the staircase shape _n
	at locations { x,x+1,…,n  }.
	The event { H(x,y)≤ h } means that
	at most h of these pipes exit the top boundary at positions ≥ y.
	Note that the color-blind model has indistinguishable pipes, and thus the “resolution”
	of their crossings
	does not change the behavior of the system (see the proof of <Ref> for more detail).

	View the horizontal coordinate j
	as time t=0,1,…,n,
	and record the pipes' coordinates as
	
    η_1(t)<…<η_k(t),     t=0,1,…,n .

	The initial condition is η_k-m+1(0)=n-m+1, 1≤ m≤ k.

	At each time step, according to the w_p^∙ (<ref>),
	each pipe first deterministically turns up.
 	When a pipe faces up, it can travel up a further
	distance distributed as a geometric random variable with parameter p,
	see (<ref>).
	However, during a time step t→ t+1, each pipe η_m cannot
	travel further than the previous pipe' location η_m-1(t)
	(this is the exclusion mechanism).
	When a pipe stops moving up, it turns right and waits till the next time step.
	Once a pipe reaches the top boundary, it exits the system.
	See <Ref>, left, for an illustration.



	

	Setting ξ_m(t) n+1-t-η_m(t),
	one readily verifies that the evolution of ξ_1(t)> … > ξ_k(t)
	is the same as that of the TASEP
	with k=n-x+1 particles
	defined in
	<Ref>.
	In particular, subtracting t from n+1-η_m(t)
	eliminates the deterministic up movement of the pipes by one at each time step.

	The top boundary for pipes (as in <Ref>, left)
	becomes the moving exit boundary for TASEP. The event
	{H(x,y)≤ h} for pipes is equivalent to the event
	that at least k-h=n-x+1-h of the TASEP particles
	have exited before time y-1. Because the
	exit times are ordered as in (<ref>),
	we get the desired first identity in (<ref>).




 §.§ Integrability of TASEP



Fix m≥ 0, and let λ=(λ_1≥…≥λ_m≥ 0 )
be a partition with at most m parts.
The Schur symmetric polynomial in m variables indexed by λ is defined as

    s_λ(a_1,…,a_m)[a_i^λ_j+m-j]_i,j=1^m/[a_i^m-j]_i,j=1^m,

where a_i are variables. The denominator is the Vandermonde determinant
∏_1≤ i<j≤ m(a_i-a_j).

Let m,t≥1.
The Schur measure <cit.>
is a probability measure on partitions with at most min(m,t) parts
depending on parameters a_1,…,a_m,b_1,…,b_t. Its probability weights
are defined as

    (λ)∏_i=1^m∏_j=1^t
    	(1-a_ib_j)
    	·
    	s_λ(a_1,…,a_m)
    	s_λ(b_1,…,b_t).

When m=t=1, the weights (<ref>) define the geometric distribution
on λ=(λ_1≥0) with parameter a_1b_1.
In general, the form of the normalizing constant in (<ref>)
follows from the Cauchy summation identity for Schur polynomials.
The infinite sum
for the normalizing constant runs
over all λ of length ≤min(m,t) and converges if |a_ib_j|<1 for all i,j.

The next statement connects TASEP to Schur measure,
which provides the necessary integrability structure for the former.


	
	For any t,m≥1, the displacement of the m-th particle
	in the TASEP with parallel updates, geometric jumps with parameter p,
	and densely packed initial configuration (as in <Ref>)
	has the same distribution as λ_m, the last part of a partition
	under the Schur measure
	(<ref>)
	with a_1=…=a_m=1 and b_1=…=b_t=p:
	
    ξ_m(t)-ξ_m(0)d=λ_m,
    		    λ∼
    		(1-p)^m t s_λ(1,…,1 _m)
    		s_λ(p,…,p _t).



<Ref> is a well-known
result in Integrable Probability.
It follows either from the
Robinson–Schensted–Knuth (RSK) correspondence with column insertion,
or by representing the TASEP as a
marginally Markovian evolution of a process on the space of semistandard Young tableaux
(equivalently, interlacing arrays / Gelfand–Tsetlin patterns).

For the simpler continuous time TASEP (when each particle waits an exponentially
distributed time and jumps to the right by 1 if the destination
is unoccupied), the connection to Schur measures is
exploited in the celebrated work <cit.>
via last-passage percolation, which is related to the row RSK.
The RSK approach can be traced back to <cit.>.
Dynamics on interlacing arrays are constructed later in
<cit.> using a non-RSK approach.

For our discrete time TASEP with geometric jumps,
the RSK was explicitly used in
<cit.>
to establish <Ref>.
A systematic treatment of discrete time
dynamics on interlacing arrays connected to Schur measures
leading to various TASEPs and PushTASEPs may be found in
<cit.>.
It includes
RSK-type and other dynamics on interlacing arrays under one roof,
as well as a generalization of the Schur structure to the level of q-Whittaker
symmetric polynomials.



 §.§ Determinantal structure



The connection
to Schur measures (<Ref>)
allows to extract law of large numbers and
fluctuation results for the TASEP particles.
The key property of Schur measures which makes this possible is
the fact that they form a determinantal point process.

Fix m,t≥ 1, and define a random point configuration
X(λ){λ_j+m-j}_j=1^m on ℤ_≥0,
where λ is distributed as (<ref>).
If λ has less parts than m, append it by zeros.
These zeros translate into a part of X(λ) of the form
{0,1,…,l } for some l.


	
	The random point configuration X(λ) on ℤ_≥0 is a
	determinantal point process. This means that for
	any r≥1 and pairwise distinct points u_1,…,u_r ∈ℤ_≥0,
	the correlation functions have the form
	
    (X(λ) contains all of the points u_1,…,u_r )=
    		[ K(u_i,u_j) ]_i,j=1^r.

	The correlation kernel K is given by a double contour integral:
	
    K(u_1,u_2)=1/(2π𝐢)^2dz  dw/z-ww^u_2-m/z^u_1-m+1(1-p/z)^m/(1-z)^t(1-w)^t/(1-p/w)^m,
    		    
    		u_1,u_2∈ℤ_≥0,

	where the contours are positively oriented simple closed curves satisfying
	p<|w|<|z|<1.


<Ref> is due to
<cit.>,
see also
<cit.>.
General discussions and many examples of determinantal point processes
may be found
in <cit.>, <cit.>,
or
<cit.>.




 §.§ Asymptotics of TASEP



Determinantal structure powers the asymptotic behavior of
the TASEP particles.
We are interested in
probabilities of the form

    _TASEP(ξ̅_m(t)≥ u),
    	    where  ξ̅_m(t)ξ_m(t)-ξ_m(0).

Indeed, since
<Ref>
reduces the distribution of H(x,y) to
probabilities of the form (<ref>),
we can forget about the moving exit boundary in TASEP.
Indeed, the parameter n of the moving boundary is absorbed into m and u.



Let the parameters
t,m,u grow to infinity proportionally to each other:

    m=⌊ L𝗆⌋ ,     
    	t=⌊ L𝗍⌋ ,     
    	u=⌊ L𝗎⌋ ,
    	      L→∞.

First, let us heuristically discuss the law of large numbers and fluctuations
of
_TASEP(ξ̅_m(t)≥ u),
without providing exact constants.
The latter are given in <Ref> below,
and we formulate the asymptotic results in <Ref>.
The proof of <Ref> using the determinantal structure
is a standard application of the steepest descent method for double contour integrals.
We provide the computations (leading to the constants in <Ref>)
in
<Ref>.



At the “hydrodynamic” scale (<ref>),
TASEP has a limit shape
(first observed in the continuous time TASEP in <cit.>, see also
<cit.>). That is, there exists
a function 𝖼(𝗆,𝗍) such that
for any ε>0, we have

    lim_L→∞_TASEP(ξ̅_m(t)≥ L𝖼(𝗆,𝗍)+ε)
    	=0,
    	    lim_L→∞_TASEP(ξ̅_m(t)≥ L𝖼(𝗆,𝗍)-ε)
    	=1.

In other words, we have convergence in probability:

    lim_L→∞
    	L^-1ξ̅_⌊ L𝗆⌋(⌊ L𝗍⌋ )
    	=𝖼(𝗆,𝗍).

Let us note a certain a priori property of the limit shape:

	
	If 𝗍<𝗆/p, we have 𝖼(𝗆,𝗍)=0.


	This follows from the fact that the leftmost hole in the TASEP configuration
	(which starts immediately to the right of ξ_1(0) at time t=0)
	travels left by at most 1 at each time step.
	The probability that the hole moves is p=(G>0),
	where G is the geometric random variable (<ref>).
	Therefore, if 𝗍<𝗆/p, then the hole has not reached the
	m-th particle by time t with probability 1-e^-C L for some C>0.
	This implies the claim.


Next, since TASEP belongs to the Kardar–Parisi–Zhang (KPZ) universality class
<cit.>, we expect
fluctuations of order L^1/3 provided that 𝗍>𝗆/p.
More precisely,
modifying m and u on the scale
L^1/3 probes the scaling window in the law of large numbers
(<ref>). That is,
there exist constants
𝗏_1(𝗆,𝗍),
𝗏_2(𝗆,𝗍)>0, such that

    lim_L→∞_TASEP( ξ̅_⌊ L𝗆- L^1/3α𝗏_1(𝗆,𝗍)⌋
    		(⌊ L𝗍⌋ )≥
    	L𝖼(𝗆,𝗍)-
    	L^1/3β𝗏_2(𝗆,𝗍))
    	=F_2( α +β)

for all α,β∈ℝ.
Here, F_2 denotes the cumulative distribution function of
the Tracy–Widom GUE distribution. The plus signs
by α and β in the right-hand side of (<ref>)
are straightforward from the monotonicity of the pre-limit probabilities.

The distribution function F_2
was first discovered
in connection with the fluctuations of the largest
eigenvalue in large complex Hermitian random matrices
exhibiting unitary symmetry <cit.>.
Subsequently,
F_2 was put into a larger class of limiting distributions arising in random
growth models and interacting particle systems, known as the KPZ universality class.
We refer to the surveys <cit.>,
<cit.>,
<cit.> for a detailed exposition.
The appearance of F_2 in the fluctuations (out of several
candidates in the KPZ universality class)
is a feature of the step initial
configuration in our TASEP.


