Paper Title: Minimal Quantum Circuits for Simulating Fibonacci Anyons.



















































 

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    |ττ̅⟩1|τ1⟩t|1τ̅⟩|11⟩⟩1/ϕ1/√(ϕ)Department of Physics, University of Illinois at Urbana-Champaign,
    1110 West Green Street, Urbana, IL 61801, USAInstitute for Quantum Information,
    RWTH Aachen University, D-52056 Aachen, GermanyCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USARudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford, OX1 3PU, UKhormozi@bnl.govComputational Science Initiative, Brookhaven National Laboratory, Upton, NY 11973, USA§ ABSTRACT 
    The Fibonacci topological order is the prime candidate for the realization of universal topological quantum computation. We devise minimal quantum circuits to demonstrate the non-Abelian nature of the doubled Fibonacci topological order, as realized in the Levin-Wen string net model.  Our circuits effectively initialize the ground state, create excitations, twist and braid them, all in the smallest lattices possible. We further design methods to determine the fusion amplitudes and braiding phases of multiple excitations by carrying out a single qubit measurement.  We show that the fusion channels of the doubled Fibonacci model can be detected using only three qubits, twisting phases can be measured using five, and braiding can be demonstrated using nine qubits. These designs provide the simplest possible settings for demonstrating the properties of Fibonacci anyons and can be used as realistic blueprints for implementation on many modern quantum architectures.
    
    Minimal Quantum Circuits for Simulating Fibonacci Anyons
        Layla Hormozi
        August 5, 2024
    ========================================================
    
    
    
    § INTRODUCTION
    
    The emergence of topological order in interacting quantum systems is one of richest phenomena in modern condensed matter physics <cit.>. The possibility of creating long-range entangled states <cit.> and quasi-particle excitations with non-Abelian statistics has opened the door to new vistas in fault-tolerant quantum information processing, both in the form of (passively) error-tolerant topological quantum processors, and (active) topological quantum error correcting codes (QECCs) <cit.>. For almost three decades, this two-way relationship has been a wellspring of creative work, from quantum algorithms with super-polynomial speedups <cit.> to topological codes such as the surface code, which are among the most promising QECCs <cit.>. 
    
    In recent years, rapid progress in simulating topological phases has been achieved, and observation of Abelian statistics has been reported in a variety of platforms. This includes minimal implementations using optical qubits <cit.> to intermediate-scale experiments in superconducting qubits <cit.>. More recently, progress has been reported on quantum simulation of certain non-Abelian phases, including dislocation-induced non-Abelian excitations in Abelian states <cit.>, Majorana states <cit.>, as well as non-Abelian states, such as those with D_4 and S_3 topological order <cit.>.
    
    Despite this progress, some of the most interesting topological phases, particularly those which can be used for universal topological quantum computation and quantum error correction, have remained out of reach. Fortunately, rapid advances in the science and engineering of controllable arrays of synthetic qubits, in various platforms, are increasingly enabling more advanced digital quantum simulation. Given efficient circuits which prepare and manipulate the ground states and quasiparticle excitations of topological phases of matter, it is now possible to directly probe their fascinating properties on a variety of nascent quantum hardwares.
    
    
    
    
    
    In this article, we describe minimal quantum circuits which demonstrate the non-Abelian properties of Doubled Fibonacci (DFib) anyons, arising as excitations of Levin and Wen's string-net model <cit.>. Fibonacci topological order <cit.>, is particularly interesting, since it is the simplest candidate for universal topological quantum computing, having braiding operations which are universal and can be implemented fault tolerantly <cit.>. As suggested above, our approach is based on digital quantum simulation: applying a series of quantum gates which effectively project into the desired eigenstate of the string-net Hamiltonian. <cit.>. 
    
    In constructing these circuits, we identify the minimal settings in which the intrinsically “topological” properties  of Fibonacci anyons (i.e., the building blocks for topological quantum computation) can be verified. In particular, each property is realized and measured using the fewest number of qubits possible, and with circuits that are sufficiently shallow to be accessible on virtually any modern quantum hardware. Though our constructions generalize to larger system sizes, asymptotic scaling is not the focus of this paper. Rather, we aim to build a specific set of primitive subroutines to enable further progress in the experimental understanding of topological phases of matter on programmable quantum devices.
    
    
    This paper is organized as follows. In Sec. <ref>, we review the essential features of both the chiral Fibonacci and the achiral DFib models and establish the notation. In Sec. <ref>, we describe our approach to initialization and measurement of DFib anyons. In Sec. <ref>, we discuss the details of constructing minimal circuits for demonstrating fusion, twisting and braiding of DFib anyons in different settings. Our circuits use, three, five and nine qubits, respectively — the smallest number possible in each case. Section <ref> concludes and provides an outlook.
    
    
    
    
    
    
    § THE FIBONACCI TOPOLOGICAL ORDER
    
    
    In this section, we first briefly review the (chiral) Fibonacci anyon model (Fib), then describe the basics of the Levin-Wen string-net model <cit.> that realizes the (achiral) DFib topological order.  
    
    
    
     §.§ The Chiral Fibonacci Model
    
    The Fibonacci anyon model <cit.> can be described by a set of anyon types, here represented by 1 and τ, where the former represents the trivial (vacuum) state and the latter the only non-trivial anyon type, carrying a quantum number known as topological charge. Similar to other quantum numbers (e.g., spin or electric charge), there are rules for combining topological charge. In the case of Fibonacci anyons, these rules are: 
    
    τ× 1 = τ,
    
    1 × 1 = 1,
    τ×τ = 1 + τ.
    
    
    The first two equations mean that combining (fusing) any object of charge 1 or τ with the vacuum state 1 will not affect the charge of that object. The third equation means that combining  the topological charges of two objects with charge τ will result in either an object with trivial charge 1 or an object with charge τ.
    
    An important consequence of the Fibonacci fusion rule is that the Hilbert space of n Fibonacci anyons, each of charge τ, has a dimension which grows according to the Fibonacci sequence (hence the name). In the asymptotic limit, the dimensionality of the Hilbert space grows as ϕ^n, where
    
    
        ϕ = √(5) + 1/2
     
    
    is the golden ratio. Since ϕ sets the growth rate of Hilbert space in this model, it is referred to as the quantum dimension of Fibonacci anyons, d_τ = ϕ. The quantum dimension of the vacuum particle is unity d_1 = 1.
    
    The order of anyon fusion defines a choice of basis for the Hilbert space. For example, given three anyons in a row, we can choose to fuse the first two, then add the third one. Equivalently, we can choose to first fuse the last two, then add the first one. These two choices define two different bases for the three-dimensional Hilbert space of three anyons. These two bases can be mapped to each other by a unitary operation known as F operator, or F-move <cit.>, defined pictorially in Fig. <ref>(a). In its simplest non-trivial form, the F  operator has the following matrix representation:
    
    
    F ≡ F^ττ m_ττ m' = [    1/ϕ 1/√(ϕ); 1/√(ϕ)   -1/ϕ;        ],
    
    
    where m,m' ∈{1, τ}.
    
    
    In the quantum computing community, Fibonacci anyons are best known for the universality of their braiding: their braid operators are generators of SU(2), and hence can be used to approximate all single-qubit gates with arbitrary accuracy <cit.>. In general, the braid operator R^ij_m describes the phases resulting from a counterclockwise exchange of two anyons with topological charges i and j, such that the total charge of the system is m, as depicted in Fig. <ref>(b). The matrix representation of the non-trivial  braiding of two Fibonacci anyons with total charge m is as follows:
    
    
    R 
    ≡ R^ττ_m 
    =  [ e^-4π i/5         0;         0 e^+3π i/5;           ].
    
    The diagonal nature of this matrix indicates that braiding two anyons does not change their total charge. The first and second diagonal elements correspond to the phase acquired by braiding two Fibonacci anyons with total charges 1 and τ, respectively. These are the braiding phases for so-called “right-handed” Fibonacci anyons; we can define a separate left-handed Fibonacci model whose braiding phases are the complex conjugate of those above. 
    
    
    
    
    
    Closely related to braiding is the concept of twisting, or a counterclockwise rotation of  an anyon around itself by 2π, as shown in  Fig. <ref>(c). The action of twisting is encoded in the  topological twist operator, or topological spin, θ_i, which is related to the braid operator as follows, 
    
    θ_i = (R^i i_1)^-1.
    
    
    It follows that braiding and twisting are related to each other as,
    
    
    (R_m^ij)^2 = θ_mθ_i^-1θ_j^-1,
    
    
    provided that 
    i and j   fuse to m.
    Here  i,j,m∈{1, τ}, θ_1 = 1 and θ_τ = e^+4π i/5. A visual depiction of this relation is shown in Fig. <ref>(d).
    
    
    Another element of the Fibonacci anyon model that will be relevant to us is the modular S-matrix, which has the form,
    
    
    S 
    = 1/D[  1  ϕ;  ϕ -1;    ].
    
    
    Here D = √(1 + ϕ^2) is the so-called total quantum dimension of the Fibonacci model. This operator describes the process in which two pairs of τ particles are created out of vacuum, then two anyons from different pairs exchange twice (one making a full wrap around the other), and finally each re-annihilates with its respective partner, forming the so-called “Hopf link.’’ A visual description of this operator is also given in Fig. <ref>(e).
    
    Finally, braiding and fusion rules can be used to derive relations to resolve string crossings. A version of this relation, which will be used extensively in the next section, is given in Fig. <ref>(f).
    
    
    
    
    
    
    There is no known exactly solvable model for the realization of the chiral Fibonacci model. However, Levin and Wen's string-net construction provides a framework for the doubled (achiral) realization of all types of topological order, including DFib. This model can be thought of as two copies of the Fibonacci model with opposite chiralities, where fusion and 
    braiding rules apply independently to the right-handed τ and the left-handed τ̅ excitations. The resulting DFib topological order, Fib⊗Fib,  has four distinct anyon types: {1,τ}⊗{1, τ̅} = { 11, τ1, 1τ̅, ττ̅}. Our realization of the DFib topological order is based on Levin and Wen's original <cit.> and extended string-net models <cit.>. In what follows we will briefly describe the properties of this model. 
    
    
    
     §.§ String-net Realization of the DFib Model
    
    
    
    
    The Levin-Wen (LW) string-net model <cit.> is defined by a set of string types on the edges of a 2D trivalent lattice and a set of self-consistent rules for combining (fusing) different string types. These fusion rules  define the string-net Hilbert space. A family of exactly solvable Hamiltonians can be introduced in this Hilbert space, which realize all achiral topological phases, and can be thought of as a generalization of Kitaev's toric code [Technically,  a string net model can be defined for any “unitary fusion category”C, resulting in an  anyon theory which is the so-called Drinfeld center Z( C).]. These Hamiltonians have the following form, 
    
    
    H = -∑_v Q_v -∑_p B_p,
    
    
    where Q_v and B_p are mutually commuting vertex and plaquette projection operators. The former acts on the three edges surrounding the vertex v, while the latter acts on the edges surrounding the plaquette p. Fig. <ref>(a) shows these operators for a hexagonal lattice. The system is in its ground state when both the vertex and plaquette operators aresatisfied with eigenvalue +1.   We say an operator is  violated if it has eigenvalue 0 rather than +1, and this corresponds to an excited state of the system.
    
    
    The operators Q_v and B_p act on the space of “string types,” which can be encoded as multi-level spins, or qudits, in general. In this picture, the vertex operator enforces certain branching rules, consistent with the fusion rules of the strings. In particular, the branching rules decree that Q_v is always violated if a single non-trivial string ends at a vertex (an open string) [This means a single anyon cannot be created out of vacuum.]. Thus, non-trivial string types that satisfy Q_v always form closed configurations (loops or nets). We will refer to the space of states for which all Q_v's  are satisfied as the “string-net space.”
    
    
     
    The plaquette operator B_p effectively measures the topological charge of plaquette p and is satisfied when the plaquette contains trivial topological charge. This operator, which also controls the dynamics of string-nets, imposes further restrictions on the string-net space. Together with Q_v these operators favor a certain superposition of loops and nets, which defines the ground state of the string-net model. In the original Levin-Wen model, various excitations result from different combinations of Q_v and B_p violations.
    
    
    The DFib topological order <cit.> is a specific instance of the LW string-net model, with two string types, corresponding to {1,τ} in the chiral Fibonacci phase. Here we identify the string types 1 and τ with qubit states |0 and |1, respectively. We will sometimes refer to an edge with string type τ, or equivalently qubit state |1, as an activated edge. The vertex operators, Q_v, essentially enforce branching rules  that are consistent with the fusion rules of the Fibonacci topological order Eq. (<ref>). As a result, Q_v is only violated when a single incoming edge at vertex v is activated. 
    
    
    The plaquette operator in this case can be written as,
    
     
    B_p = 1/D^2(1 + ϕ B_p^τ),
    
    
    where the operator B_p^τ effectively adds a loop of string type τ to the plaquette p. In fact, B_p can be thought as of an operator that adds what is known as a vacuum loop, a weighted sum of vacuum and a τ loop, to the plaquette, which can then be absorbed into the surrounding edges using a series of F-moves [For a detailed description of the process of absorbing strings through lattice edges with a series of F-moves see the original work in <cit.>]. Fig. <ref> depicts  the vacuum loop and the effect of the B_p operator on plaquette p. 
    
    
    As was mentioned above, the DFib model  has four  anyon types, { 11, τ1, 1τ̅, ττ̅}. Here 11 indicates the ground state, corresponding to no topological charge, which results from satisfying all the Q_v and B_p operators. In the original string-net model, the excitations ττ̅, τ1 and 1τ̅ result from combinations of Q_v and B_p violations. 
    
     The ττ̅ excitation consists of four components: ττ̅_11, ττ̅_1τ, ττ̅_τ1 and ττ̅_ττ, all of which have the same topological charge <cit.>. Out of these, the ττ̅_11 component can be realized just by violating B_p, thus it can be realized in the original LW model without leaving the string-net space. This excitation is essentially achiral since the phases resulting from braiding τ  and τ̅ generally cancel each other. However, due to its multi-channel fusion properties, a restricted set of braiding phases can be observed for these anyons. We will discuss this in more detail in Sec. <ref>.
     
    
    
    To demonstrate the full braiding phases of the DFib model, we need to consider the chiral excitations τ 1 or 1 τ̅. These excitations possess braiding statistics that are essentially equivalent to that of chiral Fibonacci anyons (right- and left-handed, respectively) and can therefore realize universal quantum computation <cit.>.
    
    
    To realize these excitations without violating Q_v, we use the extended LW model, which is defined on a tailed lattice with two extra edges added to each plaquette <cit.>, as shown in Fig. <ref>(e-h). The addition of these tails will allow us to correct Q_v violations while preserving the excitations, thus effectively permitting the creation of chiral excitations within the string-net space.
    
    
    
    § INITIALIZATION AND CHARGE MEASUREMENT
    
    
    
    The string-net space is defined by satisfying the vertex operator Q_v on every vertex v.  We can project to this space by measuring this operator at every vertex and then correcting where an error is detected. 
    Relatively simple circuits for measuring the vertex operator and correcting the affected vertices have been introduced <cit.>. Here, we do not explicitly show the vertex measurement/correction circuits, but we assume all Q_v operators are satisfied before applying our circuits. 
    
    
    Projection to the ground state requires further satisfying B_p operators at every plaquette, thus selecting a particular superposition of string-nets, which is a highly entangled state. Since the plaquette operators are projectors, we can ensure they are satisfied by a projective measurement, though the measurement itself is very nontrivial, as we shall now see.
    
    We follow the method introduced in <cit.>, which is inspired by the idea of entanglement renormalization <cit.>. This method is based on the observation that the F-move acting on a given edge e, which effectively redraws the lattice at that edge, and the operator B_p, which effectively measures the  charge of plaquette p,  “commute" with each other in the following sense: 
    
    
    F_e B_p = B_p' F_e.
    
    
    This means that if we measure the charge of an n-sided plaquette p with operator B_p, then apply an F-move to an adjacent edge e to reduce the size of the plaquette, the result is the same as applying the F-move first, followed by B_p' applied to the reduced-size (n-1)-sided plaquette p' (see Fig. <ref>). An important consequence of this observation is that F-moves can be used to locally modify the lattice while preserving the ground state. 
    
    Following <cit.>, we use this fact to both initialize the plaquettes and also to measure their charge. The basic idea is that by repeatedly applying F-moves to the edges of an n-sided plaquette, we can reduce it to a single tadpole: a two-edge graph defined by a a closed edge forming the “head”  and a connecting edge forming the “tail”: 
    
    
        111.
    
    
    We can use the above rule to reduce any lattice to a series of tadpoles, which can be easily initialized to the 11 (B_p = 1) state using single qubit gates , then reverse the F-moves to return to the initial lattice, now in its ground state <cit.>. Similarly, to measure the charge of a plaquette, we can reduce it to a tadpole with the same procedure, then carry out single qubit measurements of the head and tail edges to determine the charge of the original plaquette. In what follows, we discuss how this notion can be  generalized to initialize and measure the charges of other states with anyon types τ1,  1τ̅ and ττ̅. We also generalize this notion to devise a procedure for measuring the total charges of multiple plaquettes. 
    
    
    
     §.§ Excitations
      Not only do F-moves preserve plaquettes satisfying B_p = 1, but the statement that F and B_p operators commute (Eq. (<ref>)) guarantees that this holds for all eigenstates of the LW Hamiltonian. We can use this fact to both create and measure the entire DFib spectrum. 
    
    As noted earlier, in the original string-net model with no tails, the only other anyon type that can be initialized with a tadpole, without leaving the string-net basis, is ττ̅. To realize the chiral excitations τ1 and 1τ̅ without violating the vertex operator, we use the tailed lattice <cit.> in which each plaquette is equipped with two extra edges realizing a tail (see Fig. <ref>).  Similar to the original tail-less plaquette, we can reduce the tailed plaquette using a series of F-moves. The result is a generalized tadpole where an  extra inward tail is added inside the head of the tadpole:
    
    1111,
    
    
    (see also Fig. <ref>(f)). This generalized tadpole is minimally represented by four qubits and  can be used to initialize all four anyon types of the DFib model, including the four components of the ττ̅ anyon. 
    
    
    
    
    Just as the ground state can be created by initializing tadpoles with a vacuum loop, excitations can be created by superimposing τ strings to the vacuum loops <cit.>. This process is shown in Fig. <ref>. If a string passes over a vacuum loop, it results in  a generalized tadpole which generates the τ 1 excitation. The crossings between the τ strings and the vacuum loops can be resolved by using the relation given in Fig. <ref>(f). 
    
    Likewise, if a string passes under a vacuum loop, it leads to a 1τ̅ generating (generalized) tadpole. If two strings pass over and under the vacuum loop, the result is a ττ̅ generating tadpole. Note that the end points of two strings passing over and under the vacuum loop can fuse in four possible ways, resulting in the four components of the ττ̅ anyon. In the tailed lattice, these τ strings must end on the added tails to avoid Q_v violations. These are the same tails that show up in the generalized tadpoles that generate the DFib spectrum, a summary of which is also given below,
    
     =  1/D0000 +  ϕ/D1100
    1 =  1/D0111 +  ξ^2/D1011 +   √(ϕ) ξ/D1111
    t =  1/D0111 +  ξ^-2/D1011 +   √(ϕ) ξ^-1/D1111
    _11 =  ϕ/D0000 -  1/D1100
    _1τ =  1110
    _τ1 =  1101
    _ττ =  √(ϕ)/D0111 +  √(ϕ)/D1011 +   1/D ϕ1111.
    
    
    
    Here the red (shaded) lines indicate  strings of type τ (or qubit state |1), while dotted  lines depict the type 1 strings (qubit state |0), and ξ = e^3π i/5. 
    
    
    
    
    
    
    
    Thus, to create a pair of excitations of any kind, we first reduce the lattice to a series of generalized tadpoles using  F-moves. Then, depending on the desired excitation, we initialize them according the rules given in Eqs. (<ref>)-(<ref>). Finally we return to the original lattice by reversing the F-moves.  The circuits for both initialization and measurement of these generalized tadpoles are shown in Fig. <ref>. The S gate used in the figure is the modular S-matrix (Eq. (<ref>)) while the U gate is defined as,
    
    
    U = [ ξ^-1/√(ϕ)     ξ^2/ϕ;   -ξ^-2/ϕ    ξ/√(ϕ);           ].
    
    
    
    Finally, note that given the spherical boundary conditions, the excitations are always created in pairs — all of the quantum numbers of all the excitations in the entire system must fuse to the identity.   Thus we cannot create a single plaquette excitation from the ground state without also creating its partner.   In Fig. <ref> (and also in Eqs. (<ref>)-(<ref>)) for simplicity, we show only half of the process, where a string is superimposed on a single vacuum loop inside the head of a (generalized) tadpole. In general, we need to form the tadpoles such that plaquettes that share a pair of excitations will be transformed to tadpoles that share tails, which can then be initialized together. We will discuss this in more detail the following section. 
    
    
    
    
    
    
    
     §.§ Multiple Plaquettes
     
    
    In the extended LW model defined on lattices with tails where excitations are created in the absence of Q_v violations, they are essentially created on plaquettes. Thus, to determine the fusion amplitudes resulting from combing several excitations, we need a method to measure the total charge of multiple plaquettes. We discuss this problem from two different perspectives, first by building projective operators that determine the charge of two (or more) plaquettes, then designing quantum circuits which allow us to measure the total charge of multiple plaquettes with a computational basis measurement.
    
    As was noted in Sec. <ref>, the action of the plaquette operator B_p is to determine the charge content of a plaquette (whether it is trivial or not) by inserting a vacuum loop inside the plaquette, and absorbing it through the edges by a sequence of F-moves (see Fig. <ref>). This idea  can be generalized to an operator that determines the charge content of two neighboring plaquettes by inserting  a vacuum loop inside the two plaquettes. However, in this case, there is a choice of whether to insert the loop over or under the dividing edge. It turns out that these two options lead to two different operators, which we call B^o_p_i p_j and B^u_p_i p_j, where the superscripts o and u refer to over and under respectively. A visualization of these operators is depicted in Fig. <ref>. 
    
    
    Satisfying the operator B^o_p_i p_j indicates that the total charge of the right-handed anyons 1 in the two plaquettes p_i and p_j is trivial. Likewise, satisfying  B^u_p_i p_j implies that the total charge of left-handed anyons t on the corresponding plaquettes is trivial. Satisfying both of these the operators B^o_p_i p_j and B^u_p_i p_j  indicates that the total charge content of the two plaquettes is trivial. Note that this can be true even if individual plaquette operators B_p_i and B_p_j are not satisfied. This is analogous to the case where the collective charge of two τ anyons can be trivial. 
    
    
    Similar to single plaquette operators B_p, the double plaquette operators defined above are also projection operators, hence  cannot be implemented directly as a  unitary quantum circuit but instead must be implemented as a (rather complicated) projective measurement. Here we design quantum circuits to directly measure the charge of multiple plaquettes. Our approach is to generalize the single plaquette strategy — initialization and charge measurement by first reducing to a  tadpole — to a charge measurement by reducing multiple plaquettes to a series of concentric circles with each circle tied to the next by a single edge, resembling a series of concentric generalized tadpoles. Examples of this process for small lattices are shown in Fig. <ref>. 
    
    The key idea is that, through a series of F-moves, we reduce one of the plaquettes in the original lattice to the central tadpole in the concentric circles basis. The next-outermost (generalized) tadpole then contains the total charges of the original plaquette and another plaquette, which is reduced to an annulus surrounding the central tadpole. Similarly, with more plaquettes, the outer tadpoles hold the charge of all the original plaquettes which are subsequently enclosed by those tadpoles. Since the system should be thought of as living on the surface of a sphere, the outermost ring can be viewed as a tadpole as much as the innermost ring. 
    
    
    By choosing different series of F-moves, we can select different concentric tadpole bases that correspond to different combinations of plaquettes and use it as a basis for both initialization and measurement of combined charges. As an example, Fig. <ref>(b) and (c) depict two different choices of basis, suitable for measuring the charges of plaquettes p_1 ∪ p_2 or p_3 ∪ p_2. [Note that, in general, different sequences of F-moves can be used to transform a lattice into different concentric tadpoles. The  number of these choices increases in the case of tailed lattices and sometimes there are multiple sequences of F-moves that lead to seemingly similar concentric tadpoles. However, in general different choices can lead to additional twists and sometimes braids. Thus, caution needs to be exercised when selecting the F-move sequence to avoid unwanted phases. ]
    
    The equivalence of this approach to the projective operators above can be understood by recalling the action of the two-plaquette operator B_p_i p_j, which inserts a vacuum loop inside the two plaquettes. By keeping track of the F-moves which modify the lattice into concentric tadpoles, we can see that this vacuum loop effectively covers the area around two concentric tadpoles. 
    
    
    
    
    
    
    
    
    
    
    
    § SIGNATURES OF NON-ABELIAN ANYONS IN MINIMAL LATTICES
    
    
    In this section, we provide an overview of the signatures of the DFib model that we investigate in various settings, postponing the details to the following sub-sections. The basic properties that we are interested in are fusion, braiding and twisting of various excitations within the DFib model using the minimum number of qubits. Fig. <ref> depicts these operations for (chiral) Fibonacci anyons. 
    
    We restrict our study to lattices with spherical boundary conditions. The simplest lattices that can be defined on a sphere are shown in Fig. <ref>. The smallest possible lattice  consists of  two plaquettes each forming a hemisphere with a single common edge forming the meridian (see Fig. <ref>(b)). This two-plaquette lattice can be initialized into ground state or host a pair of ττ̅ = ττ̅_11 excitations, with total charge 11. The former can be realized by by applying an S gate to the single qubit on the edge initialized at |0, while the latter by applying an S gate to the qubit on the edge initialized at |1, as can be seen in Fig. <ref>.
    
    The smallest non-trivial lattice for our purposes consists of three plaquettes, separated by three edges that meet at the two poles of the sphere. A 2D depiction of this lattice, which resembles the Greek letter Θ, is shown in Fig. <ref>(c). We will use this lattice to demonstrate the fusion of two  excitations (Fig. <ref>). 
    
    The next smallest lattice with spherical boundary conditions consists of six edges and four plaquettes and is equivalent to a tetrahedron, as shown in Fig. <ref>(d).  We will use this lattice to realize a different implementation of the  fusion, which reveals additional details about the  structure of fusion channels, and discuss the difference between the two cases. 
    
    To realize the braiding properties of the DFib model (see Fig. <ref>(b)) we need to create the chiral excitations 1 or t without violating the vertex operators Q_v. This can be done using the extended tailed lattice. The simplest tailed  lattices are shown in Fig. <ref>(e-h). These lattices have additional qubits to account for the addition of a tail to each plaquette. The simplest lattice on which the creation and braiding of two pairs of chiral anyons can be demonstrated is the tailed Θ lattice, which consists of nine qubits (see Fig. <ref>(g)).
    
    We will also briefly discuss a simple procedure for revealing the topological twist phase of a single 1 anyon, which is created as part of a pair on the simplest two-plaquette lattice, or a generalized tadpole, consisting of four qubits (Fig. <ref>(c)). A simple generalization of this procedure can then be used to verify the relation between braiding and twisting.
    
    
    A note on notation: For the remainder of the paper, we will use two different notations for our states, depending on the context. The first is the qubit basis, in which we simply refer to the states of the qubits (|0 or |1) that form a particular lattice. The second notation is used when considering the concentric tadpole basis (see Fig. <ref>), where  we use a tensor product of kets that represent the sectors of the DFib model corresponding to each tadpole in this basis. 
    
    For example, in the case of the Θ lattice where the concentric tadpole basis consists of two tadpoles (see Fig. <ref>(a)), we use the notation |inn.|out. to refer to the states of the innermost and outermost tadpoles. Likewise for the tetrahedron lattice, where the concentric tadpole basis consists of three tadpoles (See Fig. <ref>(b) and (c)), we use the notation |inn.|mid.|out., which again refer to specific anyon types of the DFib model represented by each of the innermost, middle and outermost tadpoles in this basis.    
    
    In what follows, we explain the details of how various excitations can be created, braided, twisted and measured in specific lattices, and 
    present the corresponding quantum circuits. The circuit representations of the various F-moves used in this section are given in Fig. <ref>.
    
    
    
    
    
     §.§ Fusion with Three Qubits: The Θ Lattice
    
    
    The simplest setting for detecting the fusion properties of the DFib model is to realize two pairs of  excitations on a minimal lattice consisting of three plaquettes on a sphere, or the Θ lattice. 
    The process of initialization and detection of fusion channels is shown in Fig. <ref>.
    
    
    Initialization:  We start by initializing two pairs of _11 excitations on plaquette pairs p_1, p_3 and p_2, p_3. This can be done by starting from a basis in which each of the plaquettes p_1 and p_2 are reduced to single tadpole, as shown by the trivial state  shown as |ψ_00 in Fig. <ref>. Here  qubits q_1 and q_3 form the heads of the two tadpoles and qubit q_2 serves as the common tail.   We then apply single-qubit gates to the two heads, resulting in the state |ψ_0, which in qubit basis has the form,
    
    
    |ψ_0   =    |q_1q_2q_3
       =   (ϕ|0 - |1/D)|0(ϕ|0 - |1/D).
    
    
    We then apply an F-move to qubit q_2  to restore the original Θ lattice basis, resulting in the state |ψ_1.  At this point, we have created two pairs of  excitations on plaquette pairs p_1, p_3 and p_2, p_3 in the Θ lattice, such that the total charges of plaquettes p_1∪ p_3 and p_2∪ p_3 correspond to the  sector of the DFib model. In other words, we have effectively pulled two  pairs of _11 excitations out of vacuum. 
    
    
    Measurement:  We expect the total charge of plaquettes p_1 ∪ p_2, (or equivalently the charge of plaquette p_3, due to the spherical boundary conditions) to be in a superposition state consistent with the fusion rules of the Fibonacci model. To read this total charge, we need to change the basis into concentric tadpoles (see also Fig. <ref>(a)) such that, either plaquette p_1 or plaquette p_2 will be enclosed by a tadpole, which is nested inside an annulus, containing the charge of the other plaquette. This can be done by applying an F-move, acting on qubit q_3, in |ψ_2 as shown in Fig. <ref> [This could also be done with an F_1 move resulting in a shape where p_2 is enclosed by the central tadpole.]. 
    
    As was discussed in Sec. <ref>, the outer tadpole, which encloses plaquettes p_1 and p_2, can now be measured to ascertain the total charge of p_1 ∪ p_2. Note that in this case this outer boundary is also equivalent to the boundary of the single tadpole that encloses plaquette p_3 (the remainder of the sphere) and contains the same charge as p_1 ∪ p_2. Note also that while we use the term fusion in referring to combining topological charge, we don't actually fuse excitations in plaquettes p_1 and p_2. We merely change the basis such that we can measure the superposition state that contains information on fusion amplitudes, should an actual fusion occur. 
    
    
    
    Using the notation |ψ_2  = |inn.|out., this state can be written as,
    
    
    |ψ_2   =   _11
       +   (_11_11 
    - _1τ_τ1),
    
    
    which represents the anyon types contained by the inner and outer tadpoles, as shown in Fig. <ref>. We see that the inner tadpole, which contains the charge of plaquette p_1, corresponds to a single  anyon in all three terms. This is consistent with our initialization, in which plaquettes p_1 and p_2 each contained a single  excitation. 
    
    Focusing on the outer tadpole, which contains the total charge of plaquettes p_1 ∪ p_2, the first term in |ψ_2 indicates trivial total charge with coefficient 1/ϕ, resulting in the state , while the second and third terms indicate that the total charge corresponds to the charge of a single  anyon with coefficient 1/√(ϕ). Thus, the total charge of of p_1∪ p_2, signifies the following cross fusion relation,
    
    × =  + ,
    
    
    i.e., only two fusion channels appear. We will return to this point at the end of this section.
    
    Note that while we started by initializing the _11 component of the  excitation, cross fusion results in additional components of the  excitation. This can be seen in the third term of |ψ_2 (see Fig. <ref> and Eq. (<ref>)) where the excitation _1τ1 appears in the inner tadpole and _τ1 in the outer one. Since the components of  contain the same charge, this is consistent with Eq. (<ref>), where the second term can contain different combinations of the components of .  These terms can also be obtained by resolving the loops on the right hand side of the equation shown in Fig. <ref>(a) followed by the F-move.
    
    To simplify the measurement of the fusion channels of p_1 ∪ p_2 in a quantum circuit, we apply a series of two-qubit gates to |ψ_2, which effectively maps it to a product state. These two-qubit gates are two controlled-S gates acting on qubits q_2 and q_3 and a controlled-L gate, acting on qubit q_3 where,
    
    
    L = [  1/√(ϕ)    -1/ϕ;    -1/ϕ -1/√(ϕ);         ],
    
    
    resulting in the state
    
    
    |ψ_3   =    |q_1q_2q_3
       =   (|0 + |1)|10.
    
    
    Thus, the information on the fusion channels of  p_1 ∪ p_2 is effectively transferred to qubit q_1 in this state. 
    
    To detect the fusion coefficients, we can perform statistical measurements on q_1 at this stage and observe the corresponding probability distribution. Alternatively, we can apply an F gate (Eq. (<ref>)) to q_1, then use the result as the control for a CNOT gate acting on a target ancilla qubit. This F gate acts as a “filter,” so that if q_1 is in the correct superposition, the F gate maps it to |0, which does not affect the ancilla qubit in the subsequent CNOT. Thus, with this protocol we can verify the coefficient of fusion by carrying out a single measurement of the ancilla qubit. 
    
    Alternatively, we can determine the state of  |ψ_2⟩ by first measuring the state of q_3.  If q_3 is found to be in the |1⟩ state, so should q_1 and q_2 (the last term in the expression for |ψ_2⟩ in the Fig  <ref>.c).  If q_3 is found to be in the |0⟩ state, then q_1 and q_2 can be tomographed, and we should find q_2 to be in the  |ττ⟩_11 state and q_1 to be in the |1 1 ⟩ + |ττ⟩_11 state, which directly reflects the fusion in Eq. <ref>. In cases below we will always favor read-out schemes that minimize the number of measurements, although other read-out schemes are certainly possible.
    
    
    To conclude this section, we return to the observation in  Eq. (<ref>), namely the fact that in the Θ lattice, the result of cross fusing two  excitations is two fusion channels, namely  and . This might look unexpected at first sight, since the two sets of chiral excitations in the DFib model (the right-handed 1 and the left-handed t) must fuse independently from each other, thus in principle we expect the combination of two  excitations to result in four fusion channels: , 1, t and . However, recall that 1 and t are created by passing a τ string over or under the lattice, respectively. On the Θ lattice, two excitations must share the same plaquette and for a single plaquette there is no way to distinguish between “over” and “under.” Thus only the achiral excitations can exist on this plaquette. Consequently,  the combined charge of p_1∪ p_2 lead to only two  fusion  channels:  and .
    
    
    
    
    
    
    
    
    
    
    
    
    
    To explore this idea further, we create two pairs of _11 excitations on the four plaquettes of a  tetrahedral lattice, consisting of six qubits, and measure the cross-fusion channels of opposite plaquettes. We expect to see four fusion channels in this case, including the chiral excitations. We will discuss this in the next section. 
    
    
    
     §.§ Fusion with Six Qubits: Tetrahedron Lattice
    
    
    The process of initialization and measurement of the fusion channels of two pairs of excitations on the four plaquettes of a tetrahedral lattice is shown in Fig. <ref>. 
    
    Initialization:  We start by initializing two pairs of  excitations out of vacuum on pairs of plaquettes p_1, p_2 and p_3, p_4 such that the total charge of each pair corresponds to . This can be done by starting from a basis in which plaquettes p_1 and p_2 form concentric tadpoles, enclosed by a third tadpole containing p_3. In Fig. <ref>, one such starting configuration is depicted as the basis for the state |ψ_00. We then create pairs of _11 excitations on plaquettes p_1, p_2 and p_3, p_4 by initializing the corresponding generalized tadpoles. 
    
    The innermost tadpole, which represents plaquette p_1, consists of qubits q_2 and q_5, forming its head and tail, respectively. This tadpole is initialized as, 
    
    
    |q_2 q_5 = ϕ|0 - |1/D|0.
    
    
    
    The middle (generalized) tadpole, which contains the information about total charges of p_1 ∪ p_2, consists of qubits q_1 and q_6,  together forming its head, and q_5 and q_3, which form its two tails. This tadpole is initialized as,
    
    
    |q_1 q_6 q_5 q_3 = |00 + ϕ |11/D|00
    
    Finally, the outermost tadpole, containing the charge of p_1 ∪ p_2 ∪ p_3, consists of qubit q_4 as its head and qubit q_3 as its inward tail. These qubits are initialized as,
    
    |q_4 q_3 = ϕ|0 - |1/D|0.
    
    
    The resulting state in the qubit basis is the following,
    
    
    |ψ_0   =    |q_2 q_4 |q_1 q_6 |q_5 q_3
       =   (ϕ|0 - |1/D)^⊗2(ϕ|00 +ϕ |11/D)|00.
    
    
    
    
    We can also re-write the state |ψ_0 in the concentric tadpole basis, where each ket represents the topological charge of the corresponding generalized tadpole,
    
     
    |ψ_0   = |inn. |mid. |out.
    
    where,
    
    
    |inn.   = _11
    
    |mid.   = 
    
    |out.   = _11.
    
    
    
    Here the first equality means that the plaquette represented by the innermost tadpole (p_1) and the plaquette represented by the annulus surrounding the innermost tadpole (p_2) each have a charge of _11. The second equality indicates that the total charge of plaquettes p_1 ∪ p_2, represented by the middle tadpole, is ; in other words, the pair of excitations on p_1 and p_2 are effectively pulled out of vacuum. Finally, the third equality implies that the plaquette represented by the annulus surrounding the middle tadpole (p_3) and the outside plaquette representing the remainder of the sphere (p_4) each also have a charge of _11. Due to spherical boundary conditions, the total charge of these two plaquettes is also , thus the excitations on these plaquettes are also pulled out of vacuum. 
    
    To return to the original lattice basis (tetrahedron) we apply three F-moves to qubits q_3, q_5 and q_1, as shown in Fig. <ref>(b), resulting in the state |ψ_1. The circuit definitions of the F-moves can be found in Fig. <ref>. At this point, we have created two pairs of  excitations on plaquettes p_1, p_2 and p_3, p_4 such that each pair fuse to vacuum. 
    
    
    
    Measurement: To measure the fusion coefficients resulting from cross-fusing excitations from opposite pairs, e.g., p_2 ∪ p_3 and p_1 ∪ p_4, we need to switch to a concentric tadpole basis in which two plaquettes from opposite pairs are enclosed by the middle tadpole. One such basis is shown in Fig. <ref>(c), in which the middle tadpole encloses the charge of p_2∪ p_3. In this basis, the innermost tadpole contains the charge of plaquette p_1, and the outermost tadpole corresponds to the total charge of plaquettes p_1 ∪ p_2 ∪ p_3 or, equivalently, plaquette p_4. This basis change can be achieved by applying three additional F-moves acting on qubits q_4, q_3 and q_5,  resulting in the state |ψ_2 in Fig. <ref>. In the concentric tadpole notation, this state has the form,
    
    
    |ψ_2   = 1/ϕ^2_11_11
       + 1/ϕ√(ϕ)_1τ(1 + t) _τ1
       +1/ϕ[
    1/ϕ_11_11_11 
    -1/ϕ√(ϕ)( _τ1_1τ_11 + _11_1τ_τ1) 
    + 1/ϕ^2_τ1_ττ_1τ].
    
    
    
    As was noted above, the information on the fusion channels of the total charge of p_2∪ p_3 is now represented by the state of the middle tadpole in |ψ_2. To identify the components of the fusion channels, we can use Eq. (<ref>)-(<ref>) to re-write |mid. = × as follows,
    
    
    |mid.   =   1/ϕ^2
       +   1/ϕ√(ϕ)(1  + t)  
       +   ,
    
    
    where,
    
       =   _11
       -   1/ϕ√(ϕ)( _1τ + _τ 1)
       +   1/ϕ^2_ττ.
    
    
    This is consistent with our expectation that in general the two chiral sectors of the  excitation fuse independently, resulting in four fusion channels, as given in Eq. (<ref>).
    
    The charges of individual plaquettes p_3 (represented by the innermost tadpole |inn.) and p_4 (represented by the outermost tadpole |out.) remain at . However, as we see in Eq. (<ref>), for these states different components of the  sector appear in different terms. 
    
    Similar to the previous case, it is possible to design a quantum circuit to simplify the state |ψ_1 such that the expected configuration will be mapped to an un-entangled state which can then be measured for verification.
    
    
    
    
    
     §.§ Fusion and Braiding with Nine Qubits: Tailed Θ Lattice
    
    
    To detect non-Abelian braiding phases in the DFib model, we need to create the chiral excitations 1 or t. To realize these excitations without violating the vertex operators Q_v, we employ the extended tailed lattice <cit.>. The simplest nontrivial tailed lattice for this purpose would be a tailed Θ lattice, consisting of nine qubits, as shown in Fig. <ref>(g). In what follows we describe circuits to create these excitations, change the basis to measure fusion channels, then braid the excitations and finally detect the braiding phases. 
    
    
    
    
    
    
      §.§.§ Fusion
    
    
    Here we focus on creating two pairs of 1 excitations on three plaquettes, aiming at detecting the following fusion relation,
    
    1 ×1 =  + 1,
    
    
    which results from fusing excitations from opposite pairs, created out of vacuum. The general procedure resembles the case of creating  excitations in the Θ lattice described above, but the initialization and measurement processes involve details that are specific to the chiral excitations, which we describe bellow. The full circuit is depicted in Fig. <ref>.
    
    Initialization: We initialize the tailed Θ lattice in a basis in which plaquettes p_1 and p_2 are enclosed by separate (generalized) tadpoles, resulting in |ψ_0, as shown in Fig. <ref>(c). The general form of initialization circuit for  1 anyons is given in Fig. <ref>. Here we apply these circuits independently to the two tadpoles, but initialize the tail of plaquette p_3, with an F-gate. This can be understood by first imagining two tails for plaquette p_3 by using two extra qubits q_10 and q_11, then initialize each tadpole independently according to Fig. <ref>. We then combine these tails by applying an F-move to q_9 and through away the extra qubits  q_10 and q_11, leaving us with |q_9=|0 + |1. Since everything about this process is deterministic, we choose to do away with the extra qubits and directly initialize the tail qubit q_9 with an F gate. 
    
    We then apply three F-moves to |ψ_0 to return the state to the Θ lattice basis, resulting in |ψ_1. This process is depicted in Fig. <ref>(c) where the effects of the F-moves on the lattice are shown. Here the colored solid lines are merely guides to the eye, to keep track of the locations of pairs of excitations. At this point we have created two pairs of 1 excitations on plaquettes p_1, p_3 and p_2, p_3, such that each of these pairs would fuse to the vacuum state .
    
    Measurement: To measure the  amplitudes of the superposition state |ψ_1, resulting from cross-fusing the charges of plaquettes p_1∪ p_2, we change into a concentric tadpole basis where plaquettes p_1 and  p_2 are enclosed by concentric tadpoles. This can be done with the aid of three F-moves, resulting in the state |ψ_2 as shown in Fig. <ref>(d). 
    
    In this state, the outer tadpole, consisting of qubits  q_5 and q_6, which form the head, and qubits q_2 and q_9, forming its tails, contains the information on the total charge content of plaquettes p_1 ∪ p_2, or equivalently, p_3. Likewise, qubits q_1, q_3, q_4 and q_7, which form the components of the inner tadpole, contain information on the charge content of plaquette p_1. Written in the concentric tadpole basis, |ψ_2 = |inn.|out., we have, 
     
    |ψ_2 = 
    1(1/ϕ +
    1/√(ϕ)t ).
    
    
    The components of the  and 1 sectors for the inner and outer tadpoles are shown in Fig. <ref>. In terms of these components, the inner tadpole consists of three terms (Fig. <ref>(c)) and the outer tadpole consists of two terms for the  sector (Fig. <ref>(a)) and three terms for the 1 sector (Fig. <ref>(b)), leading to fifteen terms for |ψ_2.
    
    To detect the fusion amplitudes, we  first simplify |ψ_2 using the  circuits given in Fig. <ref> applied independently to the inner and outer tadpoles. Switching to the qubit basis, these circuits first map the two tadpoles to the following:
    
    
    |q_4 q_8 q_7 q_1 q_3   =    |11110
    
    |q_9 q_2 q_6 q_5   =   |0000 +  |1110.
    
    
    To further simplify, we apply CNOT gates to q_2, q_6 and q_9 to un-entangle them, resulting in 
    
    
    |ψ_3   =    |q_1 q_2 q_3 q_4 q_5 q_6 q_7 q_8 q_9
       =     |1(|0 +  |1)|0100110.
    
    
    Thus, the information on the fusion channels of the combined plaquettes p_1 ∪ p_2 is now stored in the state of a single qubit, namely q_2 [By choosing different combinations of CNOTs we could choose any of q_2, q_6 or q_9 as our flag qubit.]. 
    
    To detect these amplitudes, as before, we either carry out statistical measurements of q_2 and measure the probability distribution of each fusion channel, or apply an F gate to q_2 as a “filter,” mapping it to |0, then use the result to control a CNOT acting on ancilla qubit A. If the fusion channels are as expected, the ancilla qubit should not change. 
    
    
    
    
      §.§.§ Braiding
    
    
    Staying within the tailed Θ lattice, here we discuss how to create and braid two 1 excitations and measure the resulting phases. This process is shown in Fig. <ref>.
    
    Initialization: We repeat the same initialization as the one we used in the previous section and create two pairs of 1 excitations, on plaquettes p_1, p_3 and p_2, p_3, resulting in the state  |ψ_1 in the Θ lattice basis. 
    
    
    
    
    Braiding: To braid the charge contents of plaquettes p_1 and p_2, we simply exchange the corresponding plaquettes in a counterclockwise manner using a series of F-moves <cit.> as shown in Fig. <ref>(d), resulting in the state |ψ̃_1. Note that, while we end up in the same lattice basis as the starting point, braiding effectively changes the arrangement of qubits on the lattice. We need to keep this in mind when preparing the state for read-out. 
    
    Measurement: To measure the braid phases, we again need to change the basis into one where plaquettes p_1 and p_2 form concentric tadpoles, so that the total charges of combined plaquettes p_1 ∪ p_2 will be enclosed by the outer tadpole. This process is the same as the one in the fusion case but with different qubits, as shown in Fig. <ref>(e). Using the concentric tadpole basis,   |ψ̃_2 = |inn.|out., we get,
    
    
    |ψ̃_2 = 1 
    ( e^-4π i/5/ϕ + e^+3π i/5/√(ϕ)1
    ).
     
    
    Thus the two states of the outer tadpole have acquired the phases corresponding to a counterclockwise exchange of two τ1 anyons. 
    
    
    To verify these phases with a quantum circuit, we begin by simplifying |ψ̃_2, using the detection circuits in Fig. <ref>. This simplification effectively maps the states of the two tadpoles to the following, 
    
    
    |q_1 q_7 q_8 q_5 q_4    =    |11110
    
    | q_3 q_9 q_2 q_6   =   e^-4π i/5/ϕ|0000 +e^+3π i/5/√(ϕ)|1110.
    
    
    To further simplify this state, we un-entangle qubits q_2, q_3 and q_9 by applying two CNOT gates, resulting in 
    
    
    |ψ̃_3   =    |q_1 q_2 q_3 q_4 q_5 q_6 q_7 q_8 q_9
       =    
    |1(e^-4π i/5/ϕ|0 + e^+3π i/5/√(ϕ) |1) |1010110.
    
    
    Thus, all the information on braiding phases of the two fusion channels is now stored in q_2. For the final detection, similar to the previous cases, we design another filter, which has the form,
    
    F̃ = [     ξ^2/ϕ    ξ/√(ϕ); ξ^-1/√(ϕ)  -ξ^-2/ϕ.;           ]
    
    
    
    We believe this is the simplest setting in which to demonstrate the braiding properties of the chiral excitations of the DFib model. One may also perform these exercises on a tailed tetrahedron (Fig. <ref>(h)) with fourteen qubits, such that each of the four plaquettes contains a single 1 excitation. The results are identical to the three plaquette lattice for both fusion and braiding. 
    
    Since this article is focused on minimal models, we do not provide that circuit here. 
    
    
    
    
    
    
     §.§ Twisting with Five Qubits: Tailed Tadpole
    
    
    A chiral excitation can be twisted within a plaquette and the resulting phase can be detected using additional tricks (see, e.g., <cit.>). The simplest lattice for this purpose consists of only four qubits, forming a single generalized tadpole, which can be thought as representing two plaquettes on a sphere, one shown and the other being the outside region representing the remainder of the sphere (see Fig. <ref>(f)). The circuit for the initialization, twisting and detection of the phase is shown in Fig. <ref> (with two additional qubits to simplify readout). 
    
    
    We start by initializing  the 1 state on the four qubits forming the generalized tadpole, using the initialization circuits in Fig. <ref>. We then apply a special case of an F-move, which effectively exchanges the two tails. This F-move is defined in Fig. <ref>. Written in the original basis, this is equivalent to twisting the 1 excitation in the front plaquette in a counterclockwise manner, resulting in,
    
    θ̂1 = e^+4π i/51.
     
    
    
    To detect the overall phase, we initialize an ancilla qubit |q_0 with a Hadamard gate. Re-writing the 1 state in qubit basis as, 
    
    1    =    |q_1 q_2 q_3 q_4
       =   1/D|00(|10 + ξ^2|01 +√(ϕ)ξ|11).
    
    
    results in the initial state, 
    
    
    |ψ_0   =    |q_0|q_1q_2q_3q_4
       =   1/√(2) (|0 + |1)1.
     
    
    We then then use |q_0 as a control qubit for the application of a controlled-F-move acting on q_3, resulting in
    
    
    |ψ_1   =    |q_0|q_1q_2q_3q_4
       =   1/√(2) (|0 + e^4π i/5|1)1.
    
    
    Thus, we use the phase kickback from twisting the 1 excitation in the tadpole consisting of  qubits q_1-q_4 to impart a phase shift on the ancilla qubit |q_0. We can then measure this phase shift by using the appropriate filter, which maps the state of q_0 to |0. The filter in this case is the following gate,
    
    M = [         1  e^4π i/5; e^-4π i/5        -1;           ].
    
    Finally we can use q_0 as a control for a CNOT acting on another ancilla qubit to verify the expected phase, as in the previous cases. Since the state of the ancilla qubit q_0 is un-entangled from the qubits forming the tadpole, we do not need to apply a simplification circuit to these qubits. However for ease of checking the results, simplifying gates can be adopted from Fig. <ref> to map the 1 state to |q_1q_2q_3q_4 = |1110. 
    
    
    We end by mentioning that this procedure can easily be generalized to larger plaquettes and be used to verify the relation between braiding and twisting, Eq. (<ref>), in the nine-qubit lattice. 
    
    
    
    
    § CONCLUSIONS AND OUTLOOK
    
    
    This work was inspired by a theoretical question: what are the smallest possible lattices in which the string-net rendition of the Doubled Fibonacci model can be realized and its non-Abelian nature revealed? We answered this question by designing quantum circuits that simulate the spectrum of this model and demonstrated their fusion, twisting and braiding properties, all on minimal lattices with the fewest number of qubits possible. 
    