	
	Let 𝗆,𝗍>0. Define
	
    𝖼(𝗆,𝗍)
    		
    			0,
    			   if 𝗍≤𝗆/p;
    			
    (√(p𝗍)-√(𝗆))^2/1-p,
    			   if 𝗍≥𝗆/p.

	If 𝗍>𝗆/p, set
	
    𝗏_1(𝗆,𝗍)√(p)𝗆^1/3/𝗍^1/6(√(𝗍/p)-√(𝗆))^2/3/(√(p𝗍)-√(𝗆))^1/3,
    		    𝗏_2(𝗆,𝗍)√(p)
    		(√(p𝗍)-√(𝗆))^2/3
    		(√(𝗍/p)-√(𝗆))^2/3/(𝗆𝗍)^1/6(1-p)
    		.




	
	Let the constants 𝖼(𝗆,𝗍) and 𝗏_1,2(𝗆,𝗍) be
	given in <Ref>. Then
	
		
  * The law of large numbers (<ref>) holds for all
			𝗆,𝗍>0.
		
  * The Tracy–Widom fluctuation result (<ref>) holds
			under the condition 𝗍>𝗆/p>0.
	

The proof of <Ref> is given in <Ref>.



§ LIMIT BEHAVIOR OF GROTHENDIECK RANDOM PERMUTATIONS





 §.§ Law of large numbers for the permutation height function



In this section, we obtain the asymptotic behavior
of Grothendieck random permutations of growing order
via the
the pre-limit identity in distribution (<Ref>)
and
TASEP asymptotics (<Ref>).
As a result, we complete the proof of <Ref>
formulated in the Introduction.
First, we focus on
computations involving the law of large numbers, and
then add the fluctuation terms.
By <Ref>, we have

    _𝐰(H(x,y)≤ h)
    	=
    	_TASEP(
    		ξ_n-x+1-h(y-1)≥ n-y+2
    	),

where H is the height function of the Grothendieck random
permutation 𝐰, and
ξ_i(t) are the TASEP particles
with
the step initial configuration
ξ_i(0)=n-x-i+2, 1≤ i≤ n-x+1.
Therefore,

    _𝐰(H(x,y)≤ h)
    	=
    	_TASEP(
    		ξ̅_n-x+1-h(y-1)≥ n-y+1-h
    	),

where ξ̅_i(t)=ξ_i(t)-ξ_i(0) is the displacement.

Let h=⌊ n𝗁⌋, where 𝗁 is the
scaled permutation height function.
Let also
x=⌊ n𝗑⌋, y=⌊ n𝗒⌋,
where 𝗑,𝗒∈[0,1] are the scaled coordinates in the permutation matrix.
Note that
by (<ref>), we have
0≤𝗁≤min(1-𝗑,1-𝗒).

For each 𝗑,𝗒,𝗁, the
asymptotic location
of n^-1ξ̅_n-x+1-h(y-1) is determined
from the TASEP law of large numbers (<ref>), (<ref>).
Namely, this location is equal to

    𝖼(1-𝗑-𝗁,𝗒)=
    	(√(p𝗒)-√(1-𝗑-𝗁))^2/1-p1_p𝗒≥ 1-𝗑-𝗁
    	.

There are two cases.
If

    𝖼(1-𝗑-𝗁,𝗒)> 1-𝗒-𝗁,

then the probability (<ref>) goes to 1.
If the inequality is reversed, then this probability goes to 0.
As 𝗁 increases,
the left- and the right-hand sides of (<ref>)
increase and decrease, respectively.
Therefore, we should define

    𝗁^∘(𝗑,𝗒)inf{𝗁∈[0,min(1-𝗑,1-𝗒)]𝖼(1-𝗑-𝗁,𝗒)> 1-𝗒-𝗁}.

If the set in (<ref>) is empty, we
take the maximal possible value,
𝗁^∘(𝗑,𝗒)=min(1-𝗑,1-𝗒).

To describe 𝗁^∘ more explicitly,
introduce the northeast part of the ellipse

    ℰ_p{
    		(𝗑,𝗒)
    		
    		(𝗒-𝗑)^2/p+(𝗒+𝗑-1)^2/(1-p)=1,		1-p≤𝗑≤ 1
    	}⊂ [0,1]^2.





	
	The function
	𝗁^∘
	(<ref>) is given by
	
    𝗁^∘(𝗑,𝗒)
    		=
    		
    			1-𝗒,   (𝗑,𝗒)∈𝒜{ 0<𝗑<1-p, 𝗑/(1-p)<𝗒<1 }
    			;
    
    			1-𝗑,   
    			(𝗑,𝗒)∈ℬ{ 0<𝗑<1, 0<𝗒<(1-p)𝗑}
    			;
    
    			1+
    			2/p√((1-p) 𝗑𝗒)
    			-𝗑+𝗒/p
    			,   (𝗑,𝗒)∈𝒞{.380<𝗑<1, (1-p)𝗑<𝗒<𝗑/(1-p),
    			
    (𝗑,𝗒)belowℰ_p}
    			;
    
    			0,   
    			(𝗑,𝗒)∈𝒟{(𝗑,𝗒) above ℰ_p}
    			.

	It
	is continuous on [0,1]^2.

See <Ref>, left, for an illustration
of the subsets 𝒜,ℬ,𝒞,𝒟
in (<ref>). The graph of the function
𝗁^∘ is given in
<Ref>, right.

	In 𝒜 and 𝒞,
	we have
	𝖼(1-𝗑-𝗁^∘,𝗒)= 1-𝗒-𝗁^∘.
	Solving this for 𝗁^∘ leads to the result.
	In 𝒟, we have
	𝖼(1-𝗑-𝗁,𝗒)> 1-𝗒-𝗁
	for all 𝗁≥0, so 𝗁^∘(𝗑,𝗒)=0.
	In ℬ,
	we have
	𝖼(1-𝗑-𝗁,𝗒)< 1-𝗒-𝗁
	for all 0≤𝗁≤ 1-𝗑.
	This means that
	the set over which we take the infimum in
	(<ref>) is empty.
	This implies that
	𝗁^∘(𝗑,𝗒)=
	1-𝗑 in ℬ, which
	completes the proof.


<Ref>
and identity (<ref>)
immediately imply the law of large numbers for the
height function:

    lim_n→∞n^-1
    	H
    	( ⌊ n𝗑⌋,
    	⌊ n𝗒⌋)
    	=𝗁^∘(𝗑,𝗒),
    	    𝗑,𝗒∈[0,1],

with convergence in probability.
This completes the proof of the first part of
<Ref>
from the Introduction.



 §.§ Properties of the limiting permuton



The law of large numbers
(<ref>)
implies that the Grothendieck random permutations 𝐰∈ S_n
converge, as n→∞, to a (deterministic) permuton
prescribed by
𝗁^∘(𝗑,𝗒).
Recall that a permuton is a probability measure
on [0,1]^2 with uniform marginals.
We refer to
<cit.>,
<cit.>,
<cit.> for detailed treatment of permutons.

In detail, the permuton is connected to the height function as

    𝗁^∘(𝗑,𝗒) =
    	( X>𝗑, Y>𝗒),
    	    𝗑,𝗒∈[0,1],

where (X,Y)∈[0,1]^2 is a random point distributed according to the permuton.
Note that both X and Y are uniformly distributed on [0,1],
while their joint distribution is nontrivial.
Indeed, from (<ref>)
if follows that
𝗁^∘(𝗑,0) = 1-𝗑,
𝗁^∘(0,𝗒) = 1-𝗒,
as it should be for uniform marginals of a permuton.

Let is find the part of the permuton
that is concentrated on
the curve
ℰ_p (<ref>)
(this concentration is visible in simulations, see <Ref>).
For small ε>0, we have

    𝗁^∘(𝗑,𝗒-ε)-𝗁^∘(𝗑,𝗒)
    	=( X>𝗑, 𝗒-ε<Y≤𝗒).

Dividing this by ε=(𝗒-ε<Y≤𝗒),
taking the limit as ε→0,
and passing to the complement event,
we obtain the conditional cumulative distribution function (cdf) of X given Y=𝗒:

    (X≤𝗑| Y = 𝗒)=
    	1+
    	∂/∂𝗒𝗁^∘(𝗑,𝗒)=
    	
    		0,    (𝗑,𝗒)∈𝒜;
    
    		1,    (𝗑,𝗒)∈ℬ∪𝒟;
    p-1p+√((1-p)𝗑𝗒)p𝗒,    (𝗑,𝗒)∈𝒞.

This function has a jump discontinuity along the ellipse ℰ_p (<ref>),
that is, at the point

    𝗑_p(𝗒)
    	
    	p+𝗒
    	-2 p 𝗒+
    	2 √((1-p)  (1-𝗒) p𝗒),
    	    𝗒>1-p.

We conclude:

	
	Let 𝗒>1-p.
	The unique discontinuity of the conditional distribution of X given Y=𝗒 is
	an atom
	at 𝗑=𝗑_p(𝗒), and the atom's mass is
	
    ( X = 𝗑_p(𝗒)| Y = 𝗒)=
    			1/p
    			-[
    				(1-p) (p+𝗒
    			-2 p 𝗒+
    			2 √((1-p)  (1-𝗒) p𝗒))
    	]^1/2/p√(𝗒),

	where
	1-p<𝗒≤ 1.


In particular, from (<ref>) we have
( X = 1-p| Y = 1)
=1, as it should be.


	
	While conditionally on Y=𝗒 (where 𝗒>1-p),
	the distribution of X has an atom,
	the permuton (that is, the joint distribution of X and Y)
	does not have atoms. This follows from the fact that
	𝗁^∘(𝗑,𝗒)
	(<ref>) is continuous on [0,1]^2.