    We argued that the ground state as well as a  pair of achiral  excitations can be realized on a lattice consisting of only one qubit. We then showed that with only three qubits, we can create two pairs of  excitations in a minimal three-plaquette lattice and demonstrate the amplitudes of their fusion channels. We further showed that by moving to a lattice with six qubits, additional fusion structure resulting from combining  excitations can be revealed. 
    
    We then created two pairs of chiral 1 excitations in a three-plaquette tailed lattice, consisting of nine qubits, braided them and measured the relative braiding phases. Finally, we showed that a total of four qubits is enough to create a pair of chiral excitations (1 or t) in two plaquettes, twist one of them and verify the resulting phase with the aid of a fifth (ancilla) qubit. 
    
    The Fibonacci anyon model exhibits one the simplest, yet most exotic, forms of topological order, with possible applications in active and passive fault-tolerant quantum computation. Our goal here was to find the minimal resources required to demonstrate the non-Abelian nature of Fibonacci anyons and we believe our circuits require sufficiently small quantum volume that they can be run on any modern qubit hardware with reasonable coherence times and gate fidelity, thus making this model more accessible to the wider physics community. 
    
    The methods discussed here can  be generalized to larger lattices, where more anyons can be created and quantum gates can be carried out topologically. An important future direction would be to quantify and measure the stability of these states in the presence of noise. Another interesting direction would be the design of minimal circuits that simulate other instances of the  string-net model, resulting in other forms of topological order, especially those that require multi-level qudits.
    
    
    
    
    § ACKNOWLEDGMENT
     
    
    We thank R. Koenig, M. Levin and A. Schotte for clarifying discussions at an earlier stage of this work, and F. Burnell for discussions and comments on an earlier draft of the manuscript. We also thank the Aspen Center for Physics
    for its hospitality during the completion of part of this
    work. This project was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. LH and JP acknowledge partial support from BNL LDRD 20-022. SB and RRA were supported by the U.S. Department of Energy, Office of Science’s SULI program.  SHS is supported by EPSRC Grants
    EP/S020527/1 and EP/X030881/1.
    
    
    Note Added: During the preparation of this work, we became aware of two other independently carried-out works on a related but different simulation of DFib anyons on specialized hardware <cit.> and <cit.>. Besides using different methods, the present work is distinct in that it focuses on designing minimal circuits suitable for realization on any quantum platform.


 Paper Title: Homotopy representations of extended holomorphic symmetry in holomorphic twists.


Point-supervised Brain Tumor Segmentation with Box-prompted MedSAM
    Xiaofeng Liu, Jonghye Woo, Chao Ma, Jinsong Ouyang, and Georges El Fakhri
X. Liu, C. Ma, J. Ouyang and G. El Fakhri are with the Department of Radiology and Biomedical Imaging, Yale University, New Haven, CT 06519.
J. Woo is with the Massachusetts General Hospital and Harvard Medical School, Boston, MA 02114.

    August 5, 2024
===========================================================================================================================================================================================================================================================================================================================

§ ABSTRACT 
We argue that holomorphic twists of supersymmetric field theories naturally come with a symmetry L_∞-algebra that nontrivially extends holomorphic symmetry. This symmetry acts on spacetime fields only up to homotopy, and the extension is only visible at the level of higher components of the action. We explicitly compute this for the holomorphic twist of ten-dimensional supersymmetric Yang–Mills theory, which produces a nontrivial action of a higher L_∞-algebra on (a graded version) of five-dimensional affine space.





§ INTRODUCTION AND SUMMARY

Twisting of physical theories <cit.> has attracted great interest in the physics literature.
In particular, the pure spinor formalism <cit.> (see reviews in <cit.>), which naturally describes such theories as supergravity <cit.>, supersymmetric Yang–Mills theory <cit.> and M2-brane models <cit.>,
accommodates twisting naturally <cit.>.

Physical theories come with representations of spacetime symmetry algebras, such as (super-)Poincaré algebras and (super-)conformal algebras. It has been long known that for theories with more than four supercharges it is often difficult to manifest this symmetry `off shell', that is, without using equations of motion. The pure spinor formalism provides a means of producing off-shell supermultiplets by introducing appropriate infinite towers of auxiliary fields and furthermore shows that the on-shell supermultiplets in fact carry a homotopy representation of the spacetime symmetries; the higher components of the action then correspond to the equations of motion needed to make the symmetry algebra close.

In this paper, we argue that holomorphic twists of supersymmetric field theories naturally come with more than just the holomorphic symmetry but rather a certain L_∞-extension of holomorphic symmetry. The extension is not visible at the level of strict representations, but spacetime fields naturally form a homotopy representation of this extended symmetry. We shall treat in detail the example of the holomorphic twist of ten-dimensional supersymmetric Yang–Mills theory. 
This twisted theory is holomorphic Chern–Simons theory on ^5 <cit.>, which enjoys a manifest isl(5) = sl(5) ⋉5 symmetry. 
As we shall see, it naturally comes with the extended holomorphic symmetry L_∞-algebra

    isl(5)(𝔰𝔩(5)⋉1005)

equipped with a certain higher bracket μ_4; and this L_∞-algebra acts on [z_1,z_2,z_3,z_4,z_5] (with appropriate grading) in the L_∞-algebraic sense.
This may be seen as a non-strict L_∞-algebra action of isl(5) on the (graded version of) five-dimensional complex affine space 𝔸^5.

We work with minimal models (of both the symmetry algebra and the field content), which canonically separates the physical information and makes clear the presence of higher-order structures (L_∞-algebras and their representations), rather than a larger strict model, which is not canonical and mixes in the physical degrees of freedom together with the unphysical auxiliary fields; this ensures that all information that we recover is physical and independent of the choice of auxiliary fields.

One way to think about this is to recall that twisting is akin to dimensional reduction <cit.> in which, rather than eliminating dependence on bosonic coordinates, we eliminate dependence on fermionic coordinates (restrict to Q-closed fields for a supersymmetry Q), which results in the `pair annihilation' of bosonic and fermionic coordinates. From this perspective, we have an `as above, so below' heuristic: the actions of twisted theories resemble those of their twistings, just like dimensional reduction preserves the forms of actions.
Using the pure spinor formalism, ten-dimensional supersymmetric Yang–Mills theory may be formulated as a holomorphic Chern–Simons theory on a complex (21|16)-dimensional pure spinor superspace (with 10 complexified ordinary spacetime coordinates, 16 ordinary superspace fermonic coordinates, and 11 bosonic pure spinor coordinates). The twisted theory has the same form of a Chern–Simons theory, but this time on 5|0 dimensions, where we have killed 16|16 coordinates. Under this `dimensional reduction', the ten-dimensional 𝒩=1 super-Poincaré symmetry, which is (55|16)-dimensional, reduces to a (39|0)-dimensional extended holomorphic symmetry. This dimensional reduction corresponds to twisting the supersymmetry algebra and taking the minimal models of the symmetry algebra and its homotopy representation on the field content. The additional factor 10 in (<ref>) and the concomitant μ_4 are the `dimensionally reduced' remnants of ten-dimensional super-Poincaré symmetry.

The discussion of the present paper is limited to the kinematics, that is, ignoring interactions and considering the linearized theory. This is not an essential restriction. A discussion of the interaction terms should make use of the L_∞-algebra formalism <cit.> for scattering amplitudes;
after colour-stripping, we should get a C_∞-algebra <cit.>, on which the extended holomorphic symmetry should act, forming an example of an open–closed homotopy algebra <cit.>. This, however, we leave to a future work.

While we focus on ten-dimensional supersymmetric Yang–Mills theory as a special case, the discussion is generic and applies, in principle, to the twists of any supersymmetric field theory. However, the twists in other dimensions often produce either a strict representation (with the 𝔦𝔰𝔩(d)-representation factoring through an 𝔦𝔰𝔩(d)-representation)
or a higher representation of an L_∞-superalgebra on affine superspace (with odd coordinates); 𝔸^5 is one of the few nontrivial purely bosonic examples that carry a higher symmetry. (For more discussion, see <ref>.)

All of our discussion is classical; there may be obstructions to quantization in the form of anomalies. For our main example of the holomorphic twist of ten-dimensional supersymmetric Yang–Mills theory, the twist (five-dimensional holomorphic Chern–Simons theory) is known to have anomalies unless it is coupled in a consistent fashion to Kodaira–Spencer gravity <cit.>.



 §.§ Organization of this paper

This paper is organized as follows. <Ref> reviews the generalities of twisting L_∞-algebras and modules over them and the appearance of higher components of the spacetime symmetry algebras and higher components of nonstrict representations of L_∞-algebras, both in the untwisted and twisted cases. <Ref> then computes the higher components of the representation of supersymmetry for ten-dimensional supersymmetric Yang–Mills theory, the higher products of the corresponding twisted extended holomorphic algebra, and the higher components of its representation on the twisted supermultiplet. <Ref> briefly surveys phenomena that appear in dimensions other than ten.

In the body of the paper, we will usually refer to irreducible representations of 𝔰𝔩(5) using their Dynkin labels, supplemented by Young tableaux where they are helpful.



§ MATHEMATICAL BACKGROUND

Here we briefly review the relevant concepts of twisting of L_∞-algebras and their modules. For more detailed reviews, see <cit.>.



 §.§ L∞-algebras

An L_∞-algebra is a homotopy generalization of the concept of a Lie algebra.

An L_∞-algebra (𝔤,{μ_k}_k≥ 1) consists of a graded vector space 𝔤=⊕_i∈ℤ𝔤^i together with skew-symmetric, multilinear maps μ_k𝔤^∧ k→𝔤 of degree 2-k for k∈{1,2,3,…} that satisfy the identity

    0=∑_i+j=n
    σ∈Sh(i,j)(-1)^jχ(σ,x)μ_j+1(μ_i(x_σ(1),…,x_σ(i)),…,x_σ(i+j))=0.

In the above, Sh(j_1,…,j_k) denotes the collection of shuffles, which are permutations σ of {1,…,j_1+…+j_k} such that σ(1)<…<σ(j_1) and σ(j_1+1)<…<σ(j_1+j_2) and so on up to σ(j_1+…+j_k-1+1)<…<σ(j_1+…+j_k).
The symbol χ(σ,x) denotes the skew-symmetric Koszul sign

    x_1∧…∧ x_k
        =χ(σ,x) x_σ(1)∧…∧ x_σ(k),

  defined for homogeneous elements x_1,…,x_k∈ g inside the exterior algebra ⋀^∙ g.

In what follows, we will often leave the products {μ_k^𝔤}_k≥ 1 implicit, and simply refer to an L_∞-algebra through its underlying graded vector space. The identities (<ref>) imply that μ_1∘μ_1=0 so that g is in particular a cochain complex.


  A morphism of L_∞-algebras ϕ(,{μ_k^𝔤}_k≥ 1)⇝(h,{μ_k^𝔥}_k≥ 1) consists of skew-symmetric, multilinear component maps
  
    ϕ^(n)^∧ n→ h

of degree 1-n for n∈{1,2,…}, satisfying the following coherence relations:
  
    ∑_j∈{1,…,i}
    k_1+…+k_j=i
    σ∈Sh(k_1,…,k_j)ζ(σ,k,x)/j!μ_j^𝔥(ϕ^(k_1)( x_σ(1), …, x_σ(k_1)),
          …, ϕ^(k_j)( x_σ(k_1+…+k_j-1+1), …,  x_σ(i)))
    
          =
          ∑_j+k=i
    σ∈Sh(j,k)(-1)^k χ(σ,x) ϕ^(k+1)(μ_j^𝔤( x_σ(1), …,  x_σ(j)),  x_σ(j+1), …,  x_σ(i)),

where

    ζ(σ,k,x)χ(σ,x)(-1)^∑_1 ≤ m<n ≤ j k_m k_n+∑_m=1^j-1 k_m(j-m)+∑_m=2^j(1-k_m)∑_k=1^k_1+…+k_m-1|x_σ(k)| .

We shall sometimes omit the μ_k and just write ⇝ h. 
L_∞-morphisms compose associatively, so that one has the category whose objects are L_∞-algebras and whose morphisms are L_∞-morphisms between them.

An L_∞-morphism is an  L_∞-(quasi-)isomorphism if the first component map is a (quasi-)isomorphism of the underlying cochain complexes.



  
Homotopy transfer of L_∞-algebras. 
L_∞-algebras admit a good homotopy theory in the sense that minimal models exist and can be computed by homotopy transfer using a strong deformation retract. Let us sketch how this works. Concretely, given an L_∞-algebra ( g,{μ_k^𝔤}_k≥ 1) one can always choose a strong deformation retract, denoted by a triple (i,p,h), from the underlying cochain complex ( g,μ_1) to its cohomology H( g):

    (𝔤,μ_1) [shift left, "p"] [loop left, "h"]     ( H(𝔤),0) [shift left, "i"]

(i.e. pi=id_ H( g) and ip=id_ g-[d,h]).
Then there exists an L_∞-algebra structure on the cohomology H(𝔤) together with an L_∞-quasi-isomorphism

    e H(𝔤)⇝𝔤,

whose first component is e^(1)=i; furthermore, there exist explicit formulae for the L_∞-algebra structure of H(𝔤) and the quasi-isomorphism e in terms of (i,p,h) <cit.>, e.g. using the tensor trick <cit.>, which can be interpreted as a sum over Feynman diagrams <cit.>. For example the ternary bracket μ_3^ H( g), is (modulo relative signs) the sum


    [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (-0.6,-1.4);
         [thick] (-0.35,-1.65) – (0,-2);
         [thick] (2,0) – (0,-2);
         [thick] (0,-2) – (0,-3);
         [thick] (-1,-1)–(0,0);
         at (-.5,-1.5) h;
         at (0,-3.5) p;
         at (0,0.5) i;
         at (-2,0.5) i;
         at (2,0.5) i;
        
         [gray!50] (-1,-1) circle (0.35cm);
         [thick] (-1,-1) circle (0.35cm);
         at (-0.98,-1) μ_2;
        [gray!50] (0,-2) circle (0.35cm);
         [thick] (0,-2) circle (0.35cm);
         at (0.02,-2) μ_2;
      
      +
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (0,-2);
         [thick] (2,0) – (0.65,-1.35);
         [thick] (0,-2) – (0.4,-1.6);
         [thick] (0,-2) – (0,-3);
         [thick] (1,-1)–(0,0);
         at (0.55,-1.45) h;
         [gray!50] (1,-1) circle (0.35cm);
         [thick] (1,-1) circle (0.35cm);
         at (1.02,-1) μ_2;
        [gray!50] (0,-2) circle (0.35cm);
         [thick] (0,-2) circle (0.35cm);
         at (0.02,-2) μ_2; at (0,-3.5) p;
         at (0,0.5) i;
         at (-2,0.5) i;
         at (2,0.5) i;
      
      +
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (-0.6,-1.4);
         [thick] (-0.35,-1.65) – (0,-2);
        
         [thick] (1,-1) – (0,-2);
         [thick] (0,-3) – (0,-2);
         [thick] (1,-1)–(0,0);
         [thick] (-1,-1) – (0.3,-0.57);
         [thick] (2,0) – (0.7,-.45);
         at (-.5,-1.5) h;
      
         at (0,-3.5) p;
         at (0,0.5) i;
         at (-2,0.5) i;
         at (2,0.5) i;
    
         [gray!50] (-1,-1) circle (0.35cm);
         [thick] (-1,-1) circle (0.35cm);
         at (-0.98,-1) μ_2;
        [gray!50] (0,-2) circle (0.35cm);
         [thick] (0,-2) circle (0.35cm);
         at (0.02,-2) μ_2;
      
      +
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (0,-2);
         [thick] (2,0) – (0,-2);
         [thick] (0,0) – (0,-2);
         [thick] (0,-2) – (0,-3);
         at (0,-3.5) p;
         at (0,0.5) i;
         at (-2,0.5) i;
         at (2,0.5) i;
        [gray!50] (0,-2) circle (0.35cm);
         [thick] (0,-2) circle (0.35cm);
         at (0.02,-2) μ_3;
      .

More generally, μ_k^ H( g) is computed by a sum[The explicit relative signs between the trees can be worked out by using the aforementioned tensor trick, for example.] over all rooted trees with k leaves, where one decorates the leaves with i, the n+1-ary vertices with μ_n, the internal edges with h, and the root with p.
The L_∞-algebra structure on H( g) is called the minimal model of 𝔤; minimal models are unique up to L_∞-isomorphisms.



  
Twisting L_∞-algebras
L_∞-algebras also admit a notion of twist with respect to a Maurer–Cartan element; for reviews, see <cit.>. In the definitions below, for a L_∞-algebra ( g,{μ_k}_k≥ 1), we assume for simplicity that μ_i=0 for sufficiently large i; this can be relaxed <cit.>.

Let (𝔤,{μ_k^𝔤}_k≥ 1) be an L_∞-algebra such that μ_i=0 for sufficiently large i.
A Maurer–Cartan element Q∈𝔤^1 of 𝔤 is an element of degree 1 such that

    ∑_i=1^∞1/i!μ_i(Q,…,Q)=0.



Let (𝔤,{μ_k^𝔤}_k≥ 1) be an L_∞-algebra such that μ_i=0 for sufficiently large i.
Let Q∈𝔤^1 be a Maurer–Cartan element.
The twist of 𝔤 with respect to Q is the L_∞-algebra 𝔤_Q whose underlying graded vector space is that of 𝔤 but whose brackets μ_k^Q are

    μ_k^Q      𝔤_Q^∧ k   →𝔤_Q
          (x_1,…,x_k)   ↦∑_i≥ 01/i!μ_i+k(Q,…,Q,x_1,…,x_k).





 §.§ L∞-representations

The notion of a representation of (or module over) a Lie algebra generalizes to the setting of homotopy algebras as follows.

  
    An L_∞-representation
of an L_∞-algebra (𝔤,{μ_k^𝔤}_k≥ 1) on a graded vector space M is an L_∞-algebra structure {μ_k^𝔤⋉ M}_k≥ 1 on the direct sum 𝔤⊕ M 
such that
    
    μ_k^𝔤⋉ M(x_1⊕0,…,x_k⊕0)=μ_k(x_1,…,x_k)

     and  μ_k+1^𝔤⋉ M(x_1,… x_k,m)∈ 0⊕ M for x_1,…,x_k∈𝔤 and m∈ M, and such that μ_k^𝔤⋉ M vanishes whenever at least two of its arguments belong to 0⊕ M⊂𝔤⊕ M. We will write g ⋉ M to refer to L_∞-algebras of this form.
We write
    
    ρ^(k)(x_1,…,x_k)μ_k+1^𝔤⋉ M(x_1,…,x_k,-) M→ M.

      Observe that, in particular, ρ^(k) carries degree 1-k.
    Note that ρ^(0) defines a differential on M, making it a cochain complex. We call an L_∞-representation strict whenever ρ^(k)=0 for k>1.

The L_∞-algebra homotopy Jacobi identities (<ref>) then can be written as a series of coherence relations amongst the ρ^(k)'s and μ_j^ g's.

As with L_∞-algebras themselves, L_∞-representations admit a good homotopy theory in that minimal models exist and homotopy transfer is possible. That is, given an L_∞-algebra 𝔤 and a g-representation M,
we can always choose a strong deformation retract

    [loop left,"(h,h')"] (𝔤⊕ M,μ_1+ρ^(0) )[shift left, "(p,p')"]    [shift left, "(i,i')"]  ( H(𝔤)⊕ H(M),0)

and perform homotopy transfer of L_∞-algebra structures along this retract to obtain an L_∞-algebra on H( g)⊕ H(M), which then defines the L_∞-representation of H(𝔤) on H(M).[The induced brackets on H( g)⊕ H(M) automatically satisfy the conditions given in <ref>. Indeed, as there are no brackets in g ⊕ M that reduce the number of factors of M, no such brackets can arise through composition.][This is the minimal model for the two-coloured operad of pairs of L_∞-algebras and their L_∞-representations, rather than the minimal model for the uncoloured operad of L_∞-representations over a fixed L_∞-algebra 𝔤.]

Given an L_∞-algebra 𝔤 and a g-representation M with structure maps ρ^(k),
then it is clear by inspection that a Maurer–Cartan element Q∈𝔤 is also Maurer–Cartan element of 𝔤⋉ M and that the twist (𝔤⋉ M)_Q factorizes as (𝔤⋉ M)_Q=𝔤_Q ⋉ M_Q <cit.>, where
M_Q comes with the structure maps

    ρ_Q^(k)(x_1,…,x_k) ∑_i=0^∞1/i!ρ^(i+k)(Q,…,Q,x_1,…,x_k).




§ HIGHER SYMMETRY OF TWISTED TEN-DIMENSIONAL SUPERSYMMETRIC YANG–MILLS THEORY

In the Batalin–Vilkovisky formalism <cit.>,
the field content of a perturbative gauge theory is a graded vector space (ℱ,d_ℱ) that comes equipped with a differential.
Field theories often respect symmetry algebras such as the super-Poincaré algebra, the (super-)conformal algebra, the (super-)(anti-)de Sitter algebra, the (super-)Galilean algebra, etc. 
The action of such a symmetry algebra may be off shell (i.e. on ℱ itself) or merely on shell (i.e. only on the space of solutions to the equations of motion[Since we ignore interactions, for us this is the linearized equations of motion, but in general one should consider the interacting case.]). The symmetry algebras are usually ungraded or ℤ/2ℤ-graded (i.e. superalgebras), and correspondingly the field space is ℤ/2ℤ-graded into bosons and fermions (in addition to the ℤ-grading corresponding to ghost number).
This ℤ/2ℤ grading may be often lifted to a ℤ grading; correspondingly, the ℤ/2ℤ grading of the field space may also be lifted to a ℤ-grading.[In order to lift the /2 grading of field space to ℤ, we work with polynomials over spacetime rather than smooth functions. This simplification can be avoided; see <cit.>.] 
The ℤ-grading enables a good homotopy theory of L_∞-algebras and L_∞-representations, and in particular off-shell realizations of symmetries can be in most cases lifted to a non-strict L_∞-representation of the corresponding symmetry <cit.>.

We turn now to our main example of interest, which is the twist of ten-dimensional super-Yang–Mills theory. (For simplicity and convenience with twisting, we assume all symmetries and fields to be complexified.)

We first discuss the twisted super-Poincaré algebra itself.
Let 

    V ≅^10
 
be a ten-dimensional complex vector space equipped with a nondegenerate symmetric bilinear form.
The ten-dimensional 𝒩=(1,0) super-Poincaré superalgebra is the Lie superalgebra

    𝔬(V)⋉( S_+⊕ V)

where S_± are the two 16-dimensional Weyl spinor representations of 𝔬(V) and  denotes parity reversal.
The ℤ/2ℤ grading of the super-Poincaré superalgebra can be lifted to a ℤ-grading as the graded Lie algebra

    𝔭𝔬(V)⋉(S_+[-1]⊕ V[-2])

in which the elements are graded as twice the conformal dimension (i.e. rotations in degree 0, supertranslations in degree 1, translations in degree 2).
For convenience, we can pick a basis _μν, _α, _μ of 𝔭^0, 𝔭^1, 𝔭^2 respectively. Then the structure constants for [𝔭^1,𝔭^1]⊂𝔭^2 are

    [_α,_β]=2γ^μ_αβ_μ,

where γ^μ_αβ are the chiral gamma (or Pauli) matrices in ten dimensions, i.e. the branching for the 𝔬(V)-representation S_+⊗ S_+→ V.



 §.§ The minimal model of the holomorphic twist algebra

Suppose we pick a nonzero Maurer–Cartan element of 𝔭, i.e. a nonzero Q=Q^α_α∈ S_+ such that [Q,Q]=0, that is,

    γ^μ_αβQ^α Q^β=0.

This picks out a subspace

    L = [Q,S_+]⊂ V.

This subspace L is a maximal isotropic subspace with respect to the bilinear form on V.
Indeed, using the Fierz identity 

    2γ^μ_α(βγ^0_μ|γ)δ=-γ^μ_βγγ^0_μαδ,

we have

    Q^β Q^γγ^μ_αβγ^0_μγδ∝(Q^β Q^γγ^μ_βγ)γ^0_μαδ=0;

given now any elements ψ,χ∈ S_+, consider the elements [Q,ψ],[Q,χ]∈ L. We have

    [Q,ψ]^μ[Q,χ]_μ =(Q^β Q^γγ_αβ^μγ_μγδ)ψ^αχ^δ=0.

Thus L is indeed contained in its own orthogonal complement, i.e. it is an isotropic subspace.
Isotropy implies L≤5. Furthermore, the set of Maurer–Cartan elements of 𝔭 consists of two 𝔬(V)-orbits, namely nonzero ones and {0}; and it can be shown that, when Q0, then L=5 <cit.>. That is, L is indeed a maximal isotropic subspace.

Thus, we have the short exact sequence of vector spaces

    0→ L→ V L^*→ 0

where the quotient q is via the composition VV^*↠ L^* in which VV^* is given by the bilinear form on V. Let us choose a splitting of (<ref>) to write

    V=L⊕ L^*.

This decomposition then fixes Lie subalgebras 𝔰𝔩(L)⊂𝔤𝔩(L)⊂𝔬(V), under which the ten-dimensional representation V canonically decomposes into 𝔰𝔩(L) irreducible representations as

    V≅_sl(L) L⊕ L^*≅_sl(L) (1000)_sl(L)⊕ (0001)_sl(L)

(Here and elsewhere we write (ijkl)_sl(L) for the irreducible representation with these sl(L) Dynkin labels, i.e. for the irreducible representation with highest weight iω_1 + j ω_2 + k ω_3 + lω_4 where ω_1,…,ω_4 are the fundamental weights.)

Similarly, the adjoint representation of o(V) decomposes into irreducible sl(L)-representations as

    𝔬(V)    ≅_sl(L)⊕𝔰𝔩(L)⊕ L^∧2⊕(L^*)^∧2
       ≅_sl(L) (0000)_sl(L)⊕ (1001)_sl(L)⊕ (0100)_sl(L)⊕ (0010)_sl(L),

and the spinor representations S_± decompose as

    S_+    ≅_sl(L)⊕_i=0^2(L^*)^∧(2i)≅_sl(L)(0000)_sl(L)⊕(0010)_sl(L)⊕(1000)_sl(L),
    
        S_-    ≅_sl(L)⊕_i=0^2(L^*)^∧(2i+1)≅_sl(L) S_+^*≅(0001)_sl(L)⊕(0100)_sl(L)⊕(0000)_sl(L).

Our chosen element Q∈ S_+ spans the one-dimensional sl(L)-submodule ≅ (L^*)^∧0.
The twist of p by Q is <cit.>

    𝔭_Q = 
    ([row sep=0]
    (0000)_sl(L)["id"]    (0000)_sl(L)(L^*)^∧0[-1]
    (0010)_sl(L)(L^*)^∧2["id"]    (0010)_sl(L)(L^*)^∧2[-1]
       (1000)_sl(L)(L^*)^∧4[-1]["id"]    (1000)_sl(L)L[-2]
    (1001)_sl(L)𝔰𝔩(L)  ⋉  (0100)_sl(L)L^∧2      (0001)_sl(L)L^*[-2]).


We work in an explicit basis (^i_j,^ij,_ij,,_ij,^i,^i,_i) of 𝔭 given in <ref>. In particular, the basis elements ^i_j span gl(L), and we shall write 

    ^i_j^i_j - 1/5δ^i_j^k_k

for the basis elements of sl(L). 
 
For a reason to become apparent in the theorem below, let us note that the representation L^* ⊗ (L^∧ 2)^∧ 3 of sl(L) decomposes into irreducibles as

    L^* ⊗ (L^∧ 2)^∧ 3≅ (0021)_sl(L)⊕ (0110)_sl(L)⊕ (1001)_sl(L)⊕ (2010)_sl(L)⊕ (2002)_sl(L).

In particular, the adjoint representation sl(L) ≅ (1001)_sl(L) occurs with multiplicity one. We shall write 

    P_L^*⊗(L^∧2)^∧3→𝔰𝔩(L)

for the projector onto this irreducible component. 

The minimal model of the ten-dimensional twisted ℤ-graded 𝒩=(1,0) super-Poincaré algebra 𝔭_Q is the L_∞-algebra whose underlying graded Lie algebra is

    H(𝔭_Q) = 𝔰𝔩(L)⋉( L^∧2⊕ L^*[-2]),

and whose higher brackets μ_i (i≥3)
are all zero except for μ_4, whose only nonvanishing component is given by

    μ_4(_i,^jk,^lm,^np)
           =-δ_i^[jε^k]lmqrδ^[n_q^p]_r
        +δ_i^[jε^k]npqrδ^[l_q^m]_r
           -δ_i^[lε^m]npqrδ^[j_q^k]_r
        +δ_i^[lε^m]jkqrδ^[n_q^p]_r
           -δ_i^[nε^p]jkqrδ^[l_q^m]_r
        +δ_i^[nε^p]lmqrδ^[j_q^k]_r
        P^jklmnp;_i^r_q^q_r,

where P^jklmnp;_i^r_q is the projector

    L^*⊗(L^∧2)^∧3→𝔰𝔩(L),

and where the skew-symmetrizations are unnormalized.


From <ref>, we see that there is an evident 𝔰𝔩(L)-equivariant strong deformation retract (i,p,h) of cochain complexes 

    [loop left,"h"] ( p_Q,ad_Q) [shift left, "p"]    [shift left, "i"]  ( H( p_Q),0)

from p_Q to its cohomology

    H(𝔭_Q) = (𝔰𝔩(L)⋉ L^∧2 0 L^*).

(The remaining sl(L) irreducible representations present in <ref> participate in trivial pairs; one defines the homotopy h to act as the inverse to the differential on these.)
The Lie algebra structure μ_2 on this cohomology H( p_Q) is given by restriction.

It remains to check what higher brackets μ_i are induced by homotopy transfer. 
We are to sum over rooted binary trees in which each vertex corresponds to the binary bracket μ_2^ p(-,-) = [-,-] of p (and thus of p_Q), each internal edge to the homotopy h, each leaf to i and the root to p <cit.>.

We will use Feynman-diagrammatic terminology, referring to elements as `states' (see <cit.>). 
Recall our notation (^i_j,^ij,_ij,,_ij,^i,^i,_i) for the basis elements of 𝔭 (whose underlying graded vector space we identify with that of 𝔭_Q) as given in <ref>. We will refer to a state as intermediate if it lies in the image of h.

Using the strong deformation retract (i,p,h), we try to construct the possible intermediate states, keeping track of the representations under 𝔰𝔩(L). Using the embedding i H(p_Q)↪p_Q, we will identify the basis elements ^i_j,^ij,_i in p_Q with those of H(p_Q). The products μ_i for i>2 can then be computed in a top-down recursive fashion by starting with two elements x, y∈H(p_Q), and then compute which intermediate states are allowed by considering

    h[i(x),i(y)].

The next intermediate states are then computed by plugging (<ref>), and one element a∈ i(H(p_Q))⊕(h[i,i]), into [-,-]. If the result lies in H(p_Q), we apply p, and we are done. If not, we apply h to obtain new intermediate states and then continue the procedure of pairing (using [-,-]) the newly obtained intermediate states with each other or with previously obtained intermediate states or states in the cohomology.

Starting with two elements of H(p_Q), applying h[-,-] can only yield the intermediate states

    h[_k,^ij]=δ^i_k^j-δ^j_k^i.

Using ^i together with ^i_j,^ij,_i, we can only further create 


    [^i,^j]    = 0, 
    
    [^i,_j]    = 0, 
    
    h[^ij,^k]    = -1/2^ijklm_lm,
    
    [^i_j,^k]   =-1/5δ^i_j^k+δ^k_j^i.


Among these, [^i_j,^k] does not belong to the cohomology, i.e. it is not Q-closed, nor can it produce a new intermediate state since h([^i_j,^k])=0.
Thus the only intermediate state we can create is given by h[^ij,^k]∝_lm. Applied to (<ref>), we obtain

    h[h[_k,^ij],^lm]=h[δ^i_k^j-δ^j_k^i,^lm]
        =1/2(
        δ^i_kε^jlmnp
        -δ^j_k
        ε^ilmnp)_np.

Using the new intermediate state _ij together with the previously created intermediate state ^i and the cohomology (^i_j,^ij,_i), we can create the following new states:


    [_ij,_k]    = 0, 
    
    [_ij,^k]    = 0, 
    
    [^i_j,_kl]   =
    -δ^i_k_jl-δ^i_l_kj
    +2/5δ^i_j_kl, 
    
    [^ij,_kl]    = δ^i_k^j_l-δ^j_k^i_l-δ^i_l^j_k+δ^j_l^i_k, 
    
    [_ij,_kl]    = 0.


Now, all these states sit in degree 0 and are thus killed by h, so that none of them can create further intermediate states. The only nontrivial thing we can now do is to project to the cohomology: [^i_j,_kl] never lies in the cohomology, whereas the traceless part of [^ij,_kl] does. Applied to (<ref>), we obtain

    p[h[h[_k,^ij],^lm],^qr]
           =-1/2(δ^j_k
        ε^ilmnp
        -
        δ^i_kε^jlmnp)[_np,^qr]
       =(
        δ^j_k
        ε^ilmnp
        -δ^i_kε^jlmnp)
        (
            δ^q_n^r_p
            -δ^r_n^q_p
        )
       =δ^[j_k
        ε^i]lmnpδ^[q_n^r]_p.

After graded-skew-symmetrization among the three arguments of the form ^ij, this yields the only nonvanishing component of μ_4.

There are no other μ_i since we have systematically constructed all possible nonzero tree Feynman diagrams (by constructing all possible intermediate states that occur in them).




 §.§ Action on 𝔸⁵

The pure spinor formalism <cit.> associates certain sheaves on (a derived replacement of) the variety of Maurer–Cartan elements to off-shell representations of the super-Poincaré algebra.
In particular, for the ten-dimensional 𝒩=(1,0) super-Poincaré algebra, it associates to the structure sheaf of the Maurer–Cartan variety Specℂ[λ^α]/(γ^μ_αβλ^αλ^β) the pure spinor supermultiplet <cit.>

    M   ℂ[x^μ,θ^α,λ^α]/(γ^μ_αβλ^αλ^β)
       ≅((10000)_𝔬(V))⊗(⋀(00010)_𝔬(V))⊗(⊕_i=0^∞(000i0)_𝔬(V))

where the 𝔬(V)-representation has been specified by Dynkin labels; we use the index notation where V indices are _μ and S_+ indices are _α (hence S_-≅ S_+^* indices are ^α). The formal variables x,θ,λ transform as V,S_-,S_- respectively under 𝔬(V), which in turn determines the 𝔬(V)-representation on M.
The generators carry the degrees[In fact, this grading can be refined into a bigrading <cit.>, and the 𝔭-representation respects this bigrading. But we do not need this fact.]

    |x|   =-2     |θ|    = -1     |λ|   = 0.

 The differential is <cit.>

    d λ^α(∂/∂θ^α-γ^μ_αβθ^β∂/∂ x^μ).

The 𝔬(V)-representation of M extends to a strict representation of 𝔭 as

  

    ρ^(1)_0(_μ)   = ∂/∂ x^μ, 
    ρ^(1)_0(_α)    = ∂/∂θ^α+γ^μ_αβθ^β∂/∂ x^μ.



We can twist this 𝔭-representation to obtain a (strict) representation  of 𝔭_Q on M_Q; the action of p_Q is through (<ref>), but the differential on M_Q is now d+ρ_0^(1)(Q). The cohomology of M_Q is the ring of regular functions on 𝔸^5.

The cohomology of M_Q is

    H(M_Q)=ℂ[z^i],

where z^i=(z^1,…,z^5) is a formal variable of degree -2 transforming under 𝔰𝔩(L) as the defining representation L, such that the L_∞-representation of the subalgebra 𝔦𝔰𝔩(L) of H(𝔭_Q) is

    ρ^(1)(^i_j)   =z^i∂/∂ z^j-1/5δ^i_jz^k∂/∂ z^k,   ρ^(1)(_i)   =∂/∂ z^i,

with ρ^(k)=0 for k≥2.

This corresponds to the fact that the holomorphic twist of ten-dimensional supersymmetric Yang–Mills theory is holomorphic Chern–Simons theory <cit.>, whose space of fields is[up to issues such as holomorphic versus algebraic functions, which we ignore] the algebraic Dolbeault complex of 𝔸^5, namely ℂ[z^i,z̅_i,dz̅^i], and whose cohomology is therefore ℂ[z^i].

Furthermore, the cohomology ℂ[z^i]= H(M_Q) is included into M_Q as a 𝔰𝔩(L)-subrepresentation <cit.>. Since 𝔰𝔩(L) is simple, there exists an sl(L)-equivariant strong deformation retract of cochain complexes
  
    [loop left,"(h,h')"]  p_Q⊕ M_Q [shift left, "(p,p')"]    [shift left, "(i,i')"]  H(𝔭_Q)⊕ H(M_Q),

whose restriction to 𝔭_Q↔ H(𝔭_Q) is the strong deformation-retract (i,p,h) given in (<ref>).
Thus, by taking the minimal model of (𝔭_Q,M_Q) along the strong deformation retract (<ref>), the above 𝔦𝔰𝔩(L)-representation extends into an L_∞-representation of the entirety of H(𝔭_Q), and such minimal models are unique up to quasi-isomorphisms of L_∞-algebra representations.
This minimal model is an L_∞-representation of the L_∞-algebra H( p_Q) on H(M_Q)=ℂ[z^i].
The following theorem computes this minimal model explicitly.

The minimal model of the L_∞-representation of 𝔭_Q on M_Q obtained using the strong deformation retract (i,i';p,p';h,h') is the H(𝔭_Q)-representation on H(M_Q)=ℂ[z^i] given by

    ρ^(1)(^i_j)    = z^i ∂/∂ z^j-1/5^i_jz^k∂/∂ z^k, 
    ρ^(1)(_i)    = ∂/∂ z^i, 
    ρ^(3)(^ij,^kl,^mn)    =
    1/2(z^[iε^j]klp[mz^n]
    -
    z^[iε^j]mnp[kz^l]
           +z^[kε^l]mnp[iz^j]
    -
    z^[kε^l]ijp[mz^n]
           +z^[mε^n]ijp[kz^l]
    -
    z^[mε^n]klp[iz^j])∂/∂ z^p
        P^ijklmn;_pq^rz^pz^q∂/∂ z^r,

with all other components vanishing (in particular, ρ^(1)(^ij)=0),
where P^ijklmn;_pq^r is the projection

    (0100)_𝔰𝔩(L)^∧3=(0020)_𝔰𝔩(L)⊕(2001)_𝔰𝔩(L)→(2001)_𝔰𝔩(L)

in terms of 𝔰𝔩(L) Dynkin labels
or, in Young tableau notation,

    1,1^∧3
        =
        2,2,2⊕3,1,1,1→3,1,1,1.


First, note that for degree reasons, we can only have nonzero ρ^(k) for odd k since ℂ[z^i] and H(𝔭_Q) are all concentrated in even degree and ρ^(k) carries degree 1-k. The leading component ρ^(1) is fixed simply by restriction of the 𝔭_Q-representation ρ_0 on M_Q to i( H(𝔭_Q))⊂𝔭_Q and i'( H(M_Q))⊂ M_Q as 

    ρ^(1)(x) = p'∘ρ_0(i(x))∘ i'

for x∈ H(𝔭_Q) (so that i(x)∈𝔭_Q);
in particular, ρ^(1)(^ij)=0. Furthermore, the ρ^(k) vanish whenever one of the arguments is ^i_j except when k=1 (<ref>).

Hence, it suffices to determine ρ^(3),ρ^(5),ρ^(7),… where all arguments are either ^ij or _i. Now, the possibilities of ρ^(k) are constrained by the fact that all operations μ_k, ρ^(k), and the strong deformation retract (i,i';p,p';h,h') are 𝔰𝔩(L)-equivariant. Suppose that ρ^(k) does not vanish when fed p arguments of the form ^ij and q arguments of the form _i with p+q=k≡12. Then, representation-theoretically, it must yield a nontrivial 𝔰𝔩(L)-representation that is a direct summand of

    ( L^∧2)^∧ p⊗(L^*)^∧ q.

On the other hand, it must carry the degree (1-p-q)+2q=1-p+q, and hence be a sum of terms of the form

    z^n(∂/∂ z)^n+(1-p+q)/2   if 1-p+q≥0
    
            z^n-(1-p+q)/2(∂/∂ z)^n   if 1-p+q≤0,

where z^n refers to a product z^i_1⋯ z^i_n, and similarly for (z)^n. Since z^i transforms as L and ∂/∂ z^i as L^*, this must transform under 𝔰𝔩(L) as a direct summand of

    L^⊙ n⊗(L^*)^⊙(n+(1-p+q)/2)   if 1-p+q≥0
    
            L^⊙(n-(1-p+q)/2)⊗(L^*)^⊙ n   if 1-p+q≤0.

Thus, the two 𝔰𝔩(L)-representations (<ref>) and (<ref>) must share some nontrivial subrepresentations if the corresponding ρ^(k) is to not vanish. <Ref> shows that this is only possible for (p,q)=(3,0) and (p,q)=(4,3), corresponding to

    ρ^(3)(^ij,^kl,^mn)= P^ijklmn;_pq^r(α_0z^pz^q∂/∂ z^r+α_1z^pz^qz^s∂^2/∂ z^r∂ z^s+…)

and

    ρ^(7)(^ij,^kl,^mn,^pq,_r,_s,_t)
    =P^ijklmnpq;_rst^u_v(β_0z^v∂/∂ z^u
        +
        β_1z^vz^w∂^2/∂ z^u∂ z^w+…),

respectively, where P^ijklmn;_pq^r is the projector (0100)^∧3_𝔰𝔩(L)→(2001)_𝔰𝔩(L) as in (<ref>) and P^ijklmnpq;_rst^u_v is the projector (0100)^∧4_𝔰𝔩(L)⊗(0001)^∧3_𝔰𝔩(L)→(1001)_𝔰𝔩(L).

Now, we must solve the coherence relations. One L_∞-representation coherence relation states[In this coherence relation, in our case, terms of the form [ρ^(0),ρ^(4)], [ρ^(2),ρ^(2)], ρ^(4)(μ_1), ρ^(3)(μ_2), and ρ^(2)(μ_3) vanish.]

    [ρ^(1)(_i),ρ^(3)(^jk,^lm,^np)]=ρ^(1)(μ_4(_i,^jk,^lm,^np)).

  Substituting ρ^(1)(_i)=∂/∂ z^i and (<ref>) into (<ref>) yields
  
    [
            ∂/∂ z^i,
            P^jklmnp;_qr^s(α_0z^qz^r∂/∂ z^s+α_1z^qz^rz^t∂^2/∂ z^s∂ z^t+…)
        ]
    
        =
        P^jklmnp;_i^r_qz^q∂/∂ z^r,

  where P^jklmnp;_i^r_q is the projector (<ref>).
  Solving this yields α_0=1 and α_n=0 for n>0.
  
  Next, we have the L_∞-module coherence relation[In this coherence relation, terms of the form [ρ^(0),ρ^(8)], [ρ^(2),ρ^(6)], [ρ^(3),ρ^(5)], [ρ^(4),ρ^(3)], ρ^(1)(μ_8), ρ^(2)(μ_7), …, ρ^(8)(μ_1) vanish.]

    0=[ρ^(7)(^ij,^kl,^mn,^pq,_[r,_s,_t),ρ^(1)(_u])].

Plugging in the ansatz (<ref>) into (<ref>) yields

    0=[P^ijklmnpq;_[rst|^v_w(β_0z^w∂/∂ z^v
        +
        β_1z^wz^x∂^2/∂ z^v∂ z^x+…),∂/∂ z^|u]].

Solving this shows that the coefficients β_0,β_1,… must all vanish since non-constant-coefficient differential operators do not commute with ∂/∂ z^l.



In the L_∞-representation of the L_∞-algebra H(𝔭_Q) on ℂ[z^i] obtained by homotopy transfer from M_Q using the 𝔰𝔩(L)-equivariant strong deformation retract (i,i';p,p';h,h'), we have

    ρ^(k)(^i_j,…) = 0

if k≥2.

We are to perform a homotopy transfer of L_∞-algebras along

    [loop left,"(h,h')"]  p_Q⊕ M_Q [shift left, "(p,p')"]    [shift left, "(i,i')"]  H(𝔭_Q)⊕ H(M_Q).

Let us again use Feynman-diagrammatic terminology to refer to elements as `states'.
If ρ^(k)(^i_j,…)0, this would mean that there is at least one tree Feynman diagram with at least one external leg corresponding to ^i_j. Assuming that k≥2, we have the following possibilities.


  *  The vertex connected to this leg may be directly connected to p' as

    p'(ρ_0(^i_j)X)
        =
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (0,-2);
         [thick, dashed] (2,0) – (0,-2);
         [thick, dashed] (0,-2) – (0,-3);
         at (0,-3.5) p';
         at (2.5,0.5) X;
         at (-2,0.5) ^i_j;
      ,

where X∈ M_Q. In this case, we may assume X to be an intermediate state X=h'(X̃). (The alternative, that X lies in the cohomology, only yields ρ^(1).)

  *  The vertex connected to this leg may feed into h and connect to the rest of the tree as

    p'(… h'(ρ_0(^i_j)X)…)
        =
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (0,-2);
         [thick, dashed] (2,0) – (0,-2);
         [thick,dashed] (0,-2) – (0,-5);
        [color=white, fill=white] (0,-3.5) circle (0.5);
         at (0,-3.5) h';
         at (0,-5.5) ⋮;
         at (2.5,0.5) X;
         at (-2,0.5) ^i_j;
      ,

where X∈ M_Q may be either an intermediate state X=h'(X̃) or belong to the cohomology (X∈ i'( H(M_Q))). In either case, we have h(X)=0.

  *  The vertex connected to this leg may feed into h and connect to the rest of the tree as

    p'(… h[^i_j,x]…)
        =
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (0,-2);
         [thick] (2,0) – (0,-2);
         [thick] (0,-2) – (0,-5);
        [color=white, fill=white] (0,-3.5) circle (0.5);
         at (0,-3.5) h;
         at (0,-5.5) ⋮;
         at (2.5,0.5) x;
         at (-2,0.5) ^i_j;
      ,

where x∈𝔭_Q is either an intermediate state x=h(x̃) or belongs to the cohomology (x∈ i( H(𝔭_Q))). In either case, h(x)=0.


In the first case (<ref>), since the strong deformation retract (i,i';p,p';h,h') is 𝔰𝔩(L)-equivariant, p'(ρ_0(^i_j)h'(X̃)) can be nonzero only if p'(h'(X̃)) is already nonzero. But this cannot be the case since (i',p',h') forms a strong deformation retract, whose definition requires p'∘ h'=0.

Similarly, in the latter case (<ref>),
since the strong deformation retract (i,i';p,p';h,h') is 𝔰𝔩(L)-equivariant, h'(ρ_0(^i_j)X) can be nonzero only if h'(X)) is already nonzero, but this is not possible.

Finally, in the last case (<ref>), again, since the strong deformation retract (i,i';p,p';h,h') is 𝔰𝔩(L)-equivariant, h[^i_j,x] can be nonzero only if h(x) is already nonzero, which is not possible.



For p+q odd and p+q≥3, the 𝔰𝔩(L)-representation

    R_p,q(L^∧2)^∧ p⊗(L^*)^∧ q

has no irreducible components in common with

    R̃_p,q⊕_n=0^∞ L^⊙ n⊗(L^*)^⊙(n+(1+q-p)/2)   if  1+q-p≥0 
    ⊕_n=0^∞ L^⊙(n-(1+q-p)/2)⊗(L^*)^⊙ n   if  1+q-p≤0

except when (p,q)=(3,0) or (4,3), in which case the irreducible components in common are
(2001)_𝔰𝔩(L) and (1001)_𝔰𝔩(L)
respectively.

We must compute the tensor product appearing in R̃_p,q. For 1+q-p≥0 and any nonnegative integer n≥0, it is easy to see that

    (n000)⊗(000(n+(1+q-p)/2))
        =⊕_i=0^n(i00(i+(1+q-p)/2)).

  Similarly, for 1+q-p≤0 we have
  
    ( (n-(1+q-p)/2))000)⊗(000n)
        =⊕_i=0^n((i-(1+q-p)/2))00i).

Thus, R̃_p,q only contains irreducible representations of the form

    (i00(i+(1+q-p)/2))   if 1+q-p≥0
    ( (i-(1+q-p)/2)00i)   if 1+q-p≤0    (i∈{0,1,2,…}).

Given this, iterating over[Recall that (L^∧ 2)^∧ k and (L^*)^∧ l are zero for k>10 and l>5.] p∈{0,1,…,10} and q∈{0,1,…,5} and verifying whether an irreducible representation of the above form appears in R_p,q (using e.g. a computer algebra system), one can see that (p,q)∈{(3,0),(4,3)} are the only possible solutions. 




§ OTHER DIMENSIONS AND AMOUNTS OF SUPERSYMMETRY

The above construction works for general supersymmetry algebras and general supermultiplets, but ten-dimensional 𝒩=(1,0) super-Poincaré algebra seems to be one of the very few in having nontrivial and purely bosonic higher products, at least if one is to start from the vector supermultiplet; a glance at <cit.> shows that this is the only case in which the cohomology is simply a polynomial ring in bosonic variables.

In general, for sufficiently large dimension n, the number of spinorial components in a super-Poincaré algebra increases as 𝒪(2^n) whereas bosonic components increase as 𝒪(n^2). Indeed, already at 14 dimensions, the minimal spinor has 128 components while 𝔦𝔬(14) has 105 components. So we are restricted to 13 or fewer dimensions (unless we twist again to eliminate more supertranslations) even if one did not take into consideration no-go theorems about higher-spin theories (since we ignore dynamics here). Similarly, the 11-dimensional case (starting with the supergravity multiplet) is discussed in <cit.>. There are two possible cases. In one case <cit.>, H(𝔭_Q) contains fermionic elements. Then we expect the action of H(𝔭_Q) to contain a ρ^(2) involving the remaining fermionic elements.
In the other case <cit.>, however, we expect to see a higher action of

    H(𝔭_Q) = (𝔤_2⊕𝔰𝔩(L)⊕ V_7⊗ L⊕ℂ)⋉ L

(which should carry nontrivial μ_2 and μ_4)
on

    𝔸(L^*)=Specℂ[z_1,z_2],

where L is a two-dimensional vector space and V_7 is a seven-dimensional vector space; here, ℂ[z_1,z_2]= H(Ω^0,∙(L)⊗^∙(V)) is the cohomology of the Dolbeault–de Rham complex on two complex and seven real dimensions.

On the other hand, if there are too few dimensions, higher products may vanish. For example, for the four-dimensional 𝒩=1 super-Poincaré algebra, the twist gives a decomposition of four-dimensional complexified spacetime V as V=L⊕ L^*, where L is a two-dimensional vector space, with the twisted super-Poincaré algebra being <cit.>

    𝔭_Q=(
             (L^*)^∧2    L^∧0
    
            (𝔰𝔩(L)⋉ L^∧2)⊕𝔤𝔩(1)_R    L^∧1    L 
    𝔤𝔩(1)_tr     L^∧2    L^*
        ),

where 𝔤𝔩(1)_R is the R-symmetry and 𝔤𝔩(1)_tr is the trace part of 𝔤𝔩(L).
Following the proof of <ref>, we see that there are no higher brackets for H(𝔭_Q) by constructing all possible intermediate states: apply μ_2( L^∧2,-) to L^*[-2] to get L[-2]; applying the homotopy h yields L^∧1; but now applying another μ_2( L^∧2,-) simply kills everything.