As an application, let us consider the law of large numbers of the number of inversions
(𝐰)#{i<j𝐰(i)>𝐰(j)}, where
𝐰∈ S_n is the Grothendieck random permutation.
By a result of
<cit.> (see also Theorem 1 in the survey <cit.> or
<cit.>),
the convergence of 𝐰∈ S_n to a permuton
is equivalent to the convergence (in probability) of all pattern counting statistics
of 𝐰.
The limiting pattern counts are determined by the permuton.
In particular,
for inversions, we have

    lim_n→∞(𝐰)/n2=
    	γ_p
    	[ (X_1>X_2,  Y_1<Y_2)  or  (X_1<X_2,  Y_1>Y_2) ],
    	     p∈(0,1),

where (X_1,Y_1) and (X_2,Y_2) are independent
points of [0,1]^2
sampled from the limiting permuton.


	
	We have γ_1/2=1-π/4.

The constant γ_p for general p can be expressed
as a sum of a double and a single
integral of algebraic functions,
and we were not able to evaluate it in closed form except for p=1/2.
The integrals for general p are given in the proof below.

	Fix (𝗑,𝗒)∈[0,1]^2, and let (X,Y)∈[0,1]^2
	be a random point sampled from the permuton.
	We have
	
    [ (X>𝗑,  Y<𝗒)  or  (X<𝗑,  Y>𝗒) ]
    		=1-𝗁^∘(𝗑,𝗒)-
    		𝗁^∙(𝗑,𝗒),

	where 𝗁^∘(𝗑,𝗒)=(X>𝗑,  Y>𝗒) is given by
	(<ref>), and
	
    𝗁^∙(𝗑,𝗒)(X≤𝗑,  Y≤𝗒).

	In (<ref>),
	we used the absence of atoms
	(<Ref>) and
	ignored the difference between strict and weak inequalities.
	The function 𝗁^∙ can be computed using the
	conditional distribution (<ref>):
	
    𝗁^∙(𝗑,𝗒)=
    		∫_0^𝗒(X≤𝗑| Y = w) dw=
    		𝗑,    (𝗑,𝗒)∈𝒜;
    𝗒,    (𝗑,𝗒)∈ℬ;
    p-1/p(𝗑+𝗒)+
    			2/p√((1-p)𝗑𝗒),
    			    (𝗑,𝗒)∈𝒞;
    𝗑+𝗒-1,    (𝗑,𝗒)∈𝒟.


	By (<ref>),
	the constant γ_p (for general p) is given by
	
    γ_p=
    		𝔼[ 1 - 𝗁^∘(X,Y) - 𝗁^∙(X,Y) ],

	where the expectation is taken with respect to the permuton.
	To evaluate this expectation, we use the conditional density
	of X given Y=y which is obtained from
	(<ref>):
	
    f_X| Y=𝗒(𝗑)=
    		1/2p√(1-p/𝗑𝗒)1_(𝗑,𝗒)∈𝒞.

	Recalling that for Y=𝗒>1-p the conditional distribution of X has an atom,
	we can express the expectation as follows:
	
    γ_p=
    			1   -
    			∬_𝒞(
    				𝗁^∘(𝗑,𝗒)+𝗁^∙(𝗑,𝗒)
    			)
    			f_X| Y=𝗒(𝗑) d𝗑 d𝗒
       -
    			∫_1-p^1
    			( 𝗁^∘(𝗑_p(𝗒),𝗒)+
    				𝗁^∙(𝗑_p(𝗒),𝗒)_𝗑_p(𝗒)+
    				𝗒-1)
    			( X=𝗑_p(𝗒)| Y=𝗒) d𝗒.

	For p=1/2, a
	computer algebra system
	like 
	is able to find explicit anti-derivatives, and
	evaluate these integrals.
	We have:
	
    ∬_𝒞4  √(2𝗑𝗒) - 3 𝗑 - 3 𝗒 + 1/√(2𝗑𝗒)  d𝗑 d𝗒=π+2/8,

	and
	
    ∫_1/2^1
    		(
    			𝗒 + √((1 - 𝗒) 𝗒) - 12)
    		(
    		2 - 𝗒^-1√(𝗒+2 𝗒√((1-𝗒 ) 𝗒)))  d𝗒=π-2/8.

	Combining the last two integrals, we obtain γ_1/2=1-π/4.
	This completes the proof.




 §.§ Fluctuations of the permutation height function



Let us now apply the second part of
<Ref> to the Grothendieck random permutations, and
obtain fluctuations of the height function
around its limit shape 𝗁^∘(𝗑,𝗒).
Throughout this subsection, we assume that the point (𝗑,𝗒)
belongs to the curved triangle 𝒞
where the height function 𝗁^∘ is not linear.

Recall that by H(x,y) we denote the random height function of the
Grothendieck random permutation 𝐰∈ S_n.
By (<ref>), we have
for all r∈ℝ:

    _𝐰(
    			H(⌊ n𝗑⌋,⌊ n𝗒⌋)
    			≤
    			n𝗁^∘+ r n^1/3)
    		
       =
    		_TASEP(
    			ξ̅_⌊ n(1-𝗑-𝗁^∘-r n^1/3) ⌋
    				(⌊ n𝗒⌋ )
    				≥
    				n(1-𝗒-𝗁^∘-r n^1/3)
    		)+o(1),

where we used the shorthand notation

    𝗁^∘
    	=
    	𝗁^∘(𝗑,𝗒)
    	=
    	1+
    	2/p√((1-p) x y)
    	-x+y/p
    	.

The term o(1) in (<ref>)
is due to the fact that we removed
shifts by (+1), and combined the integer parts. These
modifications are negligible because fluctuations live on the scale n^1/3.

By the fluctuation result for TASEP,
the probability in the right-hand side of
(<ref>)
has the following behavior:

    lim_n→∞_TASEP(
    			ξ̅_⌊ n(1-𝗑-𝗁^∘-r n^1/3) ⌋
    				(⌊ n𝗒⌋ )
    				≥
    				n(1-𝗒-𝗁^∘-r n^1/3)
    			)
       =
    		F_2( r( 1/𝗏_1(1-𝗑-𝗁^∘,𝗒)
    		+
    		1/𝗏_2(1-𝗑-𝗁^∘,𝗒)) ),

where F_2 is the Tracy–Widom GUE cumulative distribution function,
and the constants 𝗏_1,𝗏_2 are given by
(<ref>).
We have

    1/𝗏_1(𝗆,𝗍)
    	+
    	1/𝗏_2(𝗆,𝗍)=
    	(√(𝗍)-√(𝗆 p)) 𝗍^1/6/𝗆^1/3
    	(√(𝗍/p)-√(𝗆))^2/3
    	(√(𝗍 p)-√(𝗆))^2/3.

Let us denote the above expression with
𝗆=1-𝗑-𝗁^∘(𝗑,𝗒)
and 𝗍=𝗒 by

    𝗏(𝗑,𝗒)(√(𝗒)-√(p(1-𝗑-𝗁^∘(𝗑,𝗒))))
    	𝗒^1/6/(1-𝗑-𝗁^∘(𝗑,𝗒))^1/3
    	(√(𝗒/p)-√(1-𝗑-𝗁^∘(𝗑,𝗒)))^2/3
    	(√(𝗒 p)-√(1-𝗑-𝗁^∘(𝗑,𝗒)))^2/3,

where 𝗁^∘(𝗑,𝗒) is given in (<ref>).
While it is not evident from (<ref>),
inside 𝒞
the function 𝗏(𝗑,𝗒)
is symmetric in 𝗑,𝗒.

With the constant 𝗏(𝗑,𝗒), the convergence
in (<ref>)
implies the desired
Tracy–Widom GUE fluctuations of the height function
H(x,y).
This completes the proof of
<Ref>.



§ RANDOM PERMUTATIONS FROM NON-REDUCED PIPE DREAMS



Here we consider a different family of random permutations
which is
obtained by sampling a non-reduced pipe dream and not resolving any double crossings.
This corresponds to setting q=1 in <Ref>
from the Introduction.
For shorter notation, throughout this section we denote these random permutations
𝐰^(1) by 𝐰.



 §.§ Exact formula



Consider the staircase shape
_n= { (i,j) i+j≤ n }
(where i and j are the row and column indices, respectively)
and
place tiles
-2pt
    < g r a p h i c s >

or
-2pt
    < g r a p h i c s >

in each box of _n independently with probabilities p and 1-p, respectively.
Let the random permutation 𝐰∈ S_n
is obtained by following the pipes starting at (i,0), and ending at (0,𝐰_i^-1),
i=1,…,n.
For example, for the pipe dream in <Ref>, this procedure gives
w=241635.

Observe that each pipe is a random walk in the bulk of
_n, equipped with the mandatory turns by
90^∘ at the elbows at the diagonal boundary i+j=n.
We begin by deriving an exact formula for the probability
distribution of 𝐰_i^-1, the outgoing position
of the i-th pipe. For general p, this distribution
involves two Gauss hypergeometric functions _2F_1, which
for p=1/2 simplifies to a sum of binomial
coefficients. We do not use these exact formulas for
asymptotic analysis, but instead, in
<Ref> below, we obtain bounds on
the expected number of inversions using the random walk
approach.






Recall the Gauss hypergeometric function

    _2F_1(a,b;c;z) = ∑_k≥ 0 (a)_k (b)_k/(c_k)z^k/k!,

where (a)_k = a(a+1)⋯ (a+k-1). Set

    F_1(i,j)   
    		(1-p)p^i+j-2_2F_1(-i+1,-j+1;1; (1/p-1)^2)
    		
       = ∑_k=0^∞i-1kj-1k (1-p)^2k+1 p^i+j-2-2k;
    
    		F_2(s,n)     n (1-p)^2p^n+s-3_2F_1(-s+2,-n+1,2, (1/p-1)^2 )
    		
       = ∑_k=0^∞s-2knk+1 (1-p)^2k+2 p^n+s-3-2k.

Note that both hypergeometric series terminate after finitely many terms.
For p=1/2, these functions simplify to binomial coefficients:

    F_1(i,j)|_p=1/2
    	=2^-i-j+1i+j-2j-1,
    	    
    	F_2(s,n)|_p=1/2
    	=2^-s-n+1n+s-2n-1.



	
	We have for all i,j=1,…,n:
	
    (𝐰^-1_i=j) = F_1(i,j) +
    			p^n 1_i+j=n+1
    			+
    			F_2(i+j-n,n)1_i+j>n+1.