§ ACKNOWLEDGEMENTS

The authors (DSHJ, HK, and CASY) were supported in part by the Leverhulme Research Project Grant RPG–2021–092. The authors thank Fridrich Valach III0000-0003-0020-1999 and Charles Strickland-Constable0000-0003-0294-1253 for helpful discussions; casy thanks Brian R. Williams0000-0001-9641-861X, and dshj thanks Ingmar Akira Saberi0000-0002-2005-938X.


§ CONVENTIONS
 
In a basis adapted to the choice of pure spinor Q∈ S_+, the super-Poincaré algebra p =  p^0 ⊕ p^1 ⊕ p^2 in ten dimensions has the basis elements

    ^i_j, ^ij, _ij   ∈ p^0,   , _ij,^i   ∈ p^1,   ^i,_i   ∈ p^2,

with ^ij=-^ji, _ij=-_ji, and _ij=-_ji.
The graded-skew-symmetric Lie brackets among these basis elements are

    [ ^i_j, ^k_l ]    = δ^k_j ^i_l -δ^i_l ^k_j    [ ^i_j, ^kl]    = δ^k_j ^il + δ^l_j ^ik
    [ ^i_j, _kl]    = - δ^i_k _jl - δ^i_l _kj   [ ^ij, _kl]    = δ^i_k^j_l - δ^j_k^i_l - δ^i_l^j_k + δ^j_l^i_k
    [ ^i_j, ^k ]    =  δ^k_j ^i    [ ^i_j, _k ]    = -δ^i_k _j 
    [ ^ij, _k ]    =  δ^j_k ^i - δ^i_k ^j   [ _ij, ^k ]    =  δ_i^k _j - δ_j^k _i
    [ ^ij , ]    = 0    [ _ij, ]    = -_ij
    [ ^ij , _kl]    = (δ^i_l δ^j_k-δ^i_k δ^j_l)    [ _ij,  _kl]    = -ε_ijklm^m 
    [ ^ij , ^k ]    = -1/2ε^ijklm_lm   [ _ij,  ^k ]    = 0 
    [ ^i_j, ]    =  + 1/2δ^i_j    [ ^i_j, _kl]    =  + 1/2δ^i_j _kl - δ^i_k _jl - δ^i_l _kj
    [ ^i_j, ^k ]    = -1/2δ^i_j ^k + δ^k_j ^i    [  , ^i ]    = ^i 
    [ ^i , _jk]    = δ^i_k _j -δ^i_j _k     [ _ij , _kl]    = - ε_ijklm^m

with all remaining brackets of basis elements vanishing. Here the indices i,j,… run over {1,2,3,4,5}, and we employ the Einstein summation convention.
Here ^i_j span gl(5). The basis elements of the subalgebras sl(5) are

    ^i_j^i_j - 1/5δ^i_j^k_k.


unsrturl




 Paper Title: Integrability of the multi-species ASEP with long-range jumps on $\mathbb{Z}$.







Integrability of the multi-species ASEP with long-range jumps on ℤ
    Eunghyun Lee[eunghyun.lee@nu.edu.kz]
 Department of Mathematics,
                                            
 School of Sciences and Humanities, 
 Nazarbayev University, 
 Kazakhstan    
    
===============================================================================================================================================================================================

§ ABSTRACT 
Let us consider a two-sided multi-species stochastic particle model with finitely many particles on ℤ defined as follows. Suppose that each particle is labelled by a positive integer l and waits a random time exponentially distributed with rate 1. It then chooses the right direction to jump with probability p or the left direction with probability q=1-p. If the particle chooses the right direction, it jumps to the nearest site occupied by a particle l'<l (with the convention that an empty site is considered as a particle with labelled 0). If the particle chooses the left direction, it follows the rule of the multi-species totally asymmetric simple exclusion process (mTASEP). We show that this model is integrable and provide the exact formula of the transition probability using the Bethe ansatz. 
 

Key words:
Exactly solvable models, Bethe ansatz, ASEP, TASEP, PushTASEP, Integrability, Symmetry




§ INTRODUCTION

In this paper, we treat a multi-species stochastic particle model with finitely many particles on the infinite lattice ℤ, which has the exclusion rule that each site cannot be occupied by more than one particle. Each particle is labelled by a positive integer l representing its species. A state of the system  can be described by specifying each particle's species and its location.  To be more specific, if  X=(x_1,…, x_N)∈ℤ^N with x_1<…<x_N represents the positions of N particles and the species of the particle at x_i is denoted by π_i, then we denote the state of the system by (X,π)=(x_1,…, x_N, π_1⋯π_N) with π = π_1⋯π_N. In a single-species model, of course, a state is simply denoted by (x_1,…, x_N).

The asymmetric simple exclusion process (ASEP) is the most well-known single-species stochastic particle model. In the ASEP on ℤ, each particle waits an exponential time with rate 1 to choose a direction to jump. It chooses the right direction with probability p or the left direction with probability q=1-p. After choosing a direction, a particle at x immediately jumps to x± 1 if the  site is empty. However, if the target site is already occupied by another particle, the particle gives up jumping and waits a random time again to jump. The technique to find the transition probability of the ASEP on ℤ using the Bethe ansatz is well known <cit.>.

In the multi-species ASEP, a higher-numbered particle has priority over a lower-numbered particle in jumping. To be more specific, if a particle l at x chooses to jump to x ± 1  already occupied by l'<l, then the particle l can indeed jump to the site by switching their positions.  However, if l'≥ l, then the particle  l gives up jumping to the site occupied by l'. This model has been extensively studied in many directions <cit.>.

Another stochastic particle model with rules as simple as those of the ASEP is the drop-push model <cit.>. In the drop-push model, each particle jumps to the next site on the right after an exponential random time with rate 1. Unlike the ASEP, a particle can jump even if the target site is already occupied by pushing the particle occupying the site. For example, if  a particle at x jumps to x+1 occupied by another particle, the particle at x jumps to x+1, pushing the particle at x+1 to x+2. In this case, if x+2 is also occupied, then the particle pushed to x+2 also pushes the particle at x+2 to x+3 (Figure <ref> illustrates the definition of the drop-push model).



 A multi-species version of the (one-sided) drop-push model was introduced as the frog model in <cit.>. Additionally, the stationary distribution of the frog model  was recently studied <cit.>. In particular, <cit.> treated various multi-species particle models in a unified way to study the stationary distribution. Another direction of extending to a multi-species version of the (one-sided) drop-push model is found in <cit.>. A straightforward extension of the  one-sided drop-push model to the one where a particle jumps to the right with rate p or to the left with rate q=1-p is not integrable in the sense that the Bethe ansatz is not applicable. However, it is known that if the jumping rates satisfy certain conditions, the two-sided model is integrable <cit.>.  A natural question arises as to whether there exists an integrable multi-species version of the model in <cit.>. Unfortunately, the authors' best efforts to find such a version have not been fruitful. Nevertheless, in this paper, we introduce a two-sided integrable multi-species model incorporating  the rule of the long-range jumps partially.

 

Definition of the model
 

In this paper, we show that if particles follow the rule of the frog model when they jump to the right and follow the rule of the multi-species TASEP when they jump to the left, then the corresponding two-sided model is integrable. Hence, we can use the Bethe ansatz to find the transition probability. For our task, overall, we will take a similar approach to the method in <cit.>. The precise definition of the model is as follows.

A particle  l at x waits an exponential time with rate 1 to jump to the right with probability p or to the left with probability q=1-p. If l chooses the right direction, it jumps to the nearest right site occupied by  l'<l (including  l'=0 which corresponds to an empty site), and  l' immediately jumps to the nearest right site occupied by l”<l'. If  l at x chooses the left direction to jump and x-1 is occupied by l'<l, then l jumps to x-1 and l'  moves to x. However, if  l'≥ l, then l gives up jumping to x-1.

We will call this model the multi-species ASEP  with long-range jumps. The definition of our model with p=1 can be motivated by the basic coupling  of two drop-push models with different configurations. The idea of viewing a low-numbered particle in the two-species ASEP as a discrepancy between the configurations of two ASEPs is well known <cit.>. Using the same idea, let η_t be a drop-push model and η_t(x) represent the state of site x at time t; that is, η_t(x) = 0 if x is unoccupied and η_t(x) = 1 if it is occupied. Let (η_t, ζ_t) be the coupled process of the drop-push models η_t and ζ_t with η_0(x) ≤ζ_0(x) for all x. If we think of a site x with η_t(x) = ζ_t(x) = 1 as being occupied by a particle of species 2, and  a site x with η_t(x) =0, ζ_t(x) = 1 as being occupied by a particle of species 1, then the evolution of the coupled process is  the same as that of our model p=1 in Definition <ref>. Similarly, the two-sided model with general p in Definition <ref> can be interpreted via the basic coupling of  two-sided corresponding single species models. The two-sided single-species model where a particle follows the rule of the drop-push model when it moves to the right and follows the rule of the TASEP when it moves to the left was studied in <cit.>, so our model can be seen as a multi-species version of the model in <cit.>.  Figure <ref> illustrates the interpretation by the basic coupling.



Notations

 

Suppose that there are N particles labelled by positive integers a_1,…, a_N, and let ℳ=[a_1,…,a_N] be the multi-set of those N integers. Let P_(Y,ν)(X,π;t) be the probability that the system will be in state (X,π) at time t given that the system is in state (Y,ν) at time 0. We view π and ν  as permutations of  ℳ of N elements. Of course, if π and ν are permutations of two different multi-sets, then  P_(Y,ν)(X,π;t)=0.

We will consider an arbitrary multi-set [a_1,…,a_N] with each a_i ∈{1,…, N} to cover  all possible compositions of species. For this purpose, let 𝐏_Y(X;t) be  the N^N × N^N matrix such that  rows and columns are labelled with 1⋯ 1, …, N⋯ N and the (π,ν)-element is P_(Y,ν)(X,π;t).  According to the definition of the model, 𝐏_Y(X;t) is a lower triangular matrix when rows and columns are labelled lexicographically from 1 ⋯ 1 to N ⋯ N.  For example, for N=2,

    𝐏_Y(X;t) =
        11    12    21    22 
    
        11     P_(Y,11)(X,11;t)      0      0     0 
    
        12     0       P_(Y,12)(X,12;t)     0      0 
    
        21     0       P_(Y,12)(X,21;t)      P_(Y,21)(X,21;t)      0 
    
        22    0      0    0      P_(Y,22)(X,22;t)
        .

Throughout this paper, 𝐈 denotes the identity matrix and 0 denotes the zero matrix. If needed, we write   𝐈_n and 0_n to specify that their dimensions are n by n. We write  𝐌_(π,ν) for  the (π,ν)-element of a matrix 𝐌.

 

Organization of the paper
 

In Section <ref>, we investigate the case of N=2 as a preliminary building block for the case of general N.  In Section <ref>, we investigate the case of N=3. The arguments for N=3  build upon the results for N=2. However, in the case of N=3, two-body reducibility and the Yang-Baxter equation are discussed for integrability. Moving from N=2 to N=3 is not a trivial extension, but if the integrability of N=3 is proved, then the integrability of general N is trivially confirmed. Hence, we provide detailed  arguments for the case of N=3 in Section  <ref> and only the results for general N  in Section <ref>. This approach also helps us understand the integrability of the model better, reducing the heavy notations for general N. In Section <ref>, we provide the exact formula for the transition probability for an arbitrary N-particle system.


§ N=2



 §.§ Master equations

Each element P_(Y,ν)(X,π;t) in (<ref>) satisfies the master equation  and the initial condition:

    P_(Y,ν)(X,π;0) =
    
    1      if (Y,ν)=(X,π)
    
    0     otherwise.

For a fixed Y, the form of the master equation for P_(Y,ν)(X,π;t) depends on X, π,ν. The master equation for P_(Y,ν)(X,π;t) when X=(x_1,x_2) with  x_1< x_2-1 is the (π,ν)-element of the matrix equation:

    d/dt𝐏_Y(x_1,x_2;t) =      p 𝐏_Y(x_1-1,x_2;t) + p 𝐏_Y(x_1,x_2-1;t) 
    
                                          +      q 𝐏_Y(x_1+1,x_2;t) + q 𝐏_Y(x_1,x_2+1;t) - 2𝐏_Y(x_1,x_2;t)

where the differentiation on the left side in  (<ref>) denotes the differentiation of each matrix element.

The master equation for P_(Y,ν)(X,π;t)  when  X=(x,x+1) is
the (π,ν)-element of the matrix equation:

    d/dt𝐏_Y(x,x+1;t) = p𝐏_Y(x-1,x+1;t) +   p𝐁 𝐏_Y(x-1,x;t) - 2p𝐏_Y(x,x+1;t) 
       + q𝐏_Y(x,x+2;t)  + q𝐁_1 𝐏_Y(x,x+1;t)- q𝐁_2𝐏_Y(x,x+1;t)

where

    𝐁 =     11     12     21     22  
    
        11     1     0     0     0  
    
        12     0     0     0     0  
    
        21     0     1     1     0  
    
        22     0     0     0     1
      ,


    𝐁_1 =     11     12     21     22  
    
        11     0     0     0     0  
    
        12     0     0     0     0  
    
        21     0     1     0     0  
    
        22     0     0     0     0
      ,

and

    𝐁_2 =     11     12     21     22  
    
        11     1     0     0     0  
    
        12     0     2     0     0  
    
        21     0     0     1     0  
    
        22     0     0     0     1
      .

Note that the (ij,lm)-element of p𝐁 represents the instantaneous rate of transition from state (x-1,x,lm) to state (x,x+1, ij), while the (ij,lm)-element of q𝐁_1 represents the instantaneous rate of transition from state (x,x+1,lm) to state (x,x+1, ij). Moreover,  (<ref>) can be expressed as:

    d/dt𝐏_Y(x,x+1;t) =     p𝐏_Y(x-1,x+1;t) +   p𝐁 𝐏_Y(x-1,x;t)  
         + q𝐏_Y(x,x+2;t)  + q𝐁 𝐏_Y(x,x+1;t)- 2𝐏_Y(x,x+1;t)

due to the relation 𝐁 - 𝐁_1 +  𝐁_2  = 2𝐈.

The matrix form of the initial condition (<ref>)  is given by:

    𝐏_Y(X;0) =
    𝐈    if X = Y, 
    0    if X≠Y.



 §.§ Bethe ansatz

In this section, we extend the  Bethe ansatz method <cit.> to find solutions of  the matrix equations  (<ref>) and (<ref>).

Recall  that 𝐏_Y(X;t) in (<ref>) was defined for the physical region, meaning for Y=(y_1,y_2) and X=(x_1,x_2) with y_1<y_2 and x_1<x_2, respectively. Let 𝐔(X;t) be the 4× 4 matrix obtained by replacing P_(Y,ν)(X,π;t) in (<ref>) with some functions U_(Y,ν)(X,π;t) defined for all  X,Y ∈ℤ^2. Suppose that 𝐔(X;t)= 𝐔(x_1,x_2;t) satisfies

    d/dt𝐔(x_1,x_2;t) =      p𝐔(x_1-1,x_2;t) + p𝐔(x_1,x_2-1;t) 
    
    +     q𝐔(x_1+1,x_2;t) + q𝐔(x_1,x_2+1;t) - 2𝐔(x_1,x_2;t)

for all x_1,x_2 ∈ℤ and
 
    𝐔(x,x;t)= 𝐁 𝐔(x-1,x;t)

for all x ∈ℤ. Then, for x_1=x and x_2 = x+1, (<ref>) equals

    d/dt𝐔(x,x+1;t) =     p𝐔(x-1,x+1;t) +   p𝐁 𝐔(x-1,x;t)  
         + q𝐔(x,x+2;t)  + q𝐁 𝐔(x,x+1;t)- 2𝐔(x,x+1;t)

which is in form of  (<ref>). Thus, if 𝐔(x_1,x_2;t)  satisfies  (<ref>) and  the so called, the boundary condition (<ref>), then it obviously satisfies (<ref>) when x_1< x_2-1, and satisfies (<ref>) when x_1=x and x_2 = x+1.

By  the ansatz of the separation of variables U_(Y,ν)(X,π;t)= U_(Y,ν)(X,π)T(t) and letting 𝐔(X) = 𝐔(x_1,x_2) be the matrix such that 𝐔(X;t) =𝐔(X) T(t) and then substituting it into (<ref>), we obtain T(t) = e^ε t  for some constant  ε (a constant with respect to t,x_1,x_2). The equations for spatial variables of (<ref>) and (<ref>) are

    ε𝐔(x_1,x_2) =      p𝐔(x_1-1,x_2) + p𝐔(x_1,x_2-1)  
          + q𝐔(x_1+1,x_2) + q𝐔(x_1,x_2+1) - 2𝐔(x_1,x_2)

 and
 
    𝐔(x,x)= 𝐁 𝐔(x-1,x).


The Bethe ansatz solution of the matrix equation (<ref>) is given by

    𝐔(x_1,x_2)= 𝐀_12ξ_1^x_1ξ_2^x_2 +𝐀_21ξ_2^x_1ξ_1^x_2

for any  nonzero complex numbers ξ_1 and ξ_2. Here, 𝐀_12 and 𝐀_21 are 4× 4 matrices  similar in structure to 𝐔(x_1,x_2), meaning that the positions of the zero elements in  𝐀_12, 𝐀_21 and 𝐔(x_1,x_2) correspond. The nonzero elements of 𝐀_12 and 𝐀_21 are constants (with respect to t, x_1,x_2). By substituting (<ref>) in (<ref>), we  find

    ε =p/ξ_1 + p/ξ_2 +  qξ_1 + qξ_2- 2.

Additionally, since (<ref>) must satisfy (<ref>), substituting (<ref>) into (<ref>) yields

    𝐀_21 =  -(𝐈 - 𝐁1ξ_2)^-1(𝐈 - 𝐁1ξ_1)𝐀_12.

It is important to note that  𝐈 - 𝐁1ξ is  block-diagonal, allowing us to  compute

    -(𝐈 - 𝐁1ξ_2)^-1(𝐈 - 𝐁1ξ_1)

block-wise.  Specifically, for the 2× 2 blocks in the center of (<ref>),
we have

    -[𝐈_2 -     12     21
    
    12    0    0 
    
    21   1ξ_2   1ξ_2]^-1[𝐈_2 -     12     21
    
    12    0    0 
    
    21   1ξ_1   1ξ_1] =     12     21
    
    12    -1    0 
    
    21    T_21     S_21

where

    S_21 =  -1-1/ξ_1/1-1/ξ_2  and  T_21 = 1/ξ_1-1/ξ_2/1-1/ξ_2.

Hence, we obtain

    𝐀_21 =
       11    12    21    22 
    
                                      11         S_21    0     0     0 
    
                                      12         0     -1     0      0
    
                                      21         0    T_21    S_21    0 
    
                                      22         0     0     0      S_21
    𝐀_12


as a condition for (<ref>) to be satisfied. Thus, 𝐔(x_1,x_2)e^ε t with  (<ref>) and  (<ref>) satisfies  (<ref>) when x_1<x_2-1, and  (<ref>) when x_1=x and x_2 = x+1. (For now, 𝐔(x_1,x_2)e^ε t is not the transition probability yet because we have not shown that it satisfies the initial condition.)

A key idea of defining our two-sided model  as in Definition <ref>  comes from the observation that the boundary condition for the one-sided multi-species model with long-range jumps (to the right direction) and the boundary condition for the multi-species TASEP (to the left direction) are the same, as  given by (<ref>).



§ N=3

For N=3, 𝐏_Y(X;t)= 𝐏_Y(x_1,x_2,x_3;t) is a 27 × 27 matrix whose rows and columns are labelled by 111,112,…, 333.  For X=(x_1,x_2,x_3) with x_1< x_2-1<x_3-2, we immediately obtain the master equation in matrix form:

    d/dt𝐏_Y(x_1,x_2,x_3;t)=     p𝐏_Y(x_1-1,x_2,x_3;t) + p𝐏_Y(x_1,x_2-1,x_3;t) + p𝐏(x_1,x_2,x_3-1;t)
    
    +     q𝐏_Y(x_1+1,x_2,x_3;t) + q𝐏_Y(x_1,x_2+1,x_3;t) + q𝐏(x_1,x_2,x_3+1;t)
        - 3𝐏_Y(x_1,x_2,x_3;t).



 §.§ Master equation for X=(x_1,x_2,x_3) with x_1=x, x_2=x+1 and x_2<x_3-1

In this case, we could find  the master equation in matrix form by finding all master equations for the elements of  𝐏_Y(X;t), but we will use another approach. Considering the positions of particles and the definition of the model, we see that the master equation for (x,x+1, x_3)  should be in the form of

    d/dt𝐏_Y(x,x+1,x_3;t) =       p𝐏_Y(x-1,x+1,x_3;t) + 𝐂 𝐏_Y(x-1,x,x_3;t)
    
    +     p𝐏_Y(x,x+1,x_3-1;t)-3p𝐏_Y(x,x+1,x_3;t)
    
    +    𝐂_1𝐏_Y(x,x+1,x_3;t) + q𝐏_Y(x,x+2,x_3;t)
    
    +     q𝐏_Y(x,x+1,x_3+1;t)-𝐂_2𝐏_Y(x,x+1,x_3;t)

for some 27 × 27 matrices 𝐂, 𝐂_1, 𝐂_2.


  §.§.§ Matrix 𝐂

The  (ijk,lmn)-element of 𝐂 is the instantaneous rate of transition from state (x-1,x,x_3,lmn) to state (x,x+1,x_3,ijk). In fact, if the transition

    (x-1,x,x_3,lmn)  ⟶  (x,x+1,x_3,ijk)

is permissible, then it occurs with rate p by the definition of the model, and if (<ref>) is not permissible, its rate is obviously 0. The permissibility of (<ref>) depends on ijk and lmn. Since x_3 is kept the same during the transition, it means that k=n. If k≠ n,  (<ref>) is not possible so its rate is 0. In the case that k=n, we may say that the permissibility of  (<ref>) depends on the permissibility of the transition from (x-1,x,lm) to (x,x+1,ij), which is actually  encoded in  (<ref>). Now, let us view 𝐂 as a 9 × 9 partitioned matrix whose rows and columns are labelled by 11·, 12·, 13·, 21·, 22·, 23·, 31·, 32·, 33·, and the (ij·, lm·)-element is a 3× 3 matrix whose rows are labelled by ij1,ij2,ij3 and columns are labelled by lm1,lm2,lm3. According to the permissibility of the transition from (x-1,x,lm) to (x,x+1,ij)  available in (<ref>) and the requirement k=n for  (<ref>), we obtain

    𝐂=     11·    12·    13·    21·    22·    23·    31·    32·    33·
    
         11·    p𝐈   0   0   0   0   0   0   0   0
    
         12·   0   0   0   0   0   0   0   0   0
    
         13·   0   0   0   0   0   0   0   0   0
    
          21·   0     p𝐈   0     p𝐈   0   0   0   0   0
    
         22·   0   0   0   0     p𝐈   0   0   0   0
    
         23·   0   0   0   0   0   0   0   0   0
    
         31·   0   0     p𝐈   0   0   0     p𝐈   0   0
    
        32·   0   0   0   0   0     p𝐈   0     p𝐈   0
    
         33·   0   0   0   0   0   0   0   0    p𝐈
    .

If we let

    𝐁=     11     12     13     21     22    23     31     32     33
    
         11     1    0     0     0     0     0     0     0    0
    
         12      0     0      0      0      0      0    0       0    0
    
         13      0     0      0     0     0      0     0     0     0
    
          21   0      1      0      1     0      0      0     0    0
    
         22      0      0      0      0      1      0      0     0    0
    
         23     0      0      0      0    0      0      0      0    0
    
         31     0     0     1     0      0     0     1     0     0
    
        32       0      0     0      0     0      1      0     1   0
    
         33     0      0     0     0      0      0     0      0    1
    ,

then we have a succinct expression:

    𝐂= p𝐁⊗𝐈.


We have already used the notation 𝐁 in (<ref>) for N=2, but here for N=3 we use it again because later the notation 𝐁 will be defined for general N consistently with (<ref>) and  (<ref>).



  §.§.§ Matrices 𝐂_1 and 𝐂_2

The  (ijk,lmn)-element of 𝐂_1 is the instantaneous rate of transition

    (x,x+1,x_3,lmn)  ⟶  (x,x+1,x_3,ijk)

which is due to the interchange of positions of particles at x and x+1. Hence, the instantaneous rate of (<ref>) is q if the transition is permissible, and otherwise, the rate is 0. Since x+1< x_3-1 in (<ref>), it means that k=n, and we see that the permissibility of (<ref>) depends on the permissibility of

    (x,x+1,lm)  ⟶  (x,x+1,ij)

which is actually encoded in  (<ref>). Using a similar argument as in Section <ref>, if we let

    𝐁_1=     11     12     13     21     22    23     31     32     33
    
         11     0    0     0     0     0     0     0     0    0
    
         12      0     0      0      0      0      0    0       0    0
    
         13      0     0      0     0     0      0     0     0     0
    
          21   0      1      0      0     0      0      0     0    0
    
         22      0      0      0      0      0      0      0     0    0
    
         23     0      0      0      0    0      0      0      0    0
    
         31     0     0     1     0      0     0     0     0     0
    
        32       0      0     0      0     0      1      0     0   0
    
         33     0      0     0     0      0      0     0      0    0
    ,

 we have

    𝐂_1 = q𝐁_1 ⊗𝐈.


We have already used the notation 𝐁_1 in (<ref>) for N=2, but here for N=3 we use it again because later the notation 𝐁_1 will be defined for general N consistently with (<ref>) and  (<ref>).