    We proceed by induction on i.
		For i=1, we have
		
    (𝐰^-1_1=j) =
        
        (1-p) p^j-1,     if  1≤ j<n;
    
    		p^n-1,     if  j=n,

    which coincides with the proposed formula. Now let i>1.
		Let X_i^(n) denote the exit position of the pipe started at (i,0),
		assuming that all squares on the diagonal i+j=n+1 are elbows.
		By considering where this pipe crosses the first row,
		we can represent
		X_i^(n)=X_i-1^(n-1) + Y,
		where Y depends on X_i-1^(n-1) and has the following
		conditional distribution
		given X_i-1^(n-1):
		
    (Y=m| X_i-1^(n-1)) =
    			
    				p,    m=0;
    
    				(1-p)^2  p^m-1,    1≤ m< n-X_i-1^(n-1);
    
    				(1-p)  p^n-X_i-1^(n-1)-1,    m=n-X_i-1^(n-1).

		Since 𝐰^-1_i=X_i^(n),
		we can write
    
    (X_i-1^(n-1)=j)· p+∑_r< j(X_i-1^(n-1)=r) ·(𝐰^-1_i =j)
    					
       
    					= 
    					(1-p)^2 p^j-r-1,     j<n;
    ∑_r< n(X_i-1^(n-1)=r)· (1-p)  p^n-r-1,      j=n.

		Using the induction hypothesis,
		we know the distribution of X_i-1^(n-1)
		in terms of F_1 and F_2.
		Therefore, it remains to verify the recursion
		(<ref>)
		for the answer (<ref>).
		This is easily done by grouping binomial coefficients,
		summing over r, and using the hockey-stick identity.





 §.§ Estimates from random walks



We are interested in the asymptotics of the (expected) number of inversions
(𝐰) in the random permutation 𝐰 obtained from the non-reduced pipe dream model.
First, we use the following known bound in terms of a
displacement (disarray)
dis(w) ∑_i=1^n |i-w_i|
=
∑_i=1^n|i-w_i^-1|:


	For any permutation w, we have
	 
    1/2dis(w) ≤(w) ≤dis(w).


Taking the expectation of (<ref>) and using linearity,
we see that it suffices to understand the asymptotic behavior
of 𝔼[|i-𝐰_i^-1|] for all i.
In principle, the exact distribution of 𝐰_i^-1
from <Ref>
should allow us to compute this expectation.
However, this is not straightforward.
Instead, we use the random walk interpretation of each individual
pipe:


	Fix ε>0 and let i<(1-ε)n. Then
	
    𝔼[|i-𝐰_i^-1|]
    		=
    		√(4i/πp/1-p) + O(1),     n→∞.



	Fix i. Let J_i,J_i-1,…,J_1,J_0
	be the (random) column coordinates of the positions of the i-th pipe
	in row i,i-1,…,1,0.
	In particular, J_i=0 and J_0=𝐰^-1_i.
	For example, in the pipe dream in <Ref>, left, we have
	J_3=0, J_2=3, J_1=3, and J_0=5 for i=3.

	We aim to upper bound the probability that
	the pipe reaches the diagonal i+j=n first time at row k_0, that is,
	J_k_0=n-k_0. Denote this event by R_k_0,
	and observe that (R_a)≥(R_b) for a<b.
	Before the pipe reaches the diagonal,
	the horizontal displacements J_k-J_k+1, k_0<k<i-1, are iid (independent
	identically distributed), and are distributed as
	
    ( {J_k-J_k+1=m}∩ R_k_0)=p1_m=0
    		+
    		(1-p)^2p^m-11_m≥ 1,     m∈ℤ_≥ 0,
    		   k_0<k<i-1.

	One readily checks that the
	expectation and variance of the random variable in the right-hand side are equal to 1
	and 2p/1-p, respectively.
	We can represent
	
    J_k_0=
    		J_k_0-J_k_0+1+J_i-1+∑_k=k_0+1^i-2( J_k-J_k+1).

	For the event R_k_0 to occur, the above sum must be
	equal to n-k_0=i-k_0+(n-i)>i-k_0+n ε.
	Since the expectations of the iid summands are equal to 1,
	by a standard large deviation estimate, this probability
	is upper bounded by e^-c_ε n for suitable
	c_ε(k_0)>0.[Note that if i-k_0≪ n, the actual bound is even stronger than this, but we do not need this precision.]
	By monotonicity, we can choose these constants such that c_ε(k_0)≥ C_ε c_ε(0)>0
	for all k_0.
	Taking the union over all k_0, we see that the probability
	that a pipe started at i ever reaches the diagonal
	is exponentially small.

	Therefore, J_0=𝐰^-1_i
	is close (with exponentially small error in probability)
	to a sum of i iid random variables
	J_k-J_k+1.
	By the Central Limit Theorem,
	this sum is approximately normal with mean i and variance
	2ip/(1-p). Subtracting i from 𝐰^-1_i, taking the absolute value,
	and using the fact that 𝔼( |Z| )=√(2/π)
	for standard normal Z, we obtain the result.


We can now bound the expected number of inversions in 𝐰 on the scale
n^3/2:

	
	Fix p∈[0,1).
	For every ε
		>0 and sufficiently large n, we have
		
    2/3√(π) (1-ε)  n^3/2√(p/1-p)≤𝔼[(𝐰)] ≤4/3√(π) (1+ε)  n^3/2√(p/1-p).



		For p=0, the permutation is identity with probability 1,
		so the bounds
		(<ref>) hold trivially.

    Let p>0 and let us calculate 𝔼[dis(𝐰)] to apply <Ref>.
		With <Ref> for i< (1-ε)^2/3n, we obtain
		for large n:
    
    𝔼[dis(𝐰)] =∑_i=1^n 𝔼[|𝐰_i^-1-i|] ≥∑_i=1^(1-ε)^2/3n2/√(π)√(p/1-p)√(i)≥2/√(π)√(p/1-p)2/3 n√(n) (1-ε),

    where the last inequality is a simple Riemann sum approximation. The upper bound follows similarly, this time extending the Riemann sum to n.


Numerical simulations suggest the following behavior of the
number of inversions:

	
	For any p∈[0,1), we have convergence in probability:
	
    lim_n→∞(𝐰)/n^3/2 = ϰ√(p/1-p).


Note that the bounds
2/(3√(π))≈ 0.376 and 4/(3√(π))≈ 0.752
in <Ref>
are at the same time bounds on ϰ.
Simulations show that ϰ is close to 0.5, but is not exactly equal to it.

The fact that the number of inversions lives on scale
n^3/2
implies a trivial permuton limit of 𝐰.
Denote
by id the
deterministic identity permuton
supported on the diagonal of [0,1]^2.


	
	For any fixed p∈ [0,1), the random permutations 𝐰∈ S_n converge in distribution
	to id as n→∞.


	For any τ∈ S_k and w∈ S_n, k< n, denote by
	t(τ,w)
	<cit.>
	the relative frequency of
	the pattern τ in w.
	This is simply the number of times the pattern τ appears in w,
	divided by nk.
	In particular, t( 21, w)=(w)/ n2.
	We have for any 2<ℓ < n
	<cit.>:
	
    t(21, w) = ∑_τ∈ S_ℓt(21, τ) t(τ, w).

	The term in the sum with τ=id vanishes since t(21, id)=0.
	For our random permutations
	𝐰∈ S_n, the left-hand side of (<ref>)
	converges to zero in probability by <Ref>
	and Markov inequality:
	
    (
    			(𝐰) ≥δ n^2)≤ C_δ n^-1/2→ 0,     n→∞

	for all δ>0.
	For a fixed ℓ, this implies that
	t(τ,𝐰)→0,
	n→∞,
	in probability
	for all idτ∈ S_ℓ, ℓ≥3. We see that all relative frequencies
	converge to zero except for t(id, 𝐰)→ 1.
	Therefore,
	by
	<cit.>,
	we get the result.


This completes the proof of <Ref> from the Introduction.



§ MAXIMAL PRINCIPAL SPECIALIZATIONS OF GROTHENDIECK POLYNOMIALS





 §.§ Grothendieck polynomials via bumpless pipe dreams



In this subsection, we outline another
combinatorial model for Schubert and Grothendieck polynomials
based on bumpless pipe dreams. Some of these definitions and results
go back to
<cit.>, and are described in detail in
<cit.> and <cit.>.

A bumpless pipe dream is a
tiling D of the n× n square
with six types of tiles:

    SE bump  -2pt
        < g r a p h i c s >
    ,   NW bump  -2pt
        < g r a p h i c s >
    ,   cross  -2pt
        < g r a p h i c s >
    ,   empty  -2pt
        < g r a p h i c s >
    ,   horizontal  -2pt
        < g r a p h i c s >
    ,   vertical  -2pt
        < g r a p h i c s >
    .

These tiles form a set of pipes labeled 1,…,n going
from the bottom boundary (left to right) to the right
boundary. See <Ref>, left, for an illustration.
We denote the set of all bumpless pipe dreams of size n by (n).


	
	Bumpless pipe dreams are the same as
	configurations
	of the six-vertex model in the n× n square
	with domain wall boundary conditions (see <cit.> or <cit.>).
	We refer to <cit.>
	or <cit.> for background and many properties
	on the six-vertex model. This object is also sometimes called
	the oscullating lattice paths model <cit.>.

	The same configurations can be identified with
	alternating sign matrices (ASMs) of size n,
	see <cit.> for a detailed exposition,
	and <cit.>, <cit.> for applications
	of the six-vertex model to the enumeration of ASMs.






	
	A permutation w(D) is associated to a bumpless pipe dream D
	by tracing the pipes from left to right, starting from the bottom boundary,
	and ignoring the second and subsequent crossings of each pair of pipes.
	See <Ref>, right, for a reading of the
	bumpless pipe dream in the left part of the figure
	which leads to w(D). The local rules for labeling the
	tiles in a bumpless pipe dream
	(which produce the permutation w(D))
	are given in
	<Ref>.
	Denote by (w)⊂(n) the set of bumpless pipe dreams
	whose associated permutation is w.