The matrix 𝐂_2 is a diagonal matrix such that the (ijk,ijk)-element is the instantaneous rate of jumping out of the state (x,x+1,x_3,ijk) due to particles' jumps to the left. The rate of the jump of the particle k at x_3 to the left direction is q and the rate of the jump of i at x and j at x+1 is encoded in (<ref>). Hence, if we let

    𝐁_2=     11     12     13     21     22    23     31     32     33
    
         11     1    0     0     0     0     0     0     0    0
    
         12      0     2      0      0      0      0    0       0    0
    
         13      0     0      2     0     0      0     0     0     0
    
          21   0      0      0      1     0      0      0     0    0
    
         22      0      0      0      0      1      0      0     0    0
    
         23     0      0      0      0    0      2      0      0    0
    
         31     0     0     0     0      0     0     1     0     0
    
        32       0      0     0      0     0      0      0     1   0
    
         33     0      0     0     0      0      0     0      0    1
    ,

then we obtain

    𝐂_2 = q𝐁_2 ⊗𝐈 + q𝐈_27.


We have already used the notation 𝐁_2 in (<ref>) for N=2, but here for N=3 we use it again because later the notation 𝐁_2 will be defined for general N consistently with (<ref>) and  (<ref>).

Moreover, observing that 𝐁_2 + 𝐁 - 𝐁_1 = 2𝐈_9,  we see that (<ref>) is equivalent to

    d/dt𝐏_Y(x,x+1,x_3;t)  =  p𝐏_Y(x-1,x+1,x_3;t) + p(𝐁⊗𝐈) 𝐏_Y(x-1,x,x_3;t) 
        + p𝐏_Y(x,x+1,x_3-1;t) +q(𝐁⊗𝐈)𝐏_Y(x,x+1,x_3;t) + q𝐏_Y(x,x+2,x_3;t)
        +q𝐏_Y(x,x+1,x_3+1;t)-3𝐏_Y(x,x+1,x_3;t).



 §.§ Master equation for X=(x_1,x_2,x_3) with x_1+1<x_2, x_2=x and x_3=x+1

Considering the positions of particles and the definition of the model, we see that the master equation  should be in the form:

    d/dt𝐏_Y(x_1,x,x+1;t) =       p𝐏_Y(x_1-1,x,x+1;t) + p𝐏_Y(x_1,x-1,x+1;t)
    
    +    𝐄 𝐏_Y(x_1,x-1,x;t)-3p𝐏_Y(x_1,x,x+1;t)
    
    +      q𝐏_Y(x_1+1,x,x+1;t)+ 𝐄_1𝐏_Y(x_1,x,x+1;t)
    
    +     q𝐏_Y(x_1,x,x+2;t)-𝐄_2𝐏_Y(x_1,x,x+1;t)

for some 27 × 27 matrices 𝐄, 𝐄_1, 𝐄_2.

First, we find the matrix 𝐄. The (ijk,lmn)-element of 𝐄 is the instantaneous rate of  the transition

    (x_1,x-1,x,lmn)  ⟶  (x_1,x,x+1,ijk).

If the transition (<ref>) is permissible, then it occurs with rate p by the definition of the model. If (<ref>) is not permissible, its rate is obviously 0.     Since x_1 is kept the same during the transition (<ref>), it means that i=l. If  i≠ l, then  (<ref>) is not possible, so its rate is  0. In the case that i=l,  we may say that the permissibility of   (<ref>) depends on the permissibility of the transition from (x-1,x,mn) to (x,x+1,jk), which is actually encoded in the matrix 𝐁 in (<ref>).

Now, let us view 𝐄 as a 3× 3 partitioned matrix whose rows and columns are labelled by 1**, 2**, 3*. The (i**, l**)-element is a 9 × 9 matrix whose rows and columns are labelled by i11,i12,…, i33 and l11,l12,…, l33, respectively. Then, using 𝐁 in (<ref>) and the requirement i =l for  the transition (<ref>), we obtain

    𝐄=    1**     2**     3** 
    
        1**     p𝐁   0   0
    
        2**     0    p𝐁   0
    
        3**     0   0     p𝐁
     = 𝐈⊗ p𝐁.

Using similar arguments as in Section <ref>, we obtain:
 
    𝐄_1 =  𝐈⊗ q𝐁_1

 with  𝐁_1 in (<ref>), and
 
    𝐄_2 = 𝐈⊗ q𝐁_2  + q𝐈_27

with 𝐁_2 in (<ref>). Moreover, observing that 𝐁_2 + 𝐁 - 𝐁_1 = 2𝐈_9, we see that (<ref>) is equivalent to

    d/dt𝐏_Y(x_1,x,x+1;t) =   p𝐏_Y(x_1-1,x,x+1;t) + p𝐏_Y(x_1,x-1,x+1;t)
       + p(𝐈⊗𝐁)  𝐏_Y(x_1,x-1,x;t)+  q𝐏_Y(x_1+1,x,x+1;t)
        +q(𝐈⊗𝐁)𝐏_Y(x_1,x,x+1;t)+ q𝐏_Y(x_1,x,x+2;t)-3𝐏_Y(x_1,x,x+1;t).



 §.§ Master equation for X=(x,x+1,x+2)

Considering all possible states that can transition to states with (x,x+1,x+2), we see that  the master equation should be in the form:

    d/dt𝐏_Y(x,x+1,x+2;t) =     p𝐏_Y(x-1,x+1,x+2;t) + 𝐀𝐏_Y(x-1,x,x+2;t) 
    
    +    𝐀_1 𝐏_Y(x-1,x,x+1;t) -3p 𝐏_Y(x,x+1,x+2;t) 
    
    +     q𝐏_Y(x,x+1,x+3;t) + 𝐀_2 𝐏_Y(x,x+1,x+2;t)
    
    -    𝐀_3 𝐏_Y(x,x+1,x+2;t)

for some  27 × 27 matrices  𝐀,𝐀_1, 𝐀_2, 𝐀_3.

The (ijk, lmn)-element of 𝐀 is the instantaneous rate of transition from  (x-1,x,x+2,lmn) to  (x,x+1,x+2,ijk).  Using the same argument as in Section <ref>, we find that 𝐀 = 𝐂 =  p𝐁⊗𝐈.

The (ijk, lmn)-element of 𝐀_1 is the instantaneous rate of transition

    (x-1,x,x+1,lmn)   ⟶  (x,x+1,x+2,ijk)

We could find all matrix elements of  𝐀_1 by checking the permissibility of (<ref>), but we will take a different approach by considering an alternative perspective on the definition of our model. Specifically, we will interpret the rule of the long-range jump as consecutive two-body interactions.

 (equivalent to Definition <ref>)   A particle  l at position x waits an exponential time with rate 1 and then jumps to x+1 with probability p or to x-1 with probability q=1-p. When the particle jumps to x ± 1, if x ± 1 is already occupied by a particle l', then x ± 1 instantaneously accommodates both l and l'  such that max(l,l')  is positioned to the left of min(l,l') at x ± 1, and min(l,l') immediately jumps to (x ± 1) +1.

Figure <ref> and <ref> illustrate some transitions as described in Definition <ref>.




According to Definition <ref>, the transition (<ref>) can be interpreted as

    (x-1,x,x+1,lmn)   (x,x,x+1,l'm'n')
         (x,x+1,x+1,l”m”n”)  (x,x+1,x+2,ijk)

where  and  occur instantaneously. We see that  for (<ref>) to be possible it must be that n = n', l' = l” and l”m”n”=ijk. Using the arguments in Section <ref> and <ref>, the permissibility of   is represented by 𝐁⊗𝐈 and  the permissibility of   is represented by 𝐈⊗𝐁. It is obvious that  is represented by the identity matrix.  Hence, we conclude that L

    𝐀_1 =  p(𝐈⊗𝐁)(𝐁⊗𝐈).

The (ijk,lmn)-element of 𝐀_2 is the instantaneous rate of transition from  (x,x+1,x+2,lmn) to  (x,x+1,x+2,ijk). This transition is possible when either m>l or n>m according to the rule of the mTASEP to the left direction. Hence, using the arguments in Section <ref>, we have

    𝐀_2 = q(𝐁_1 ⊗𝐈) +  q(𝐈⊗𝐁_1).

𝐀_3 is a diagonal matrix such that the (ijk,ijk)-element represents the rate of jumping out of the state (x,x+1,x+2,ijk) due to particles' jumps to the left direction. Recall that q𝐁_2 ⊗𝐈 describes the rate of jumping out of the state (x,x+1,x_3,ijk) with x+1 < x_3-1 due to the first particle's jump  to the left  and the second particle's jump  to the left (which is possible only when j>i) and 𝐈⊗ q𝐁_2  describes the rate of jumping out of the state (x_1,x,x+1,ijk) with x_1<x-1 due to the second particle's jump to the left  and third particle's jump to the left (which is possible only when k>j). Hence, it must be that

    𝐀_3 = q𝐁_2 ⊗𝐈  + 𝐈⊗ q𝐁_2 - q𝐈_27.

Again, using that 𝐁_2 + 𝐁 - 𝐁_1 = 2𝐈_9, we see that (<ref>) is equivalent to

    d/dt𝐏_Y(x,x+1,x+2;t) = p𝐏_Y(x-1,x+1,x+2;t) + p(𝐁⊗𝐈) 𝐏_Y(x-1,x,x+2;t) 
       +p(𝐈⊗𝐁)(𝐁⊗𝐈) 𝐏_Y(x-1,x,x+1;t) +q𝐏_Y(x,x+1,x+3;t) 
       +  q(𝐁⊗𝐈 +  𝐈⊗𝐁) 𝐏_Y(x,x+1,x+2;t)- 3 𝐏_Y(x,x+1,x+2;t).



 §.§ Two-body reducibility

We obtained four different forms  of the master equation  (<ref>), (<ref>), (<ref>), and (<ref>) for 𝐏_Y(X;t), depending on X=(x_1,x_2,x_3). Now, we will show that (<ref>), (<ref>), and (<ref>) can be derived from the equation in the form of  (<ref>) and the boundary conditions that describe two-particle interactions.

We naturally extend the notations 𝐔(X;t) and  U_(Y,ν)(X,π;t) for N=2 in Section <ref>  to N=3. Suppose that 𝐔(X;t) = 𝐔(x_1,x_2,x_3;t) satisfies
 
    d/dt𝐔(x_1,x_2,x_3;t)=     p𝐔(x_1-1,x_2,x_3;t) + p𝐔(x_1,x_2-1,x_3;t) + p𝐔(x_1,x_2,x_3-1;t)
    
    +    q𝐔(x_1+1,x_2,x_3;t) + q𝐔(x_1,x_2+1,x_3;t) + q𝐔(x_1,x_2,x_3+1;t)
        - 3𝐔(x_1,x_2,x_3;t)

for all x_1,x_2,x_3 ∈ℤ. If 𝐔(x_1,x_2,x_3;t) also satisfies

    (𝐁⊗𝐈) 𝐔(x-1,x,x_3;t) =  𝐔(x,x,x_3;t)   for all x,x_3 ∈ℤ,

then for x_1=x, x_2 = x+1, x_2<x_3-1, (<ref>) becomes
  
    d/dt𝐔(x,x+1,x_3;t)  =  p𝐔(x-1,x+1,x_3;t) + p(𝐁⊗𝐈) 𝐔(x-1,x,x_3;t) 
        + p𝐔(x,x+1,x_3-1;t) +q(𝐁⊗𝐈)𝐔(x,x+1,x_3;t) + q𝐔(x,x+2,x_3;t)
        +q𝐔(x,x+1,x_3+1;t)-3𝐔(x,x+1,x_3;t)

which is in the form of (<ref>). Similarly, if  𝐔(x_1,x_2,x_3;t) satisfies

    ( 𝐈⊗𝐁)𝐔(x_1,x-1,x;t) =𝐔(x_1,x,x;t)   for all x_1,x ∈ℤ,

then (<ref>) becomes
 
    d/dt𝐔(x_1,x,x+1;t) =   p𝐔(x_1-1,x,x+1;t) + p𝐔(x_1,x-1,x+1;t)
       + p(𝐈⊗𝐁)  𝐔(x_1,x-1,x;t)+  q𝐔(x_1+1,x,x+1;t)
        +q(𝐈⊗𝐁)𝐔(x_1,x,x+1;t)+ q𝐔(x_1,x,x+2;t)-3𝐔(x_1,x,x+1;t).

which is in the form of (<ref>).

Now, we find a condition for (<ref>) to be in the form of (<ref>). For this purpose, we substitute x_1=x,x_2 = x+1, x_3 = x+2 in (<ref>) and compare it with (<ref>). Then, we can see that

    p𝐔(x,x,x+2;t) + p𝐔(x,x+1,x+1;t) +  q𝐔(x+1,x+1,x+2;t)  + q𝐔(x,x+2,x+2;t) 
       = p(𝐁⊗𝐈) 𝐔(x-1,x,x+2;t) +p(𝐈⊗𝐁)(𝐁⊗𝐈) 𝐔(x-1,x,x+1;t) 
       + q(𝐁⊗𝐈 +  𝐈⊗𝐁)𝐔(x,x+1,x+2;t)

for all x ∈ℤ if and only if  (<ref>) is in the form of (<ref>). To satisfy (<ref>), we set:

    p𝐔(x,x,x+2;t) + p𝐔(x,x+1,x+1;t) = 
       p(𝐁⊗𝐈) 𝐔(x-1,x,x+2;t) +p(𝐈⊗𝐁)(𝐁⊗𝐈) 𝐔(x-1,x,x+1;t)

and

    q𝐔(x+1,x+1,x+2;t)  + q𝐔(x,x+2,x+2;t) = 
        q(𝐁⊗𝐈 +  𝐈⊗𝐁)𝐔(x,x+1,x+2;t).

Now, we will show that  (<ref>) and (<ref>)  are satisfied if the boundary conditions (<ref>) and (<ref>)  are satisfied. It is easy to see that (<ref>) is satisfied by  (<ref>) and (<ref>). Let us show that  (<ref>) is satisfied by (<ref>) and (<ref>). The first term on the left side of (<ref>) is the same as the first term on the right side of (<ref>) by (<ref>). Also, the second term on the left side of (<ref>) is the same as the second term on the right side of (<ref>) because

    ( 𝐈⊗𝐁)(𝐁⊗𝐈)𝐔(x-1,x,x+1;t) = (𝐈⊗𝐁)𝐔(x,x,x+1;t)

by (<ref>), and

    (𝐈⊗𝐁)𝐔(x,x,x+1;t) = 𝐔(x,x+1,x+1;t)

by (<ref>).


 §.§ Bethe ansatz solution

In the previous section, we showed that if 𝐔(X;t) satisfies (<ref>), (<ref>) and (<ref>), then it satisfies the master equation  (<ref>), (<ref>), (<ref>), or (<ref>) for each X in the corresponding physical region. In this section, we find such a solution 𝐔(X;t) that satisfies (<ref>), (<ref>) and (<ref>). Extending the idea and the notations in Section <ref>,  we separate variables in the form U(X;t) = U(X)T(t). This leads to the following spatial equations derived from (<ref>), (<ref>) and (<ref>):

    ε𝐔(x_1,x_2,x_3) =     p𝐔(x_1-1,x_2,x_3) + p𝐔(x_1,x_2-1,x_3) + p𝐔(x_1,x_2,x_3-1) 
    
    +     q𝐔(x_1+1,x_2,x_3) + q𝐔(x_1,x_2+1,x_3) + q𝐔(x_1,x_2,x_3+1)
    
    -     3𝐔(x_1,x_2,x_3)

and

    (𝐁⊗𝐈) 𝐔(x-1,x,x_3)    =    𝐔(x,x,x_3),
    
    ( 𝐈⊗𝐁)𝐔(x_1,x-1,x)    =    𝐔(x_1,x,x).

The Bethe ansatz solution for  (<ref>) is given by

    𝐔(x_1,x_2,x_3) = ∑_σ∈ S_3𝐀_σξ_σ(1)^x_1ξ_σ(2)^x_2ξ_σ(3)^x_3

where ξ_1,ξ_2,ξ_3 are nonzero complex numbers. The sum is taken over all permutations σ=σ(1)σ(2)σ(3) in the symmetric group S_3, and 𝐀_σ is a 27 × 27 matrix with the same structure as  𝐔(x_1,x_2,x_3) (that is, zero elements in  𝐀_σ and  𝐔(x_1,x_2,x_3) occupy the same positions), whose nonzero elements are constants with respect to t,x_1,x_2,x_3. Substituting (<ref>) into (<ref>), we find that

    ε = p/ξ_1 + p/ξ_2 + p/ξ_3 + qξ_1 +qξ_2 + qξ_3 - 3.


We now substitute (<ref>) into (<ref>) and (<ref>) to derive conditions on 𝐀_σ that ensure (<ref>) and (<ref>) are satisfied. First, substituting (<ref>) into (<ref>) and simplifying,  we obtain

    (𝐀_123+𝐀_213)+ (𝐀_132+𝐀_312)+ (𝐀_231+𝐀_321)  
       =(𝐁⊗𝐈)((𝐀_1231ξ_1+𝐀_2131ξ_2)+ (𝐀_1321ξ_1+𝐀_3121ξ_3)+ (𝐀_2311ξ_2+𝐀_3211ξ_3)).

We set

    𝐀_123+𝐀_213 =     (𝐁⊗𝐈)(𝐀_1231ξ_1+𝐀_2131ξ_2), 
    𝐀_132+𝐀_312 =     (𝐁⊗𝐈)(𝐀_1321ξ_1+𝐀_3121ξ_3),
    𝐀_231+𝐀_321 =     (𝐁⊗𝐈)(𝐀_2311ξ_2+𝐀_3211ξ_3),

to satisfy (<ref>). Equivalently, we have

    𝐀_213 =     -(𝐈_27 - 1ξ_2𝐁⊗𝐈)^-1(𝐈_27 - 1ξ_1𝐁⊗𝐈)𝐀_123,  
    𝐀_132 =     -(𝐈_27 - 1ξ_1𝐁⊗𝐈)^-1(𝐈_27 - 1ξ_3𝐁⊗𝐈)𝐀_312,  
    𝐀_321 =     -(𝐈_27 -1ξ_3𝐁⊗𝐈)^-1(𝐈_27 - 1ξ_2𝐁⊗𝐈)𝐀_231.

Recall that  in (<ref>) 𝐈 = 𝐈_3.  Performing the matrix multiplications in (<ref>), we obtain

    -(𝐈_27 - 1ξ_β𝐁⊗𝐈)^-1(𝐈_27 - 1ξ_α𝐁⊗𝐈)
      =    -(𝐈_9⊗𝐈 - 1ξ_β𝐁⊗𝐈)^-1(𝐈_9⊗𝐈 - 1ξ_α𝐁⊗𝐈) 
     =    -((𝐈_9 - 1ξ_β𝐁)⊗𝐈)^-1((𝐈_9 - 1ξ_α𝐁)⊗𝐈)
     =     -((𝐈_9 - 1ξ_β𝐁)^-1⊗𝐈)((𝐈_9 - 1ξ_α𝐁)⊗𝐈) 
     =    -((𝐈_9 - 1ξ_β𝐁)^-1(𝐈_9 - 1ξ_α𝐁))⊗𝐈.

Note that 𝐁 can be made block-diagonal by reordering labels of rows and columns in the order of 11,22,33,12,21,13,31,23,32. Using the results from (<ref>), we then obtain

    -(𝐈_9 - 1ξ_β𝐁)^-1(𝐈_9 - 1ξ_α𝐁)  = 𝐑_βα

where 𝐑_βα is a 9 × 9 matrix with the (ij,lm)-element given by

    (𝐑_βα)_(ij,lm) =
    
    S_βα      if ij = lm and i≥ j
    
    -1      if ij = lm and i<j
    
    T_βα      if ij = ml and i>j
    
    0      otherwise.

Here, S_βα and T_βα  are defined in (<ref>).
Thus, (<ref>) can be  more succinctly written as

    𝐀_βαγ =  (𝐑_βα⊗𝐈)𝐀_αβγ.

Therfore, if 𝐀_σ satisfies (<ref>), then the Bethe ansatz solution (<ref>) will also satisfy (<ref>). Similarly, we can demonstrate that if

    𝐀_αγβ = (𝐈⊗𝐑_γβ)𝐀_αβγ,

then the Bethe ansatz solution (<ref>) will satisfy (<ref>). However, to ensure consistency between the expressions in (<ref>) and (<ref>), it must be verified that
 
    (𝐑_γβ⊗𝐈)(𝐈⊗𝐑_γα)(𝐑_βα⊗𝐈) = (𝐈⊗𝐑_βα)(𝐑_γα⊗𝐈)(𝐈⊗𝐑_γβ)

 and
 
    (𝐑_βα⊗𝐈)(𝐑_αβ⊗𝐈) = 𝐈_27=( 𝐈⊗𝐑_βα)( 𝐈⊗𝐑_αβ).

The proofs of (<ref>) and (<ref>) are provided in Appendix <ref>, using the approach outlined in Section 2.4 of <cit.>.


§ GENERAL N≥ 2

In this section, we extend the results for N=2,3 to general N≥ 2.


 §.§ Master equations

First, we generalize the matrices 𝐁 in (<ref>) and   (<ref>) for N=2,3   to general N.

We define 𝐁 as the N^2 × N^2 matrix with rows and columns labelled lexicographically by 11,12,…, NN,  such that the (ij,lm)-element is given by

    𝐁_(ij,lm)
    = 
    1   if ij = lm and i≥ j,
    
    1   if ij = ml and i>j,
    
    0   otherwise,


Now, we define a matrix that describes the permissibility of transition for a two-particle sector consisting of the l-th and (l+1)-th particles.

For l=1,…, N-1, we define N^N × N^N matrix

    𝐁^(l)   :=   𝐈  ⊗  ⋯  ⊗  𝐈_(l-1) times  ⊗  𝐁  ⊗  𝐈  ⊗  ⋯  ⊗  𝐈_(N-l-1) times;

where 𝐈  is the N × N identity matrix.

Then, 𝐁^(l) is an N^N × N^N matrix with rows and column labelled lexicographically  by π and ν, respectively, with π, ν = 1⋯1, …, N⋯ N. The (π,ν)-element of 𝐁^(l) describes the permissibility of  transition

    (x_1,…, x_l-1,x-1, x,x_l+2,…, x_N, ν)   ⟶    (x_1,…, x_l-1,x, x+1,x_l+2,…, x_N, π)

where π_i = ν_i for i ≠  l,l+1.

We generalize 𝐁_1 in (<ref>) and (<ref>), and 𝐁_2 in (<ref>) and (<ref>) to general N.

We define 𝐁_1 and 𝐁_2 as  N^2 × N^2 matrices with rows and columns labelled lexicographically by 11,12,…, NN,  such that their (ij,lm)-elements are given by

    (𝐁_1)_(ij,lm)
    = 
    1   if ij = ml and i> j,
    
    0   otherwise.

and

    (𝐁_2)_(ij,lm)
    = 
    1   if ij = lm and i≥ j,
    
    2   if ij = lm and i< j,
    
    0   otherwise,

respectively. In a same manner to Definition <ref>, we define 𝐁_1^(l) and 𝐁_2^(l).

The (π,ν)-element of 𝐁_1^(l) describes the permissibility of  transition

    (x_1,…, x_l-1,x, x+1,x_l+2,…, x_N, ν)   ⟶    (x_1,…, x_l-1,x, x+1,x_l+2,…, x_N, π)

where π_i = ν_i for i ≠  l,l+1.  The (π,π)-element of the N^N × N^N diagonal matrix  q𝐁_2^(l)   represents  the rate of jumping out of the state

    (x_1,…, x_l-1,x, x+1,x_l+2,…, x_N, π)

due to the jumps of π_l and π_l+1 to the left when x-1 is empty.

We extend the notations 𝐔(X;t) and U_(Y,ν)(X,π;t)  used for N=2,3 to general N. If 𝐔(X;t) satisfies
 
    d/dt𝐔(X;t) =    ∑_i=1^N p𝐔(x_1,⋯, x_i-1,x_i-1,x_i+1,⋯, x_N;t)  
    
                                 +    ∑_i=1^N q𝐔(x_1,⋯, x_i-1,x_i+1,x_i+1,⋯, x_N;t)
    
                                 -    N 𝐔(X;t)

and the boundary conditions
 
    𝐔(x_1,⋯,x_l-1,x,x,x_l+2,⋯, x_N;t) =𝐁^(l) 𝐔(x_1,⋯,x_l-1,x-1,x,x_l+2,⋯, x_N;t)

 for all l=1,…, N-1, then 𝐔(X;t) satisfies the master equation corresponding to any given X in the physical region. For example,  for X=(x,x+1,…,x+N-1), (<ref>) becomes

    d/dt𝐔(X;t) =     p𝐔(x-1,x+1,…, x+N-1;t) 
       +  p𝐁^(1)𝐔(x-1,x,x+2,…,x+N-1 ;t)  
       +  p𝐁^(2)𝐁^(1)𝐔(x-1,x,x+1,x+3,…,x+N-1 ;t)  
       + ⋯ +  p(𝐁^(N-1)⋯𝐁^(1))𝐔(x-1,x,…,x+N-2 ;t) 
       + q𝐔(x,x+1,…, x+N-2, x+N;t) 
       +  q(𝐁^(1) + ⋯+ 𝐁^(N-1))𝐔(X;t)- N𝐔(X;t)

by the boundary conditions (<ref>). If we use the fact that 𝐁_2 + 𝐁 - 𝐁_1 = 2𝐈_N^2, then (<ref>) can be rewritten as   the master equation involving  𝐁^(l), 𝐁_1^(l), 𝐁_2^(l).


 §.§ Bethe ansatz solution

   The Bethe ansatz solution to (<ref>) is given by
 
    𝐔(X;t) = ∑_σ∈ S_N𝐀_σ∏_i=1^N(ξ_σ(i)^x_i e^ε(ξ_i)t)

 where
 
    ε(ξ_i) = p/ξ_i +qξ_i - 1.

 To ensure that  (<ref>) satisfies (<ref>) for each l=1,⋯, N-1,   we substitute (<ref>) into (<ref>) and simplify to obtain
 
    ∑_σ∈ S_N(𝐈_N^N - 1ξ_σ(l)𝐁^(l)) 𝐀_σ = 0

 for each l.

 Let A_N be the subgroup of all even permutations of S_N, and  let T_i ∈ S_N be the simple transposition that exchanges the i-th and the (i+1)-th elements. We can then express  (<ref>) as
 
    0 =  ∑_σ∈ A_N(𝐈_N^N - 1ξ_σ(l)𝐁^(l)) 𝐀_σ
     + ∑_σ∈ A_N(𝐈_N^N - 1ξ_σ(l+1)𝐁^(l)) 𝐀_T_lσ.

 To satisfy (<ref>), we set
 
    0 = (𝐈_N^N - 1ξ_σ(l)𝐁^(l)) 𝐀_σ
     + (𝐈_N^N - 1ξ_σ(l+1)𝐁^(l)) 𝐀_T_lσ.

for each σ∈ A_N. Consequently, we obtain
 
    𝐀_T_lσ =     -(𝐈_N^N - 1ξ_σ(l+1)𝐁^(1))^-1 (𝐈_N^N - 1ξ_σ(l)𝐁^(l)) 𝐀_σ
    
     =     ( 𝐈 ⊗ ⋯ ⊗ 𝐈_(l-1) times ⊗ 𝐑_σ(l+1)σ(l) ⊗ 𝐈 ⊗ ⋯ ⊗ 𝐈_(N-l-1) times  )𝐀_σ

 where the second equality in (<ref>) is derived similarly to  (<ref>), and  𝐑_σ(l+1)σ(l) is the N^2 × N^2 matrix whose rows and columns are labelled by 11,…, NN, with its (ij,lm)-elements given as  in (<ref>).
  
Let 𝐑_βα be N^2 × N^2 matrix with rows and columns  labelled lexicographically by 11,…, NN, where  the (ij,lm)-element is given by (<ref>). We define
 
    𝐓_l,βα:= 𝐈  ⊗  ⋯  ⊗  𝐈_(l-1) times  ⊗  𝐑_βα  ⊗  𝐈  ⊗  ⋯  ⊗  𝐈_(N-l-1) times

where 𝐈 is the N× N matrix.
 
The following  proposition follows from the arguments presented above.

If

    𝐀_T_lσ =  𝐓_l,σ(l+1)σ(l)𝐀_σ

for each σ∈ S_N, then the boundary condition (<ref>) is satisfied.

Since {T_1,⋯, T_N-1} generates S_N, for any σ∈ S_N, there exists an expression

    σ = T_l_k⋯ T_l_2T_l_1

for some l_1,…, l_k ∈{1,…, N-1}.  The following lemma provided a condition for (<ref>) to be satisfied, which is evident  from the construction.

Let σ be given by (<ref>) and let σ^(i) =  T_l_i⋯ T_l_1, i=1,…, k with σ^(0) denoting the identity permutation. If

    𝐀_σ = 𝐀_ T_l_k⋯ T_l_2T_l_1 := 𝐓_l_k,σ^(k-1)(l_k+1)σ^(k-1)(l_k) × ⋯ × 𝐓_l_1,σ^(0)(l_1+1)σ^(0)(l_1),

then (<ref>) is satisfied.

For 𝐀_σ in (<ref>) to be well-defined,  the following relations  must also be satisfied:

    𝐓_i,αβ𝐓_i,βα = 𝐈;
       𝐓_i,βγ𝐓_i+1,αγ𝐓_i,αβ = 𝐓_i+1,αβ𝐓_i,αγ𝐓_i+1,βγ.

In fact, Lemma <ref> and Lemma <ref> demonstrate that these relations are satisfied.


 §.§ Transition probabilities

In the previous section, we claimed that 𝐔(X;t) in (<ref>) with 𝐀_σ defined by (<ref>) satisfies the master equation for each X in the corresponding physical region. This holds true when performing contour integrals over all ξ_i variables with appropriate contours. This approach is well known for various integrable particle models <cit.>. The  transition probability P_(Y,ν)(X,π;t) of our model can also be expressed as a contour integral, analogous to the formula for the multi-species ASEP in <cit.>.

Let C_R_i be a counterclockwise circle centered at the origin with radius R_i and suppose that 1<R_1< ⋯ <R_N. Let 𝐀_σ be given by (<ref>) and ε(ξ_i) by (<ref>). For any arbitrary initial state (Y,ν), the transition probability P_(Y,ν)(X,π;t) is given by:

    P_(Y,ν)(X,π;t) = (1/2π i)^N∫_C_R_N⋯∫_C_R_1∑_σ∈ S_N(𝐀_σ)_(π,ν)(∏_i=1^Nξ_σ(i)^x_i - y_σ(i) - 1e^ε(ξ_i)t)dξ_1 … dξ_N.



It is sufficient to verify the initial condition. Since the denominators of the nontrivial terms in the matrix 𝐑_βα that is contained in the expression of 𝐀_σ match those of S_βα^† with μ=1 and λ = 0 in Section 4 of <cit.>, the proofs of Corollary 2 and Theorem 1 in <cit.> essentially imply that (<ref>) satisfies the initial condition.



§ DISCUSSION

We have defined a multi-species stochastic particle model (distinct from the multi-species ASEP) on ℤ, where particles move either  to the right or the left: a particle jumping to the right moves to the nearest site occupied by a lower-numbered particle (or an empty site) and pushes the lower-numbered particle  forward (if any);  a particle jumping  to the left follows the rules of the conventional multi-species TASEP. We have shown that  this model is integrable and  derived the formula for the transition probability for an arbitrary initial state. A key motivation of defining this model such a way that the model can be integrable is the symmetry between the one-sided model with long-range jumps and the multi-species TASEP in the opposite direction, in the sense that their boundary conditions are compatible when using the Bethe ansatz.

Given the form of the matrices in (<ref>), it is anticipated that P_(Y,ν)(X,π;t) with π = ν can be expressed as a determinant (See <cit.> for related results in the multi-species TASEP context). Generally, the integrand in (<ref>) can be a sum of numerous products due to  𝐀_σ being a product of matrices as described in (<ref>).  However, for certain special initial permutations ν of species, such as 12⋯ N, 12⋯2, 1⋯ 12 and so on, it seems that (𝐀_σ)_(π,ν) can be expressed as a single product  (See <cit.> for related results for the multi-species ASEP). Also, it would be interesting to investigate whether the methods used in this paper could serve as a machinery to develop other new integrable multi-species particle models as well as multi-species versions of other known single-species models such as <cit.>. Furthermore, these methods might help find the transition probabilities for some known multi-species particle models mentioned in <cit.>.

Finally, we have not yet identified an integrable multi-species model where particles make long-range jumps in both directions, that is, a multi-species version of the model in <cit.>. Developing such a multi-species version and finding its transition probability remain an area for future research.







§ PROOF OF THE CONSISTENCY CONDITIONS

We use the same approach as in Section 2.4 of <cit.>.

If 𝐑_βα is the N^2 × N^2 matrix defined by (<ref>), then  𝐑_βα𝐑_αβ is the identity matrix.


Let ij and kl be labels of rows and columns with ij, kl = 11,…, NN. For i =j,
 
    (𝐑_βα𝐑_αβ)_(ii,kl) =    ∑_(mn)(𝐑_βα)_(ii,mn)(𝐑_αβ)_(mn,kl)
    
     =    (𝐑_βα)_(ii,ii)(𝐑_αβ)_(ii,kl) = 
     S_βαS_αβ=1    if kl = ii,
    
     S_βα· 0=0    if kl ≠ ii.

For i>j,

    (𝐑_βα𝐑_αβ)_(ij,kl) =    ∑_(mn)(𝐑_βα)_(ij,mn)(𝐑_αβ)_(mn,kl)
    
     =    (𝐑_βα)_(ij,ij)(𝐑_αβ)_(ij,kl) + (𝐑_βα)_(ij,ji)(𝐑_αβ)_(ji,kl)
    
     =    
      S_βα· S_αβ + T_βα· 0 =1     if kl = ij,
    
     S_βα· T_αβ+T_βα·(-1) = 0    if kl = ji
    
     S_βα·0 + T_βα· 0 = 0    if kl ≠ ij,ji.

 For i<j,
 
    (𝐑_βα𝐑_αβ)_(ij,kl) =    ∑_(mn)(𝐑_βα)_(ij,mn)(𝐑_αβ)_(mn,kl)
    
     =    (𝐑_βα)_(ij,ij)(𝐑_αβ)_(ij,kl)+(𝐑_βα)_(ij,ji)(𝐑_αβ)_(ji,kl)
    
     =    
      (-1)·(-1)+  0· T_βα =1     if kl = ij,
    
      (-1)· 0+0· S_αβ = 0    if kl = ji
    
     (-1)· 0 +0·  0= 0    if kl ≠ ij,ji.


 (Yang-Baxter equation)
Let N ≥ 3. If 𝐑_βα is the N^2 × N^2 matrix defined by (<ref>) and 𝐈 be the N × N matrix,

    (𝐑_γβ⊗𝐈)(𝐈⊗𝐑_γα)(𝐑_βα⊗𝐈) = (𝐈⊗𝐑_βα)(𝐑_γα⊗𝐈)(𝐈⊗𝐑_γβ).



Both (𝐈⊗𝐑_βα) and (𝐑_βα⊗𝐈) are N^3 × N^3 matrices with rows and columns labelled with 111,⋯, NNN, and each label is a permutation of a certain multi-set [i,j,k]. If ijk and lmn are permutations from two different multi-sets, then

    (𝐈⊗𝐑_βα)_(ijk,lmn) = 0 = (𝐑_βα⊗𝐈)_(ijk,lmn).

 Hence, if we reorder the columns and rows so that all permutations  of a given multi-set are grouped, then both (𝐈⊗𝐑_βα) and (𝐑_βα⊗𝐈) become block-diagonal. Let

    (𝐑_βα⊗𝐈)_[i,j,k]

be the block on the diagonal of the matrix 𝐑_βα⊗𝐈  whose  rows are columns are labelled with permutations of [i,j,k]. We similarly define  (𝐈⊗𝐑_βα)_[i,j,k]. It suffices to show  (<ref>) blockwise, that is,

    (𝐑_γβ⊗𝐈)_[i,j,k](𝐈⊗𝐑_γα)_[i,j,k](𝐑_βα⊗𝐈)_[i,j,k] = (𝐈⊗𝐑_βα)_[i,j,k](𝐑_γα⊗𝐈)_[i,j,k](𝐈⊗𝐑_γβ)_[i,j,k]

for each multi-set [i,j,k].  Each multi-set [i,j,k] of integers 1,…, N is one of the forms [i,i,i], [i,i,j] with i< j, [i,i,j] with i> j and [i,j,k] with i<j<k.  First, for the multi-set [i,i,i], (<ref>) is simply

    S_γβS_γαS_βα  = S_βαS_γαS_γβ

which is obviously true because

    (𝐑_βα⊗𝐈)_[i,i,i] = S_βα = (𝐈⊗𝐑_βα)_[i,i,i].

For [i,i,j] with i<j, we can obtain

    (𝐑_βα⊗𝐈)_[i,i,j]  =
        iij     iji     jii 
    
    iij     S_βα    0     0  
    
    iji    0    -1     0
    
    jii     0    T_βα    S_βα

and

    (𝐈⊗𝐑_βα)_[i,i,j]=    iij     iji     jii 
    
    iij     -1    0     0  
    
    iji    T_βα   S_βα    0
    
    jii     0    0     S_βα.

Using these matrices, we can verify (<ref>) by direct matrix computation. Similarly, (<ref>) is verified for [i,i,j] with i>j. Finally, for [i,j,k] with i<j<k, we obtain

    (𝐑_βα⊗𝐈)_[i,j,k]  =     ijk     ikj     jik     jki     kij     kji 
    
        ijk     -1     0     0     0     0     0 
    
        ikj     0     -1     0     0     0     0 
    
       jik      T_βα    0     S_βα    0    0      0 
    
       jki     0     0     0     -1     0      0
    
       kij     0      T_βα    0     0    S_βα    0 
    
       kji     0     0     0     T_βα    0     S_βα

and

    (𝐈⊗𝐑_βα)_[i,j,k]  =     ijk     ikj     jik     jki     kij     kji 
    
        ijk     -1     0     0     0     0     0 
    
        ikj     T_βα    S_βα    0     0     0     0 
    
       jik     0      0     -1     0    0      0 
    
       jki     0     0     T_βα    S_βα    0      0
    
       kij     0      0     0     0    -1      0 
    
       kji     0     0     0     0      T_βα    S_βα
    .

Again, we can verify (<ref>) by  directly performing matrix multiplication.






Acknowledgement This research is funded by Nazarbayev University under the faculty-development competitive research grants program
for 2024 – 2026 (grant number 021220FD4251). The author is grateful to Axel R. Saenz for valuable discussions, and to Temirlan Raimbekov and Nazgul Tileukabyl for their assistance in writing the manuscript.

 

Conflict of Interest The author declares that there is no conflict of interest.
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 Paper Title: Perfect Wave Transfer in Continuous Quantum Systems.







per.moosavi@fysik.su.se

Department of Physics, Stockholm University, 10691 Stockholm, Sweden



christandl@math.ku.dk





gmgraf@phys.ethz.ch

Institute for Theoretical Physics, ETH Zurich,
Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland



spyros.sotiriadis@physics.uoc.gr









§ ABSTRACT 
We study the perfect transfer of information in 1+1D continuous quantum systems. This includes effective descriptions of inhomogeneous spin chains, for which the notion of perfect state transfer in quantum information was introduced, and here phrased in terms of waves. We show that reflection symmetry is necessary for perfect wave transfer (PWT) in any inhomogeneous conformal field theory, and even sufficient when restricted to one-particle excitations. To determine if or when it is sufficient more generally, we first break conformal invariance and study a broad class of 1+1D bosonic theories. We show that the question can then be posed as an inverse Sturm-Liouville problem that determines when the bosonic theory exhibits PWT. We demonstrate how to uniquely solve this problem, which also shows that reflection symmetry is sufficient for the special case with conformal invariance. Using bosonization, our continuum results extend these notions to interacting quantum field theories.




Perfect Wave Transfer in Continuous Quantum Systems

    Spyros Sotiriadis
    
August 1, 2024

=====================================================







Introduction.—
phantomsection
tocsectionIntroduction


Non-photonic quantum communication enables the efficient transmission of information from one quantum register to another <cit.>.
This is desirable for quantum networks and computation <cit.>, not least since it
allows for the perfect transfer of states even without dynamical control <cit.>.
First shown theoretically for communication channels in the form of a spin chain with spatially varying hopping, such perfect state transfer (PST) was extended to spins on graphs and other networks <cit.> and demonstrated experimentally <cit.>.
However, the notion has remained squarely in the discrete realm.

In this paper, we extend PST to continuous systems in one space and one time dimension (1+1D).
In so doing, a rephrasing in terms of waves becomes natural, motivating the term perfect wave transfer (PWT).
As for states in PST, a system on the interval [-L/2, L/2] of length L with suitable boundary conditions (BCs) exhibits PWT if the propagation of any initial wave, after some time T, amounts to its reflection; see Fig. <ref>.
Concretely: The system enjoys PWT if

    ⟨(x, 0) ⟩ = ⟨(-x, T) ⟩,

for all time-evolved observables (x, t) and any expectation value ⟨·⟩. 
We focus on realizations that do not need dynamical control, but the same problem is also interesting for driven systems <cit.>.

Related recent advances in the continuum include non-equilibrium studies of inhomogeneous versions of conformal field theory (CFT).
These are conformally-invariant quantum field theories deformed so that the usual constant propagation velocity becomes a profile v(x) <cit.>.
Among their applications, such inhomogeneous CFTs effectively describe gapless quantum many-body systems with mesoscopic inhomogeneities, such as spin chains with spatially varying hopping, trapped ultracold atoms, and arrays of quantum circuits; cf. <cit.>.
Special cases include the square-root profile v(x) = v √(1 - (2x/L)^2) for some constant v > 0 <cit.>—the continuum counterpart of the inhomogeneous spin chain in <cit.> that first exhibited PST—and profiles produced by Möbius transformations <cit.>.
Note that for PWT, as for other transport properties of finite systems, the BCs <cit.> are crucial.
Furthermore, inhomogeneous CFT can be viewed as a theory in curved spacetime.
As such, besides applications to quantum communication in solid-state devices <cit.>, our work also relates to relativistic quantum information <cit.>.
Since the same theories are also natural to study on the half or full infinite line, our results may likewise be relevant for long-distance quantum communication <cit.>.






Engineering PST/PWT is at  heart a spectral problem, amounting to whether the propagator coincides with the parity operator at certain times.
We address this engineering question for quantum field theories describing a continuous quantum communication channel.
Remarkably, for bosonic theories <cit.>, we arrive at an inverse problem for Sturm-Liouville operators, which we solve and whose solution determines if the channel enjoys PWT.
Using bosonization <cit.>, our results extend the notion of PST/PWT to a large class of systems, including interacting fermionic quantum field theories.


Inhomogeneous CFT.—
phantomsection
tocsectionInhomogeneous CFT


To begin with, we study the continuum limit of the inhomogeneous spin chains in <cit.>.
In the gapless regime, the continuous counterpart is an inhomogeneous CFT with a velocity profile v(x) > 0, x ∈ [-L/2, L/2] <cit.>.
The simplest example is that of inhomogeneous free fermions, given by the Hamiltonian (up to an overall constant and setting ħ = 1)

    H_iFF≡∫_-L/2^L/2 x v(x)/2[ ψ^†_+ (-) ∂_x ψ_+
      + ψ^†_-∂_x ψ_-
      + ]

with fields ψ_±(x) of right(+)/left(-) fermions satisfying {ψ_±(x), ψ^†_±(x') } = δ(x-x') and {ψ_±(x), ψ^(†)_∓(x') } = 0.
Here, ⋯ denotes Wick (normal) ordering with respect to the ground state |Ω⟩ (which for bilinears in the fields amounts to an additive renormalization).
For simplicity, we focus on the theory (<ref>), while general CFTs are discussed in the Supplementary Material (SM) <cit.>.

We assume open (Neumann) BCs at x = ± L/2.
These can be incorporated by standard unfolding methods; cf. <cit.>:
Couple right and left movers by identifying ψ_-(x) = ψ_+(L-x), giving a chiral theory of only right-moving fermions on [-L/2, 3L/2] with periodic BCs and velocity v(x) for x ∈ [-L/2, L/2] and v(L-x) for x ∈ [L/2, 3L/2].
This implies zero particle flow at x = ± L/2 since the two fermion densities ρ_±(x) = ψ^†_±(x)ψ_±(x) are equal at each boundary and the current is ∝ρ_+ - ρ_-.

PST was probed in <cit.> using a time-dependent overlap between one-particle states that are reflections of each other.
Its continuum counterpart is to study states created by fields

    ψ(x)
    ≡^ xψ_+(x) + ^- xψ_-(x),

where  denotes the Fermi momentum.
Eq. (<ref>) corresponds to the low-energy linearization of the spin-chain dispersion relation around its Fermi points ±, followed by recentering the resulting fields so they describe right/left-moving excitations around zero momentum.
As a preparatory subset of possible states, consider

    |Ψ_j⟩≡∫_-L/2^L/2 x ξ_j(x) ψ^†(x) |Ω⟩     (j = 1,2)

for smooth enough functions ξ_j(x), interpreted as distributions or “wave packets” of fermionic excitations in each state, and study the time-dependent overlap

    F(t)
    ≡⟨Ψ_2| ^- H_iFF t |Ψ_1⟩.

The states (<ref>) form a restricted class as only one-particle excitations are possible and as right/left movers have the same particle distributions; cf. the more general treatment in the SM <cit.> where we lift the latter restriction.

To probe PWT for one-particle excitations as done in <cit.>, we restrict to specular waves, ξ_2(x) = ξ_1(-x), and compare F(t) in Eq. (<ref>) with ⟨Ψ_1|Ψ_1⟩.
We say that:
There is one-particle PWT if there exists T > 0 so that

    |F(T)|
    = |⟨Ψ_1|Ψ_1⟩|

for all initial waves ξ_1(x) in Eq. (<ref>) [If Eq. (<ref>) holds for any state |Ψ_1⟩ in the Fock space, then general PST/PWT (i.e., for any number of excitations) would be enjoyed.].

Eq. (<ref>) amounts to a comparison between two-point correlation functions at equal and non-equal times.
Such can be computed in any inhomogeneous CFT by mapping them to the corresponding homogeneous correlations <cit.>.
The details are given in the SM <cit.> and boil down to comparing spacetime points after conformal transformations to coordinates given by

    y = y(x) ≡∫_0^x s v_0/v(s),
        1/v_0≡1/L∫_-L/2^L/2 s/v(s).

The outcome of the comparison shows that reflection symmetry is essential:


An inhomogeneous CFT exhibits one-particle PWT 

if and only if v(x) is even.


The corresponding times are odd multiples, (2m+1)T for any integer m, of the time T = L/v_0 that excitations take to cross the entire system's length.
For later usage, we note that y ∈ [-L/2, L/2] if v(x) is even.

As an example, consider the square-root profile v(x) = v √(1 - (2x/L)^2), which gave the effective low-energy theory of the spin chain in <cit.>.
This continuum theory exhibits PWT, just like for any even profile v(x), in contrast to <cit.> where that profile was exceptional in this respect.
Notably, the square-root profile is positive everywhere except at x = ± L/2.
Thus, strictly speaking, v(x) should be regularized before using the tools in <cit.>, but since the integral defining 1/v_0 in Eq. (<ref>) converges, it has no consequence for PWT. Oppositely, the limiting case of inhomogeneous CFT given by Möbius transformations so that v(x) decays to zero at the boundaries cannot exhibit PWT since 1/v_0 diverges.


Bosonic theories.—
phantomsection
tocsectionBosonic theories


The above result and observations should not come as a surprise.
After all, if one tries to compare discrete and continuous theories, then an inhomogeneous CFT can only describe low-energy excitations of a given lattice model.
Moreover, one can show <cit.> that any Hamiltonian (with pure point spectrum) must commute with the parity operator for PWT to be possible, i.e., v(x) must be even.
The general structure of the spectrum due to conformal invariance in 1+1D <cit.> opens up the possibility that it might also be sufficient; e.g., the example (<ref>) has a linear spectrum and is one-particle reducible, suggesting that if PWT holds for one-particle excitations then this should extend to many particles.
The remainder of this paper concerns to what extent this is true more generally.
Namely:
What are necessary and sufficient conditions for PWT (<ref>) for any number of excitations?

To answer this question, we first break conformal invariance. 
Examples include 1+1D fermions with non-local interactions <cit.>, while here we focus on bosons, specifically inhomogeneous Tomonaga-Luttinger liquids (TLLs) <cit.>.
This theory is formulated using a compactified bosonic field φ(x) (modulo 2π) with conjugate Π(x) satisfying
[∂_x φ(x), Π(x')] = ∂_x δ(x-x').
The Hamiltonian is (up to an overall constant)

    H_iTLL≡∫_-L/2^L/2 x/2π v(x)
      (
          [πΠ(x)]^2/K(x)
          + K(x)[∂_x φ(x)]^2
        )
       ,

where, besides v(x) > 0, the so-called Luttinger parameter K(x) > 0 also depends on x.
For simplicity, we assume that both functions are smooth.
When K(x) = K is constant, Eq. (<ref>) is an inhomogeneous CFT, and when K = 1, it is the bosonized version of the free fermions (<ref>) <cit.>.
However, K'(x) ≠ 0 locally couples right and left movers non-trivially, thus they do not commute and the theory is not conformal <cit.>.
Again, we impose open BCs, namely zero current at the boundaries:

    (x) |_x = ± L/2 = 0,
      where  (x)
    ≡
    - v(x)K(x)/π∂_x φ(x)

is the total particle current.
Unless stated otherwise, we also assume that K'(x)|_x = ± L/2 = 0 so that right and left movers are only coupled via the BCs (<ref>) at those points.


Sturm-Liouville (SL) theory.—
phantomsection
tocsectionSturm-Liouville (SL) theory


Following <cit.>, we use SL theory to find eigenfunctions, using which the fields can be expanded and the Hamiltonian (<ref>) diagonalized.
(This generalizes <cit.> as v(x), K(x) are unrelated and differs from the approach in <cit.>.)
Indeed, using integration by parts and Eq. (<ref>),

    H_iTLL
    =
    ∫_-L/2^L/2 x/2π w(x)
      (
          [ πΠ(x)/w(x)]^2
          + φ(x) φ(x)
        )

with the SL operator

    ≡
    - 1/w(x)[ ∂_x p(x) ∂_x + q(x) ],
 


    w(x)
    ≡K(x)/v(x),
      
    p(x)
    ≡
    v(x)K(x),
      
    q(x)
    ≡
    0,

where  acts on the space  of smooth [More precisely, twice continuously differentiable.] functions u(x) on [-L/2, L/2] that satisfy v(x)K(x) ∂_x u(x)|_x = ± L/2 = 0, i.e., Neumann BCs when v(x), K(x) > 0.

We want to solve the SL problem

    u = λ u,
      
    u ∈.

A distinction is warranted: Eq. (<ref>) is regular if certain technical conditions are satisfied [Here, these amount to v(x), K(x) being strictly positive and v(x), K(x), ∂_x p(x) [and q(x) if non-zero] continuous on the whole interval [-L/2, L/2].]; otherwise it is irregular.
Its importance stems from properties that hold for regular problems <cit.>:



  * 
The spectrum consists of eigenvalues λ_n that are real and ordered: 0 ≤λ_0 < λ_1 < …. 