	A bumpless pipe dream is called reduced if no two pipes
	cross twice.
	Denote the set of reduced bumpless pipe dreams of size n by (n),
	and by (w)⊂(n) the subset
	whose associated permutation is w.


For a bumpless pipe dream D (reduced or not), denote by
(D) and (D) the set of NW bumps and empty tiles in D, respectively
(see (<ref>) for the notation).
For a tile in a bumpless pipe dream, we denote its
horizontal and vertical coordinates by (k,l), 1≤ k,l≤ n.
Here k increases from left to right, and l increases from top to bottom.

 
	For any permutation w∈ S_n, we have
	
    𝔊^β_w(x_1,…,x_n ) = β^-ℓ(w)∑_D ∈(w)∏_(k,l)∈(D)β x_k ∏_(k,l)∈(D) (1+β x_k).

	In particular, for β=0, we have
	for the Schubert polynomials:
	
    𝔖_w(x_1,…,x_n ) = ∑_D∈(w)∏_(k,l)∈(D) x_k.




 
	Recall the notation (<ref>).
	We have
	
    v_n(β)    =   ∑_D ∈(n)β^#(D)-ℓ(w(D)), 
       =∑_P ∈(n)β^#(P)-ℓ(w(P))(1+β)^#(P),

	where w(D) and w(P) are the
	permutations associated to the pipe dream D and bumpless pipe dream P, respectively.


	These identities follow from the formulas for Grothendieck polynomials in <Ref>.
For (<ref>), we immediately get

    v_n(β) = ∑_w 𝔊_w^β(1^n) = ∑_D ∈ PD(n)β^#(D)-ℓ(w(D)).

For (<ref>), we have

    v_n(β)= ∑_w 𝔊_w^β(1^n)    = ∑_P ∈ BPD(n)β^-ℓ(w(P))∑_D ∈(w)∏_(k,l)∈(D)β∏_(k,l)∈(D) (1+β )
       = ∑_P ∈ BPD(n)β^#(P)-ℓ(w(P))(1+β)^#(P).

This completes the proof.



Gao and Huang  <cit.> gave a bijection between reduced pipe dreams and reduced bumpless pipe dreams that preserves the corresponding weight of the diagrams in the corresponding expansions (<ref>) and (<ref>) of Schubert polynomials. This bijection was recently generalized to the case of pipe dreams and bumpless pipe dreams with marked NW-bumps and Grothendieck polynomials by Huang–Shimozono–Yu <cit.>.



 
By the product
formula for the number of alternating sign matrices of size n
<cit.>, <cit.>, we have
#(n) = ∏_k=0^n-1(3k+1)!/(n+k)!. Moreover,
since (n)⊂(n), we have

    u_n(0) ≤ v_n(0) ≤∏_k=0^n-1(3k+1)!/(n+k)!.

From the asymptotics of the number of AMSs of order n
(which is relatively straightforward using the product formula),
we conclude that

    lim sup_n→∞1/n^2log_2 u_n(0)   ≤ 0.37.

The asymptotic behavior of the number of bumpless pipe dreams
(six-vertex model configurations)
with periodic boundary is originally due to
Lieb <cit.>. The domain wall case
with general six-vertex parameters is treated
by Korepin–Zinn-Justin <cit.>.




 §.§ Asymptotics of Grothendieck random permutations via domino tilings



Consider a random bumpless pipe dream 𝐃∈(n)
with the distribution

    _2-ASM(𝐃=D) = 2^-n2 2^#(D),    
    	D∈(n).

Via the bijection with Alternating Sign Matrices
(<Ref>),
this distribution is equivalent to choosing each alternating
sign matrix of size n with probability
proportional to 2^#(number of -1's).
We call this distribution the 2-enumerated ASM model,
and it is equivalent (via a two-to-one bijection) to
the uniform probability distribution on
domino tilings of the Aztec diamond of size n-1.
This connection, as well as
the asymptotics of uniformly random domino tilings,
has been a subject of extensive research in
statistical mechanics and integrable probability,
see, for example,
<cit.>, <cit.>, and
<cit.>, <cit.>, <cit.>.

The results from the previous
<Ref> immediately imply:

	
	The Grothendieck random permutation 𝐰∈ S_n
	with the parameter p=1/2
	(see <Ref> for the definition)
	coincides with the permutation w(𝐃)
	associated
	(as in <Ref>)
	to the
	random bumpless pipe dream 𝐃∈(n)
	with the distribution (<ref>).



	
	For general p, an analog of <Ref>
	would involve a more complicated weighting of bumpless pipe dreams
	(six-vertex configurations). This weighting is still in the
	free-fermion family of six-vertex weights.
	At the level of domino tilings, the p-weighting corresponds to a
	deformation of the uniform model
	in which vertical and horizontal dominoes have different weights.
	This model is also well-studied,
	for example, see <cit.>.
	For simplicity, here we focus only on the case p=1/2.


Our asymptotic results for
Grothendieck random permutations (<Ref>)
readily
translate into results about the behavior of the permutations
w(𝐃) coming from the 2-enumerated ASM model.
On the other hand, the asymptotic behavior of the bumpless pipe dream 𝐃
in various regimes
has also been extensively studied,
and some of these results can be
brought back to Grothendieck random permutations.
Let us present just one example leading to asymptotically Gaussian fluctuations:


	
	Let 𝐰∈ S_n be the Grothendieck random permutation with p=1/2.
	Let 𝐰_n be the image of n under 𝐰.
	We have
	
    𝐰_n-n/2/√(n/4)d⟶𝒩(0,1),    
    		n→∞.

	Here 𝒩(0,1) is the standard normal distribution.


	Let
	𝐥_n be the position of the single NW bump in the
	rightmost column of 𝐃.  In
	<Ref>, we have
	𝐥_n=5.
	By <Ref>, 𝐰_n has the
	same distribution as 𝐥_n.
	In the language of domino tilings
	(see the map in, e.g., <cit.>), 𝐥_n is the position of the
	first particle in the particle process associated to the domino tiling.
	By
	<cit.>,
	𝐥_n satisfies a central limit theorem as in
	(<ref>). Note that our standard deviation
	√(n/4) matches the normalization in <cit.>
	as the latter work deals with GUE random matrices
	whose diagonal elements have variance 2.
	This completes the proof.



	
	<Ref> can also be proven directly by looking at the trajectory of the
	pipe of color n in the random pipe inside the staircase shape
	(<Ref>), as this trajectory is a random walk.




 §.§ Exact formulas for elementary layered permutations



We are now in a position to discuss principal specializations
of Grothendieck polynomials on layered permutations.
We begin with a few exact formulas, for which we need some notation.

Let 𝖢_n1/n+12nn denote the
nth Catalan number which counts the number of
Dyck paths of size n. Let _n denote the n-th
little Schröder number
<cit.> and
_n the n-th large Schröder number
<cit.>. It is
known that _n=2_n for n≥ 1.

The Narayana polynomial is defined as

    ℒ_n(x) ∑_d ∈Dyck(n) x^v(d),

where the sum is over Dyck paths of size n and v(d) is the number of valleys of the path d. The coefficients of ℒ_n(x) are the Narayana numbers N(n,k)=1/knknk-1 <cit.>. Note that 𝖢_n=ℒ_n(1) and s_n=ℒ_n(2). The polynomials ℒ_n(x) have the following generating function:

    ∑_n≥ 0 y^n ℒ_n(x) = 1-y(1+x)-√((1-y(1+x))^2-4xy^2)/2y.



Consider the following
elementary layered permutation
(see also (<ref>) for notation):

    w_0(k;n)id_k × w_0(n) = (1,…,k, k+n,k+n-1,…,k+1)
    	 ∈ S_k+n.

Recall the notation Υ_w(β)
(<ref>).
A
formula due to Fomin and Kirilov <cit.> relates the
Schubert principal specialization Υ_w_0(a;b)(0)
to Proctor's formula <cit.> counting bounded
plane partitions of staircase shape:


	
	For nonnegative integers k and n, we have
    
    Υ_w_0(k;n)(0) = 𝔖_w_0(k;n)(1,…,1)  = [𝖢_n-2+i+j]_i,j=1^k  = ∏_1≤ i<j≤ n2k+i+j-1/i+j-1.



In <cit.>, an analogous formula for
the β=1
Grothendieck polynomials is proven:


	
	For nonnegative integers k and n, we have
	
    Υ_w_0(k;n)(1) = 𝔊^β=1_w_0(k;n)(1,…,1) = 2^-k2[_n-2+i+j]_i,j=1^k
    		= 2^-k+12[_n-2+i+j]_i,j=1^k.



A general determinantal formula for all β may be derived
from <cit.>:
  
	For nonnegative integers k and n, we have

    Υ_w_0(k;n)(β)  =  (1+β)^-k2[ℒ_n+i+j-2(1+β)]_i,j=1^k.



The determinants
in
<Ref>
can be efficiently computed via the Dodgson condensation (the Desnanot–Jacobi identity).
We use it to get the numerical data for β=1
presented in
<Ref>.



 §.§ Asymptotically maximal specializations for layered permutations



Here we prove that principal
specializations of β=1
Grothendieck polynomials on layered permutations
(see (<ref>) for the definition)
attain the asymptotically largest
value. That is, we prove <Ref> from the Introduction.


Consider the compositions b^(n) = (…, b_2, b_1) of n
such that

    b_1 + … + b_i-1 + (2 + √(2))b_i ≤ n  for  i = 1,2,…,
     while b_1 + … + b_i-1≤ n - 4,

and b^(n) has o(n) parts.
Then for the layered permutations w(b^(n)) we have

    lim_n →∞1/n^2log_2
    Υ_w(b^(n)) (1)
    = 1/2,

which is asymptotically maximal.



	In particular, one can take compositions to be geometric
	b_i ∼ (1 - α) α^i - 1 n, for any α∈ [1/√(2), 1).


In the rest of this <Ref>,
we prove <Ref>. First, observe that the Grothendieck
specializations enjoy the following basic factorization property:

    Let u ∈ S_k, w ∈ S_n. We have
    
    Υ_u × w(β) = Υ_u(β) ·Υ_id_k × w(β).




	This follows from properties of pipe dreams
	corresponding to permutations
	of the form u × w.
	See <cit.> or <cit.>.


Let us denote (recall the notation (<ref>))

    F(k, n) Υ_w_0(k;n)(1),      G(k,n)   2^k2· F(k,n)=
    	[_n-2+i+j]_i,j=1^k,

where the last equality follows from <Ref>.



An important step in obtaining the asymptotics of Υ_w(1) for layered permutations is to understand the behavior of the function F(k,n). A similar analysis <cit.> was possible in the Schubert case (β = 0) thanks to an explicit product formula (<Ref>). For other β's, we have determinantal (but not product) formulas (<Ref> and <Ref>). However, in the case β=1 we are able to asymptotically analyze these determinants via the correspondence with 2-enumerated ASMs.


Let
k ∈ (n/√(2), n]. We have

    log_2 F(k, n - k) = n^2/2 - k^2/2 - O(n),       n →∞.


In (<ref>) and throughout the proofs below in this subsection,
we assume that the constants in the O(·) notation are nonnegative.

	Let 𝐃∈ PD(n) be the
	random bumpless pipe dream with the
	distribution (<ref>).
	Then we have
	
    F(k,n-k)=2^ n2(w(𝐃)=w_0(k;n-k)).

	Let us first consider a different event,
	
    R_k,n-k{restriction of w(𝐃) to the last n-k entries equals
    				(n,n-1,…,n-k+1)}.

	That is, under R_k,n-k, the first k entries can be an arbitrary
	permutation of {1,…,k }, while the last n-k entries are in the fully reversed order.
	Geometrically, event (<ref>)
	means that the bumpless pipe dream 𝐃
	develops a right triangle of side n-k in the lower right corner
	whose diagonal boundary is made
	of the SE bumps -2pt
    < g r a p h i c s >
, and
	the rest of the triangle
	entirely consists of crosses -2pt
    < g r a p h i c s >
.
	See <Ref> for an illustration.



	The 2-enumerated ASM model is equivalent to uniform domino tilings of the Aztec diamond,
	which develops an arctic circle phenomenon <cit.>. In terms of bumpless pipe dreams, this means that the pipe configuration outside of the circle inscribed in the n× n square is nonrandom (frozen). The four frozen phases adjacent to the four vertices of the square are different, but the one in the lower right corner consists entirely of crosses.
	By the variational principle for domino tilings
	<cit.>,
	as long as our triangle of side n-k is outside the arctic circle
	(which is equivalent to n/√(2)<k≤ n), the condition
	in R_k,n-k does not lower the asymptotics of the probability on the
	scale e^-cn^2.
	Moreover,
	
    (R_k,n-k)=1-e^-O(n),      n→∞.

	Indeed, the complementary event to R_k,n-k
	is that there is at least one non-cross in the lower right triangle of side n-k.
	By <cit.>,
	this event has probability e^-O(n).

	Returning to F(k,n-k),
	using the factorization property
	(<Ref>) and the fact that v_k(1)=2^k2, we have
	
    2^ n2-O(n)∼∑_w∈ S_kΥ_w × w_0(n-k)(1)
    		=
    		∑_w∈ S_kΥ_w(1) · F(k,n-k)=
    		2^ k2· F(k,n-k).

	This implies the desired asymptotic equivalence.


	
	The recent work <cit.>
	provides asymptotics of
	log_2 F(k, n - k) for all k, not necessarily in the range k>n/√(2).
	For k<n/√(2), the leading term
	is strictly smaller than n^2/2,
	which agrees with the variational principle
	<cit.>.
	We do not need the regime k<n/√(2) for <Ref>.




Let us denote k_0  n and

    k_i  n - b_1 - … - b_i,
       i ≥ 1,

so that b_i = k_i-1 - k_i.
We have by <Ref> and (<ref>) that

    Υ_w(…, b_2, b_1)(1)    = Υ_w(…, b_3, b_2)(1) · F(n-b_1,b_1)  
       = ∏_i ≥ 1 F(n - b_1 - … - b_i, b_i) = ∏_i ≥ 1 F(k_i, k_i-1 - k_i).

From the inequalities on the b_i's, we get

    k_i/k_i-1 = n - b_1 - … - b_i/n - b_1 - … - b_i-1∈ (1/√(2), 1).

Hence by <Ref>, we have

    log_2 F(k_i, k_i-1 - k_i) = k_i-1^2/2 - k_i^2/2 - O(k_i-1).

Therefore,

    1/n^2log_2 Υ_w(…, b_2, b_1)(1,…, 1)    = ∑_i ≥ 11/n^2log_2 F(k_i, k_i-1 - k_i) 
       = ∑_i ≥ 11/n^2(k_i-1^2/2 - k_i^2/2 - O(k_i-1) ) 
    
       = 1/2 - 1/n^2 O(∑_i ≥ 1 k_i-1).


Since k_i-1 = O(n) and there are o(n) elements in our composition, we have

    ∑_i ≥ 1 k_i = o(n^2),  and so 1/n^2 O(∑_i ≥ 1 k_i-1) = o(1),








which completes the proof of <Ref>.






	
	Recall the notation
	(<ref>),
	(<ref>).
	<Ref> establishes that on layered permutations,
	the Grothendieck specializations Υ_w(1)
	reach their asymptotic maximum: u_n'(1)∼ 2^ n2.
	However, it is not clear whether
	the absolute maximum, u_n(1),
	is attained on layered permutations.
	Dennin
	<cit.> verified that for n≤ 9, one has
	u_n(1)=u_n'(1).

	We present numerical data of
	optimal (i.e., achieving the exact, non-asymptotic maximum in u_n'(1))
	layered
	permutations for n≤ 500 in
	<Ref>.
	We also comment on why these optimal layered permutations
	have parts whose behavior does not fall under the assumptions of <Ref>.




 §.§ Bounds on the constant for Grothendieck specializations with general β



In <cit.>, the authors conjectured that for β fixed there exists a limit

    c(β)lim_n→∞1/n^2log_2 υ_n(β).

If the limit exists, it must satisfy
for β≥ 0:

    1/4log_2(2+β) ≤ c(β) ≤1/2log_2(2+β),

see <cit.>.
The limit
(<ref>)
exists when β=1, see (<ref>),
and we have c(1)=1/2.
Here we
improve the bounds (<ref>) in certain regimes:



 
	If the limit c(β) (<ref>) exists, then it must satisfy the following bounds.
	For 0 < β≤ 1, we have
	
    1/4max{log_2(2+β), 2log_2(1+β)}≤ c(β) ≤1/2min{log_2(2+β), log_2(1+1/β)}.

For β≥ 1, we have

    1/4max{log_2(2+β), 2log_2(1+1/β)}≤ c(β) ≤1/2log_2(1+β).

				See <Ref> for an illustration of the bounds.


For 0<β≤ 1 and
D∈(n), we can bound 1 ≤β^-ℓ(w(D))≤β^-n2.
Thus by (<ref>), we have

    ∑_D ∈(n)β^#(D)≤ v_n(β) ≤β^-n2∑_D ∈(n)β^#(D),

which leads to
(1+β)^n2≤ v_n(β) ≤(1+β/β)^n2.
Taking the logarithms and comparing with (<ref>) gives the result.

For β≥ 1 and D∈(n), let us estimate
β^-n2≤β^-ℓ(w(D))≤ 1. Thus by (<ref>), we have

    β^-n2∑_D ∈(n)β^#(D)≤ v_n(β) ≤∑_D ∈(n)β^#(D),

yielding
(1+β/β)^n2≤ v_n(β) ≤ (1+β)^n2,
which leads to the
desired bounds.








§ ADDITIONAL REMARKS AND OPEN PROBLEMS





 §.§ Optimal permutations achieving the exact maxima u_n(1) and u_n(β)


This project was motivated by Stanley's original question
from <cit.> on determining the family of
permutations w for which the Schubert specialization 𝔖_w(1^n) is
asymptotically maximal. This question as well as the
asymptotic value remain widely open (see <cit.> for data up to n≤ 10).
Dennin <cit.> extended this
question to Grothendieck specializations 𝔊_w^β=1(1^n),
but one can ask the same question for general β.
Presumably, answering this
question would give rise to a family of optimal permutations
w^β, which for β=0 would answer the original
Schubert specialization question.

Among the candidates for
such a family w^β are the layered permutations
<cit.>,
<cit.>
(see
<Ref> for numerical evidence for β=1).
When β=1,
Dennin verified that
layered permutations indeed achieve the maximum in u_n(1) for n≤ 9.
Our results of
<Ref> suggest that layered
permutations are indeed good candidates
as they at least attain the asymptotic maximum in u_n(1).

To obtain the asymptotics of Grothendieck specializations
on layered permutations for general β, one
has to study the asymptotics of the following Hankel
determinants:

    F(k,n,β)(1+β)^-k2[ℒ_n+i+j-2(1+β)]_i,j=1^k,

where ℒ_n(x) is the Narayana polynomial, see (<ref>)
and <Ref>.
For β=0, these determinants admit a product formula
and were analyzed in <cit.>.
We could access the special case β=1 due to the connection to
2-enumerated ASMs and domino tilings.
It would be interesting to find a deformation of the
domino tiling model related to the determinants (<ref>).



 §.§ Typical vs optimal permutations


One of our main results, <Ref>,
concerns “typical” Grothendieck
permutations.
By comparing the
permutation matrices in <Ref>, center,
and <Ref>, left, we see that
typical permutations are far from layered.
However, the
Shannon–McMillan–Breiman theorem (e.g., see
<cit.>)
suggests that probabilities of typical Grothendieck random permutations
also achieve the asymptotic maximum.
For our permutations, this is not yet a rigorous statement,
as one has to prove the asymptotic uniformity of the probability weights
of typical Grothendieck permutations.



 §.§ Local structure of Grothendieck permutations


What is the local structure of the Grothendieck random permutations?
In <Ref>, left and center, one
can observe a certain local layered structure.
Understanding this local structure goes beyond the permuton convergence.
Can the local structure be
understood in terms of descent statistics, major index, or
pattern statistics other than the overall pattern counts?
For instance, layered permutations are the
ones maximizing the packing density of the
pattern 132 <cit.>.
Potential tools
for this analysis may be found in
<cit.>. Alternatively,
one could translate the local behavior of random domino
tilings of the Aztec diamond <cit.>
to the local behavior of Grothendieck permutations.



 §.§ Random and maximal permutations for domain wall six-vertex model


The Bumpless Pipe Dreams provide a surprising connection
of Schubert calculus and K-theory to
the six-vertex model, one of the central objects in
statistical mechanics.
The analysis of typical and maximal permutations for Schubert
and Grothendieck specializations prompts the analogous questions
about the six-vertex model.

Namely, consider the six-vertex
model with
domain wall boundary conditions and
general Boltzmann weights a,b,c (for the 2-enumerated
ASMs, we have a=b=1 and c=√(2)).
Represent a random configuration of the six-vertex model
as a collection of pipes, and associate to it a random
permutation 𝐰 by resolving the intersections
as in
<Ref>.
What is the behavior of 𝔼[(𝐰)] and
how does it vary depending on the weights of the six-vertex
model? Do the permutations 𝐰 have a permuton limit?



 §.§ q-analogues and generalized principal specialization


Zhang <cit.> extended the analysis in <cit.> to the generalized principal Schubert specialization 𝔖_w(1,q,q^2,…), when q is a  root of unity. The analogue of layered permutations is the multi-layered ones <cit.>. It would be interesting to extend our results to the Grothendieck specializations
	𝔊^β_w(1,q,q^2,…) at a root of unity.
	However, having q complex makes connections to probabilistic models less clear.







§ DATA FOR MAXIMAL SPECIALIZATIONS ON LAYERED PERMUTATIONS





 §.§ Dodgson condensation and data


Principal specializations Υ_w(b)(β) of
Grothendieck polynomials with general β on layered permutations
(see <Ref> for notation)
can be maximized using dynamic programming, similarly to how this is done
for Schubert specializations in <cit.>.
To implement it, one needs an efficient way to compute the
determinants
D_β(n,k)[ℒ_n+i+j-2(1+β)]_i,j=1^k
from
<Ref>.
A recursion
follows from
the Dodgson condensation
(Desnanot–Jacobi identity)
<cit.>, <cit.>:


	For k,n∈ℤ_≥0,
	we have

    D_β(n,k)  = 1/D_β(n+2,k-2)(D_β(n+2,k-1)D_β(n,k-1) - D_β(n+1,k-1)^2),

with initial conditions D_β(n,0)=1, D_β(n,1)=ℒ_n(1+β), and D_β(0,k)=(1+β)^k2.


	Let M(n,k)=[ℒ_n+i+j-2(1+β)]_i,j=1^k, so
	D_β(n,k)= M(n,k). Clearly,
	
    D_β(n,0)=1,    
    		D_β(n,1)=ℒ_n(1+β).

	For n=0, the
	permutation w_0(0;k)=12⋯ k is the identity, and so
	𝔊^β_w_0(0;k)=1. By
	<Ref>, we have

    1 = Υ_w_0(n,k)(β) = (1+β)^-k2[ℒ_i+j-2(1+β)]_i,j=1^k.

Thus, D_β(0,k)= M(0,k)=(1+β)^k2.

For 1≤ i,j,p,q≤ k, let M_i,j^p,q denote the matrix obtained from M
by deleting rows i and j and columns p and q.
Similarly, let M_i^p be obtained from M by deleting row i and column p.
By the Dodgson condensation, we have

    (M(n,k))(M(n,k)^1,k_1,k)  = (M(n,k)^1_1)(M(n,k)^k_k) - (M(n,k)^k_1)(M(n,k)^1_k).

Observe that

    M(n,k)^1,k_1,k   = M(n+2,k-2),   M(n,k)^1_1 = M(n+2,k-1),
       M(n,k)^k_k   =M(n,k-1),   M(n,k)^k_1=M(n,k)^1_k=M(n+1,k-1).

This implies the result.


The implementation of the recursion for D_β(n,k) combined with dynamic programming
yields the data in <Ref>.
This data agrees with the asymptotic behavior f(n)→1/2
obtained in <Ref>.







 §.§ Optimal layered permutations and asymptotic maximum



Observe that the optimal layered permutations given in
<Ref> do not fall into the
description of sequences given in <Ref>.
For example, for n=700, the numerically optimal layered permutation
has parts b^(700)_opt=(1, 3, 5, 9, 14, 21, 33, 49, 73, 107, 157, 228),
whereas a sequence b^(700) from <Ref>
would satisfy the bound b_1=n-k ≤ 700/(2 + √(2)) ≈
205 on its largest part.
The parts of b^(700)_opt are suspiciously close to
the Fibonacci sequence (see <Ref>).
Let us explain this discrepancy.

Optimal
layered permutations can be constructed inductively via the
optimization

    max_b_1 + b_2 + ⋯ = nΥ_w(…, b_2, b_1)(1) =
    max_kmax_b_2 + b_3+⋯ = k{Υ_w(…,b_3, b_2)(1) · F(k, n-k)
    },

where b_1 = n-k.
Assume that max_b_2 +b_3+ ⋯ = kΥ_w(…,
b_3,b_2)(1) = 2^ k2· g(k), where g(k) =
2^-o(k^2). To extract the first part b_1, we thus need to
find

    max_k
    	{ 2^ k2· g(k) · F(k, n-k) }
    	=
    	max_k {g(k)· G(k,n-k)},

see (<ref>) for the notation.
On the other hand, in the proof of <Ref> (in particular, see
<Ref>),
we estimated the asymptotic behavior of
F(k,n-k), and
found that the coefficient
by n^2 in log_2 F(k,n-k) does not depend on k as
long as k>n/√(2).
These two problems lead to different choices of k.
See also
the
illustration in <Ref>.





The discrepancy between optimal layered permutations and the ones appearing in
<Ref> should be of order o(n) in the large n limit. Indeed,
assume that k/n<1/√(2). Then, as in the proof of
<Ref> (see also <Ref>),
one can show that log_2 F(k,n-k)∼α n^2-k^2/2-O(n) for some
α<1/2, so
2^ k2 g(k)  F(k, n-k)∼ 2^α n^2-o(n^2),
which is not maximal.
We believe that the discrepancy of the part sizes
in optimal layered permutations is of order
o(n) as n→∞.



§ LIMIT SHAPE AND FLUCTUATIONS IN TASEP FROM CONTOUR INTEGRALS



In this section we compute the constants
𝖼,𝗏_1,𝗏_2
from <Ref>, and
show how to establish
<Ref> by an asymptotic analysis of the correlation kernel.
We omit tedious but standard
estimates,
as they are
very similar to those in <cit.>
or <cit.>.



 §.§ Fredholm determinant



Recall that ξ̅_m(t)=ξ_m(t)-ξ_m(0) is the displacement of the m-th particle
at time t.
By <Ref>,
we have

    _TASEP(ξ̅_m(t)≥ u)=
    	(λ_m≥ u),

where λ_m is a point of the determinantal point process
X(λ)={λ_j+m-j }_j=1^m⊂ℤ_≥0 with the correlation kernel
(<ref>):

    K(u_1,u_2)=1/(2π𝐢)^2dz  dw/z-ww^u_2/z^u_1+1( z-p/w-p)^m( 1-w/1-z)^t,
    	    
    	u_1,u_2∈ℤ_≥0,

where the contours are positively oriented simple closed curves satisfying
p<|w|<|z|<1.

	
	The probability
	in the right-hand side of
	(<ref>) is equal to the Fredholm determinant
	of the kernel K on {0,1,…,u-1}:
	
    (λ_m≥ u)=
    		[ Id-K ]_≤ u-1
    		1+∑_ℓ=1^∞(-1)^ℓ/ℓ!∑_i_1,…,i_ℓ=0^u-1[ K(i_a,i_b) ]_a,b=1^ℓ,

	where Id is the identity operator.


	The event {λ_m≥ u} means that in the determinantal
	point process X(λ), no particles
	are located in {0,1,…,u-1 }.
	It is a well-known property of determinantal point processes that this
	probability is given by a Fredholm determinant, see e.g.
	<cit.>.
	This statement can be traced back to the inclusion–exclusion principle
	(e.g., see <cit.>).
	Note that the sum over ℓ in (<ref>) is finite, as
	for u>ℓ, the ℓ×ℓ determinant must have identical rows.
	Further details on Fredholm determinants may be found in
	<cit.> or <cit.>.


Let us assume that
m=⌊ L𝗆⌋,
t=⌊ L𝗍⌋,
u=⌊ L𝗎⌋,
where L→∞. By <Ref>,
it suffices to consider the case 𝗍≥𝗆/p,
which we assume throughout the present
<Ref>.



 §.§ Steepest descent and law of large numbers



The asymptotic analysis of K(u_1,u_2) is done by
steepest descent (see, e.g., <cit.> for an accessible introduction).
The integrand in (<ref>) has the form

    1/z-ww^u_2/z^u_1+1( z-p/w-p)^m( 1-w/1-z)^t
    	=exp{ L (
    	S(z;u_1/L)
    	-S(w;u_2/L)
    	)
    	+O(1)
    	}/z(z-w),

where

    S(z;𝗎)
    	-𝗎log z+
    	𝗆log(z-p)
    	-𝗍log(1-z).

The term O(1)
arises from dropping integer parts in m and t.
It is
negligible in the limit because the determinantal process
will be scaled from discrete to continuous space.

The point λ_m is at the left edge of the determinantal process.
Therefore, to catch its asymptotic location 𝗎=𝖼(𝗆,𝗍),
we need to find a double critical point of the function
z↦ S(z;𝗎).


	
	Let us explain the need for the double critical point
	in more detail.
	By (<ref>), we have
	K(u,u)=(u∈ X(λ)), which is the density
	of the random point configuration X(λ).
	The equation S'(z;𝗎)=0 for single critical points
	is quadratic, and its discriminant is
	a function of p and our parameters (𝗆,𝗍,𝗎).
	Fox fixed 𝗆 (viewed as a parameter of the determinantal
	process and not the scaled index on the last part λ_m),
	there are three possibilities depending on (𝗍,𝗎):
	
		
  * If the discriminant is negative,
			then
			K(u,u) converges to a value strictly between 0 and 1. Therefore,
			at the scaled time 𝗍 around the scaled location
			𝗎 there is a random configuration of points from X(λ)
			of density strictly between zero and one.
		
  * If the discriminant is positive, then
			K(u,u) converges to 0 or 1,
			and around 𝗎 there is either a densely packed, or an empty
			region of points from X(λ).
		
  * The case of zero discriminant (double critical points)
			is at the transition between the two cases, and
			thus
			corresponds to the edge of the random configuration X(λ).
	


Solving the double critical point equations S'(z;𝗎)=S”(z;𝗎)=0 in
z,𝗎, we find

    z_cr=
    	√(p𝗍)-√(𝗆)/√(𝗍/p)-√(𝗆)
    	,    𝗎_cr=
    	(√(p𝗍)-√(𝗆))^2/1-p.

Here out of two solutions to quadratic equations, we picked
the one with the smaller 𝗎_cr which corresponds to the left edge of X(λ).
Note that the solution 𝗎_cr in (<ref>) is the same as
the constant 𝖼(𝗆,𝗍)
from <Ref>, which is not a coincidence.

Let us deform the integration contours in
the kernel (<ref>) to pass near the double critical point
z_cr. This point lies inside the interval (0,p), while
the original contours satisfy p<|w|<|z|<1.
Due to high multiplicity poles in the integrand,
the w contour should not be deformed through p, and
the z contour should not be deformed through 0 and 1.

We deform the contours as follows. First,
deform w such that the new w contour goes around p and passes to the right of z_cr.
Then, deform z through w, such that
the new z contour goes around 0 and passes to the left of z_cr.
Dragging z through w picks up a residue at z=w, which needs to be integrated over the new w contour.
However, taking the residue at z=w eliminated the pole at w=p, so the
new single integral is zero.
This implies that the contour deformation does not produce extra terms in the kernel.
See <Ref>
for an illustration.



To arrive at the law of large numbers λ_m/L →𝖼(𝗆,𝗍)
(recall that this is the same as 𝗎_cr),
it remains to show that changing 𝖼
by ε or (-ε) makes the Fredholm determinant (<ref>) go to 0 or 1,
respectively. To show convergence to 1, it suffices to
estimate |K(u_1,u_2)|≤ e^-δ L for
u_1,u_2≤ L (𝖼-ε), which follows from the steepest descent analysis.
To show convergence to 0,
it suffices to look at the density function K(u,u) for u around L(𝖼+ε).
In the limit, this density
is strictly positive, and so the event
λ_m≥ L (𝖼+ε) has probability tending to zero.
We omit further details.



 §.§ Asymptotic fluctuations



Consider the probabilities of fluctuations as in <Ref>:

    _TASEP( ξ̅_⌊ L𝗆- L^1/3α𝗏_1⌋
    		(⌊ L𝗍⌋ )≥
    	L𝖼 -
    	L^1/3β𝗏_2
    )= [
    		Id
    		-
    		K^L𝗍,
    		L𝗆 - L^1/3α𝗏_1]_≤ L𝖼-
    	L^1/3β𝗏_2 +o(1).

Assume that 𝗆 and 𝗍 are fixed
(recall that 𝖼,𝗏_1,𝗏_2 depend on them).
In the right-hand side of
(<ref>),
we dropped integer parts as
they do not affect the limit to continuous space.

One can show that in the
series
(<ref>)
for the
Fredholm determinant (<ref>),
terms where
i_a< L𝖼-s L^1/3 for at least one a=1,…,ℓ
(and s sufficiently large but finite), are negligible in the limit.
Thus,
it suffices to consider
K̃(ũ_1,ũ_2)
K^L𝗍,
L𝗆 - L^1/3α𝗏_1
(L𝖼+ũ_1 L^1/3,L𝖼+ũ_2 L^1/3),
where ũ_1,ũ_2∈ℝ.
This computation will also produce
the normalization
constants 𝗏_1 and 𝗏_2
for the Tracy–Widom GUE fluctuations.

In K̃,
deform the contours to pass near the double critical point
z_cr=z_cr(𝗆,𝗍) given by (<ref>).
The contribution outside a small neighborhood of z_cr vanishes in the limit.
In the neighborhood,
make a change of variables

    z=z_cr+L^-1/3z̃/ϱ,
    	    
    	w=z_cr+L^-1/3w̃/ϱ,

where ϱ>0 is to be determined.
The new contour z̃ goes from
e^-2π𝐢/3∞ to -1 to
e^2π𝐢/3∞,
and w̃ goes from
e^-π𝐢/3∞ to 0 to
e^π𝐢/3∞.
The function
(<ref>)
is Taylor expanded in L as

    L( S(z;𝖼+L^-2/3ũ_1 )-S(z_cr;𝖼+L^-2/3ũ_1 ) )
    		
       =
    		-z̃^3/3ϱ^3p^3/2/√(𝗆𝗍)(√(𝗍/p)-√(𝗆))^4/(1-p)^3 z_cr
    		+
    		z̃/ϱ z_cr( -ũ_1
    			+
    			α𝗏_1(√(p𝗍/𝗆)-1)/1-p).

Let us pick ϱ>0 such that the coefficient
by z̃^3 is (-1/3),
that is,

    ϱ=
    	√(p)/(1-p) z_cr(√(p𝗍)-√(𝗆))^2/3
    	(√(𝗍/p)-√(𝗆))^2/3/(𝗆𝗍)^1/6.

We also have
in the integrand:

    dz  dw/z(z-w)∼ -L^-1/3/ϱ z_crdz̃ dw̃/w̃- z̃.

Adding the difference
S(z_cr;𝖼+L^-2/3ũ_2 )-S(z_cr;𝖼+L^-2/3ũ_1 )
inside the exponent in the integrand (<ref>)
does not change the correlation kernel, since it is a
gauge transformation of the form
K̃(ũ_1,ũ_2)↦K̃(ũ_1,ũ_2)f(ũ_2)/f(ũ_1),
where f(·) is a nonvanishing function.
Thus, we have

    K̃(ũ_1,ũ_2)f(ũ_2)/f(ũ_1)∼
    	-L^-1/3/ϱ z_cr∫_e^-2π𝐢/3∞^e^2π𝐢/3∞
    	dz̃∫_e^π𝐢/3∞^e^-π𝐢/3∞
    	dw̃ exp[ w̃^3/3
    			-
    			z̃^3/3
    			+
    			z̃ U_1
    			-
    			w̃ U_2
    	]/w̃-z̃,

where

    U_1,21/ϱ z_cr(
    	-ũ_1,2
    		+
    		α𝗏_1(√(p𝗍/𝗆)-1)/1-p)
    	.

The integral in (<ref>) together with the minus sign from the prefactor
is equal to the celebrated Airy_2 kernel
𝖠(U_1,U_2)
(e.g., see <cit.>).
The Tracy–Widom GUE cumulative distribution function
is the Fredholm determinant of the Airy_2 kernel, that is,
F_2(r)=[Id-𝖠]_≥ r, r∈ℝ.

The additional prefactor (L^1/3ϱ z_cr)^-1
in (<ref>)
combined with the sums in
the pre-limit Fredholm determinant (<ref>)
turns them into Riemann integrals of the Airy_2 kernel.
The normalization by
ϱ z_cr
is absorbed by
a change of variables:

    1/L^1/3ϱ z_cr∑_i=⌊ L^1/3ũ⌋≤ -L^1/3β𝗏_2
    	g(ũ)
    	∼1/ϱ z_cr∫_-∞^-β𝗏_2
    	g(ũ) dũ=
    	∫_1/ϱ z_cr(
    		β𝗏_2
    		+
    		α𝗏_1(√(p𝗍/𝗆)-1)/1-p)
    	^+∞𝗀(U) dU.

Here we incorporated the bound
L𝖼- L^1/3β𝗏_2 from (<ref>)
into the bound for i, and shifted
all indices by L𝖼.
The function g(ũ) represents
terms like K(·, i_a)K(i_a,·) in the
Fredholm determinant (<ref>)
arising when expanding the individual ℓ×ℓ
determinants as sums over permutations.
Here ũ and i_a are related as
i_a=⌊ L^1/3ũ⌋.
We also set 𝗀(U)=𝗀(ũ).

We see that
the following
convergence of Fredholm determinants holds:

    lim_L→∞[
    		Id
    		-
    		K^L𝗍,
    		L𝗆 - L^1/3α𝗏]_≤ L𝖼
    	=
    	[
    		Id
    		-
    		𝖠]_≥ r,

where

    r=
    	1/ϱ z_cr(
    		β𝗏_2
    		+
    		α𝗏_1(√(p𝗍/𝗆)-1)/1-p).

We see that setting

    𝗏_1=
    		𝗏_1
    		(𝗆,𝗍)
    		   ϱ z_cr(1-p)/√(p𝗍/𝗆)-1
    		=
    		√(p)𝗆^1/3/𝗍^1/6(√(𝗍/p)-√(𝗆))^2/3/(√(p𝗍)-√(𝗆))^1/3
    		;
    𝗏_2=
    		𝗏_2
    		(𝗆,𝗍)
    		   ϱ z_cr
    		=
    		√(p)/(1-p)(√(p𝗍)-√(𝗆))^2/3
    		(√(𝗍/p)-√(𝗆))^2/3/(𝗆𝗍)^1/6

turns the right-hand side of (<ref>) into the Tracy–Widom GUE distribution F_2(α+β).
This completes the proof of <Ref>.





alpha





A. H. Morales, Université du Québec à Montréal, Montreal, QC, Canada and University of Massachusetts Amherst, Amherst, MA, USA

E-mail: 



G. Panova, University of Southern California, Los Angeles, CA, USA

E-mail: 



L. Petrov, University of Virginia, Charlottesville, VA, USA

E-mail: 



D. Yeliussizov, Kazakh–British Technical University, Almaty, Kazakhstan

E-mail: 