  * 
Each λ_n has a unique square-integrable eigenfunction u_n(x) with n isolated zeros in [-L/2, L/2].


  * 
The u_n(x) form a complete orthonormal basis for square-integrable functions on [-L/2, L/2] with weight w(x):
∫_-L/2^L/2 x  w(x) u_n(x) u_n'(x)
= δ_n,n'.


Furthermore, assuming that the integral defining 1/v_0 in Eq. (<ref>) converges:




  * 
The λ_n can at most grow as n^2 for large n, i.e., λ_n = O(n^2).


The last follows by Weyl's law; see the SM <cit.> and, e.g., <cit.>.
Simply put, it gives a semi-classical estimate of the number of eigenvalues below a given large λ > 0 for Laplace-type operators.

Unless otherwise stated, our SL problem will be regular.
If irregular, the above properties are not guaranteed, while in certain cases they survive.
Note also that, since q(x) = 0 here, u_0(x) = const solves Eq. (<ref>) with λ_0 = 0.

The stated properties allow us to expand

    φ(x)
    = ∑_n = 0^∞φ_n u_n(x),
        Π(x)
    = ∑_n = 0^∞Π_n w(x) u_n(x),

whereby we obtain [ φ_n, Π_n'] = δ_n,n' and can diagonalize H_iTLL in Eq. (<ref>).
Indeed, the latter becomes

    H_iTLL
    = 1/2π[ πΠ_0]^2
      + ∑_n = 1^∞
        E_n a^†_n a_n

with eigenenergies

    E_n = √(λ_n),

where we expressed φ_n = √(π/2E_n)( a_n + a^†_n) and Π_n = -√(E_n/2π)( a_n - a^†_n) for n ≥ 1 in bosonic operators a^†_n, a_n with [ a_n, a^†_n'] = δ_n,n'.
Note that φ_0 (and zero modes in general) should be treated separately, but this will have no consequence here.
In the SM <cit.>, we also discuss SL operators (<ref>) with q(x) ≠ 0, corresponding to an inhomogeneous (possibly imaginary) mass M(x) by adding v_0^2 M(x)^2 φ(x)^2 inside the big parentheses in Eq. (<ref>).

The time evolution of any observable, such as φ(x), will contain terms like a^†_n^ E_n t u_n(x) and a_n^- E_n t u_n(x).
Thus, as shown in the SM <cit.>, to exhibit PWT as defined by Eq. (<ref>), all u_n(x) must have definite parity, i.e., be invariant under reflections x ↦ -x up to a sign, which must equal the value of each phase factor
^± E_n T for a common T > 0.
Moreover, the u_n(x) have definite parity if and only if v(x), K(x) [and q(x) if non-zero] are even, i.e., H_iTLL is reflection symmetric.
In conclusion, PWT boils down to requirements on the spectrum:


An inhomogeneous TLL exhibits PWT if and only if v(x) and K(x) are even and all eigenvalues are of the form E_n = π m_n/T for certain integers m_n.


From Properties <ref> and <ref> and since λ_0 = 0 here, these integers m_n satisfy 0 = m_0 < m_1 < … with m_n even (odd) if n is even (odd).
Moreover, from Property <ref>, they scale as m_n ∼ n for large n, i.e., m_n = n + o(n).

Lastly, to probe PWT based on Eq. (<ref>), we note that the results above allow one to compute correlation functions for inhomogeneous TLLs; see the SM <cit.>.
In particular, the ground-state φφ correlation function is

    ⟨Ω| φ(x,t)φ(x',t') |Ω⟩
    = ∑_n = 1^∞π/2 E_n
      u_n(x) u_n(x') ^- E_n(t-t').

Thus, for a spectrum satisfying the PWT requirements,
⟨Ω| φ(x,T)φ(x',0) |Ω⟩ = ⟨Ω| φ(-x,0)φ(x',0) |Ω⟩,
meaning that, for fixed x', the correlations at t = T are reflections of those at t = 0.
In Fig. <ref>, we plot Eq. (<ref>) for K(x) = 1, which, by bosonization, is related to log F(t) in Eq. (<ref>), in that it encodes the same information; see the SM <cit.>.


Inverse Sturm-Liouville problem.—
phantomsection
tocsectionInverse Sturm-Liouville problem


Reversing the logic, the PWT requirements on the spectrum yield an inverse spectral problem for SL operators with Neumann BCs
[and q(x) = 0]:


Given a spectrum E_n = π m_n/T for integers m_n as above, determine the even functions v(x) and K(x).


A related prominent problem is the inverse scattering occurring in the theory of integrable PDEs, whose solution is given by the Gelfand-Levitan-Marchenko equation.
However, that case is formulated on an infinite line, while here we are interested in a finite interval.

Before a general discussion, we consider two examples:

The inverse SL problem has a known solution when the spectrum is exactly linear:
For K(x) = K > 0 constant but v(x) > 0 general, the SL operator is = - v(x) ∂_x v(x) ∂_x = -v_0^2 ∂_y^2 using Eq. (<ref>) with eigenenergies E_n = π n v_0/L and eigenfunctions u_n > 0(x) = √(2 v_0/L)cos( nπ [L + 2y]/2L ) and u_0(x) = √(v_0/L).
This is precisely the conformal case of non-interacting right/left movers, enjoying PWT for any even v(x) > 0.

In the previous example, the SL operator was regular.
Relaxing this condition by taking v(x) = v √(1 - (2x/L)^2) and K(x) = K [ 1 - (2x/L)^2 ]^α for α > -1/2 and constants v, K > 0, we get an irregular problem for which Properties <ref>–<ref> remain valid; the BCs being that a solution is sufficiently well-behaved for Eq. (<ref>) to hold.
The eigenfunctions are given by Gegenbauer polynomials C^(α)_n(2x/L) <cit.> up to normalizations, which have definite parity and satisfy Eq. (<ref>) with λ_n = (π v_0/L)^2 n(n + 2α) for v_0 = 2v/π.
Clearly, only the case α = 0 (Chebyshev polynomials of the first kind) exhibits PWT, while all α≠ 0 do not, including Legendre polynomials (α = 1/2) and Chebyshev polynomials of the second kind (α = 1) <cit.>.
The respective φφ correlations are plotted in Fig. <ref>.

To solve the inverse SL problem in general, we write the operator (<ref>) in Liouville normal form, as is always possible for regular problems <cit.>: Let (y) ≡ u(x) √(K(x)) for y = y(x) given by Eq. (<ref>) and define 











    q̂(y)
    ≡
    - q(x)/w(x) + 1/√(K(x)) v_0^2 
     ∂_y^2 √(K(x)).

The eigenvalue problem (<ref>) then becomes

    v_0^2 ∂_y^2 (y) + [λ - q̂(y)] (y) = 0

with unchanged BCs
[For even v(x) > 0, they are v_0∂_y(y) - v(x) Λ(x) (y) = 0 at y = ± L/2 with Λ(x) ≡∂_xln√(K(x)) <cit.>, which imply Neumann BCs if K'(x)|_x = ± L/2 = 0.] and if reflection symmetric in x, then so in y.
Clearly, _n(y) have the same eigenvalues as the respective u_n(x) and similarly form a complete orthonormal basis but with identity weight.

Results in the literature often require two sets of eigenvalues for different BCs to get a unique solution to the inverse problem.
However, one set suffices for reflection-symmetric SL operators and certain BCs, including ours.
Indeed, <cit.> and <cit.> say that an even q̂(y) is uniquely determined by the spectrum for Neumann BCs.
Thus, viewing q̂(y) as given and since q(x) = 0, Eq. (<ref>) becomes a second-order differential equation for √(K(x(y))).
By general uniqueness results [assuming q̂(y) is continuous] and since K(x) must be even, this determines K(x) up to changing coordinates (<ref>) and an overall rescaling (discarding solutions that are not positive).

The results <cit.> do not reconstruct q̂(y).
However, since we know that K(x) = K gives an SL problem with m_n = n and q̂(y) = 0, a constant Luttinger parameter is the unique solution for the inverse problem given an exactly linear spectrum.
Beyond this special case, nothing a priori prevents the spectrum from deviating from the leading n behavior as long as the requirements on the m_n remain satisfied.
This would then give rise to a class of possible bosonic theories that exhibit PWT, where, by uniqueness, q̂(y) is non-zero, thus K(x) non-constant.
While not usually analytically tractable, there are constructive approaches to find q̂(y) <cit.>.


Discussion.—
phantomsection
tocsectionDiscussion


We introduced the notion of PWT and showed that it is exhibited by one-particle excitations in inhomogeneous CFT whenever the velocity profile v(x) > 0 is even.
Breaking conformal invariance, by considering inhomogeneous TLLs with a Luttinger-parameter profile K(x) > 0, we showed that PWT for general excitations amounts to an inverse spectral problem, which we solved.
This continuum result for bosons is remarkably similar to earlier results: 
For spin chains with nearest neighbor hopping, the corresponding class of operators can be represented as Jacobi matrices, and the resulting inverse eigenvalue problem was studied <cit.>.
As in the continuum, two sets of eigenvalues are generally needed for uniqueness <cit.>, while one set suffices for matrices symmetric with respect to the anti-diagonal (corresponding to reflection symmetry) <cit.>.

By bosonization, our results for inhomogeneous TLLs cover fermionic theories with position-dependent interactions <cit.>.
Indeed, even when K(x) = K is constant, as long as K ≠ 1, the theory describes an interacting quantum many-body system, albeit at low-energies.

Lastly, we showed that an inhomogeneous TLL with constant K(x) = K enjoys PWT for general excitations whenever v(x) > 0 is even.
Such a theory is an example of an inhomogeneous CFT.
It would be interesting to study PWT for any number of excitations in general CFTs, and also extend to systems on the half or full infinite line.




  
*Acknowledgments.



We are thankful to Pawel Caputa and Shinsei Ryu for motivating discussions on related topics.
P.M. gratefully acknowledges financial support from the Wenner-Gren Foundations (Grant No. FT2022-0002).
M.C. acknowledges financial support from the European Research Council (ERC Grant Agreement No. 818761), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and the Novo Nordisk Foundation (Grant No. NNF20OC0059939 ‘Quantum for Life’), and thanks the NCCR SwissMAP of the Swiss National Science Foundation and the Section of Mathematics at the University of Geneva for their hospitality.












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[SM(TLLs)]SM
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[Note1()]Note1
  note If Eq.  holds
  for any state |Ψ _1⟩ in the Fock space, then general PST/PWT
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[Ghosh(2004)]Ghosh:2004
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[Note2()]Note2
  note More precisely, twice continuously
  differentiable.Stop
[Note3()]Note3
  note Here, these amount to v(x), K(x) being strictly positive
  and v(x), K(x), ∂ _x p(x) [and q(x) if non-zero] continuous on
  the whole interval [-L/2, L/2].Stop
[Zettl(2010)]Zettl:2010
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[NIS(3 15)]NIST:DLMF
  @noop  title NIST Digital Library of Mathematical
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  (year Release 1.2.0 of 2024-03-15), note F. W. J.
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  eds.Stop
[Note4()]Note4
  note For even v(x) > 0, they are v_0∂ _y𝔲(y) - v(x) Λ (x) 𝔲(y) = 0 at y =
  ± L/2 with Λ (x) ≡∂ _x
  oln√(K(x)) <cit.>, which imply
  Neumann BCs if K'(x)@ifvmode @ @ @  | tone @size 
  .@ _x = ± L/2 = 0.Stop
[Borg(1946)]Borg:1946
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Supplemental Material


phantomsection
tocpartSupplemental Material



This supplemental material has three parts.
Part A gives details for two-point correlations in inhomogeneous 1+1D CFT with open (Neumann) boundary conditions (BCs).
Part B describes how to diagonalize general inhomogeneous Tomonaga-Luttinger liquids (TLLs) using Sturm-Liouville (SL) theory, leading to a formulation of perfect wave transfer (PWT) as an inverse SL problem, including extension to massive TLLs.
Part C contains details for correlation functions in inhomogeneous TLLs.






§ PART A: CORRELATIONS IN INHOMOGENEOUS 1+1D CFT WITH OPEN BOUNDARIES





The general inhomogeneous CFT Hamiltonian is (setting ħ = 1)

    H_iCFT≡∫_-L/2^L/2 x  v(x) [ T_+(x) + T_-(x) ],

where T_±(x) are the right/left-moving components of the stress-energy tensor in light-cone coordinates, whose Fourier transforms yield two commuting copies of the Virasoro algebra.
In position space, they satisfy the equal-time commutation relations

    [ T_±(x), T_±(x') ]
    = ∓ 2∂_xδ(x-x') T_±(x')
      ±δ(x-x') ∂_x'T_±(x')
      ±c/24π∂_x^3δ(x-x')

and [ T_±(x), T_∓(x') ] = 0.
Note that, at non-equal time, T_± should be understood to only depend on the “right/left-moving coordinate” x^∓; for now, we only view these as labels, distinguishing dependencies on right/left components.

Besides the T_±(x) fields, an important class of operators are Virasoro primary fields Φ(x^-, x^+), defined by how they behave under conformal transformations.
The latter are given by orientation-preserving circle diffeomorphisms f_+, f_- and primary fields transform as

    Φ(x^-, x^+)
    →
    f'_+(x^-)^Δ_Φ^+ f'_-(x^+)^Δ_Φ^-Φ(f_+(x^-), f_-(x^+)),

where (Δ_Φ^+, Δ_Φ^-) denote the field's conformal weights.
To impose open BCs at x = ± L/2, as in the main text, we will assume that primary fields come in pairs Φ_+, Φ_- with conformal weights satisfying Δ^+_Φ_+ = Δ^-_Φ_-, Δ^-_Φ_+ = Δ^+_Φ_-, since this allows one to couple excitations so that right movers are reflected to left movers when reaching x = L/2, and vice-versa at x = -L/2.
To ease notation, we will use Δ^+_Φ≡Δ^+_Φ_+, Δ^-_Φ≡Δ^-_Φ_+ for each such pair.

Before continuing, we mention two examples:



  * 
For the inhomogeneous free fermions in the main text, the components of the stress-energy tensor are
T_±(x)
= (1/2) [
    ψ^†_±(x)
    (∓∂_x ) ψ_±(x)  + ]
  - π/ 12L^2,
and the primary fields of the theory are the fermionic fields ψ_±(x) themselves, whose conformal weights are
Δ^+_ψ_+ = Δ^-_ψ_- = 1/2, Δ^-_ψ_+ = Δ^+_ψ_- = 0.


  * 
For an inhomogeneous TLL with constant Luttinger parameter K(x) = K, one has
T_±(x)
= (1/4π K)
  [ πΠ(x) ∓ K ∂_x φ(x) ]^2
  - π/ 12L^2,
and the primary fields include vertex operators constructed as exponentials of the bosonic fields.
Applied to the Luttinger model <cit.>, these vertex operators are (renormalized) fermionic fields ψ_± with conformal weights Δ^+_ψ_+ = Δ^-_ψ_- = (1+K)^2/8K, Δ^-_ψ_+ = Δ^+_ψ_- = (1-K)^2/8K.
(Clearly, free fermions are recovered for K = 1.)



As in the main text, open BCs can be incorporated by standard unfolding methods; cf. <cit.>:
Couple right- and left-moving excitations by identifying

    T_-(x) = T_+(L-x),
        Φ_-(x^-, x^+) = Φ_+(L-x^+, L-x^-)

for all pairs Φ_+, Φ_- of Virasoro primary fields.
It follows that we can equivalently consider a chiral inhomogeneous CFT of only right movers on the interval [-L/2, 3L/2] of length 2L with velocity

    (x)
    ≡
      v(x)    for  x ∈ [-L/2, L/2], 
    
      v(L-x)    for  x ∈ [L/2, 3L/2]

and with every quantity satisfying periodic BCs.
Note that the Hamiltonian (<ref>) can then be written as

    H_iCFT≡∫_-L/2^3L/2 x (x) T_+(x).


For later reference, we define

    (x) 
    ≡∫_0^x s _0/(s),
        1/_0≡1/2L∫_-L/2^3L/2 s/(s),
        _t^±(x)
    ≡^-1( (x) ±_0t ).

Naturally,  lies in the universal cover _+(S^1_2L) of the group of orientation-preserving diffeomorphisms of the circle S^1_2L with circumference 2L. 
Given any v(x), one can show the following properties:

    (L-x) = (x),
        (L-x) = (L) - (x),
        ^-1( (L)-y ) = L - ^-1(y),
        _t^±(L - x) = L - _t^∓(x).

Lastly, we introduce the undeformed Hamiltonian

    H_0 = ∫_-L/2^3L/2 x _0 T_+(x),

whose ground state we denote by |Ω_0⟩.
Similarly, we let |Ω⟩ denote the ground state of H_iCFT.
Following <cit.>, the latter are related to their undeformed counterparts by

    U_+() H_iCFT U_+()^-1
    = H_0 + const,
        
    |Ω⟩
    = U_+()^-1 |Ω_0⟩,

where U_+() is a projective unitary representation of  on the Hilbert space of the CFT.

Similar to the main text, we define

    |Φ_j⟩≡∫_-L/2^L/2 x ( ^- k xξ^+_j(x) Φ_+^†(x,x) + ^ k xξ^-_j(x) 
     Φ_-^†(x,x) ) |Ω⟩     (j = 1,2)

for smooth enough functions ξ^±_j(x), interpreted as distributions or “wave packets” of right/left-moving excitations in each state, and some k ∈ℝ in order to accommodate for  in Eq. (<ref>).
(The latter will have no effect on PWT, and one might as well set k = 0.)
As in the main text, this is a restricted class of states which only allow for one-particle excitations, while we eased other restrictions in the sense that the excitations are not necessarily fermionic and the distributions of right/left movers can be different.

To probe PWT for one-particle excitations, we seek to compute

    ⟨Φ_j| Φ_j⟩
    = ∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ( ^ k x_2ξ^+_j(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_j(x_2)Φ_-(x_2,x_2) ) 
    ×( ^- k x_1ξ^+_j(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_j(x_1) Φ_-^†(x_1,x_1) )
    |Ω⟩

and

    F(t)
    ≡⟨Φ_2| ^- H_iCFTt |Φ_1⟩
    = ∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ( ^ k x_2ξ^+_2(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_2(x_2)Φ_-(x_2,x_2) )
      ^- H_iCFTt
    ×( ^- k x_1ξ^+_1(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_1(x_1) Φ_-^†(x_1,x_1) )
    |Ω⟩,

in order to eventually compare |F(t)| with |⟨Φ_1| Φ_1⟩| for specular waves, ξ^±_2(x) = ξ^∓_1(-x).
Using the unfolding relations (<ref>), we obtain

    ⟨Φ_j| Φ_j⟩
    = ∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ( ^ k x_2ξ^+_j(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_j(x_2)Φ_+(L-x_2,L-x_2) ) 
    ×( ^- k x_1ξ^+_j(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_j(x_1) Φ_+^†(L-x_1,L-x_1) )
    |Ω⟩

and

    F(t)
    = ^- Ct∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ^ H_iCFTt( ^ k x_2ξ^+_2(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_2(x_2)Φ_+(L-x_2,L-x_2) )
      ^- H_iCFTt
    ×( ^- k x_1ξ^+_1(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_1(x_1) Φ_+^†(L-x_1,L-x_1) )
    |Ω⟩,

where we used that H_iCFT |Ω⟩ = C |Ω⟩ for some real constant C that encodes the Casimir energy.
(This only gives rise to an overall phase that one could safely ignore.)
Clearly, as stated in the main text, the above amounts to evaluating two-point correlations for primary fields, specifically those of right movers in a chiral CFT on [-L/2, 3L/2].

First, as a preliminary step, for primary fields with periodic boundary conditions on the system with length 2L, conformal invariance implies that the undeformed two-point correlations are

    ⟨Ω_0| Φ_+(x_2^-, x_2^+) Φ^†_+(x_1^-, x_1^+) |Ω_0⟩
    = 1/2π
      G^+_ΦΦ(x_2^- - x_1^-)^2Δ^+_Φ
      G^-_ΦΦ(x_2^+ - x_1^+)^2Δ^-_Φ,

where G^±_ΦΦ(x) are 2L-periodic functions.
For example, as can be seen from, e.g., <cit.>, the correlations for right-moving free fermions (cf. Example <ref> above) on the system with length 2L and periodic BCs are

    ⟨Ω_0| ψ_+(x_2) ψ^†_+(x_1) |Ω_0⟩
    = 1/2π G^+_FF(x_2 - x_1)

with

    G^+_FF(x)
    ≡πexp( -π/2L x )
    /
      2L sin( π/2L [x + 0^+] )
    ,

where we used that Δ^+_ψ = 1/2, Δ^-_ψ = 0.
Similarly, for the Luttinger model (cf. Example <ref> above), the correlations for the (renormalized) right-moving fermionic fields are

    ⟨Ω_0| ψ_+(x_2^-, x_2^+) ψ^†_+(x_1^-, x_1^+) |Ω_0⟩
    = 1/2π
      G^+_FF(x_2^- - x_1^-)^(1+K)^2/4K
      G^+_FF(x_1^+ - x_2^+)^(1-K)^2/4K,

where we used that

    G^-_FF(x)
    ≡
      -πexp( +π/2L x )
    /
      2L sin( π/2L [x - 0^+] )
    
    = πexp( - π/2L [-x] )
      /
        2L
        sin( π/2L [- x + 0^+] )
      
    = G^+_FF(-x).


Second, by results spelled out in Proposition 3.1 in <cit.> for inhomogeneous CFT, the time dependence in F(t) means that those correlations are evaluated at spacetime points given by ^±_t(x) in Eq. (<ref>).
In addition, the action of U_+() from the expression (<ref>) for |Ω⟩ also leads to a conformal transformation of the fields.
It follows from this and the properties (<ref>) that

    ⟨Φ_j| Φ_j⟩
    = 1/2π∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 '(x_2)^Δ^+_Φ'(x_1)^Δ^+_Φ'(x_2)^Δ^-_Φ'(x_1)^Δ^-_Φ
    ×(
       ^ k (x_2-x_1)ξ^+_j(x_2)ξ^+_j(x_1)
    G^+_ΦΦ((x_2) - (x_1))^2Δ^+_Φ
    G^-_ΦΦ((x_2) - (x_1))^2Δ^-_Φ
        +
    ^ k (x_2+x_1)ξ^+_j(x_2)ξ^-_j(x_1)
    G^+_ΦΦ((x_2) + (x_1) - (L))^2Δ^+_Φ
    G^-_ΦΦ((x_2) + (x_1) - (L))^2Δ^-_Φ
        +
    ^- k (x_2+x_1)ξ^-_j(x_2)ξ^+_j(x_1)
    G^+_ΦΦ( - (x_2) - (x_1) + (L))^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) - (x_1) + (L))^2Δ^-_Φ
        +
    ^- k (x_2-x_1)ξ^-_j(x_2)ξ^-_j(x_1)
    G^+_ΦΦ( - (x_2) + (x_1))^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) + (x_1))^2Δ^-_Φ)

and

    F(t)
    = ^- Ct1/2π∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 '(x_2)^Δ^+_Φ'(x_1)^Δ^+_Φ'(x_2)^Δ^-_Φ'(x_1)^Δ^-_Φ
    ×(
       ^ k (x_2-x_1)ξ^+_2(x_2)ξ^+_1(x_1) 
    G^+_ΦΦ((x_2) - (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((x_2) - (x_1) + _0t)^2Δ^-_Φ
        +
    ^ k (x_2+x_1)ξ^+_2(x_2)ξ^-_1(x_1) 
    G^+_ΦΦ((x_2) + (x_1) - (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((x_2) + (x_1) - (L) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2+x_1)ξ^-_2(x_2)ξ^+_1(x_1) 
    G^+_ΦΦ( - (x_2) - (x_1) + (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) - (x_1) + (L) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2-x_1)ξ^-_2(x_2)ξ^-_1(x_1) 
    G^+_ΦΦ( - (x_2) + (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) + (x_1) + _0t)^2Δ^-_Φ),

where we used that
(_t^±(x) )
= (x) ±_0t
and
(_t^±(L - x) )
= (L) - (_t^∓(x))
= (L) - (x) ±_0t.

Finally, for specular waves, ξ^±_2(x) = ξ^∓_1(-x), we obtain

    F(t)
    = ^- Ct1/2π∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 '(-x_2)^Δ^+_Φ'(x_1)^Δ^+_Φ'(-x_2)^Δ^-_Φ'(x_1)^Δ^-_Φ
    ×(
       ^ k (x_2-x_1)ξ^+_1(x_2)ξ^+_1(x_1)
    G^+_ΦΦ( - (-x_2) - (x_1) + (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (-x_2) - (x_1) + (L) + _0t)^2Δ^-_Φ
        +
    ^ k (x_2+x_1)ξ^+_1(x_2)ξ^-_1(x_1)
    G^+_ΦΦ( - (-x_2) + (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (-x_2) + (x_1) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2+x_1)ξ^-_1(x_2)ξ^+_1(x_1)
    G^+_ΦΦ((-x_2) - (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((-x_2) - (x_1) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2-x_1)ξ^-_1(x_2)ξ^-_1(x_1)
    G^+_ΦΦ((-x_2) + (x_1) - (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((-x_2) + (x_1) - (L) + _0t)^2Δ^-_Φ),

after a change of variable and rearranging.

By comparing the terms appearing in Eqs. (<ref>) and (<ref>), it follows that |F(t)| for certain t equals |⟨Φ_1| Φ_1⟩| for all ξ^±_1(x) if and only if (x) is even, meaning (x) must be odd.
This in turn implies (L) = L and _0 = v_0 given by Eq. (<ref>), which means that equality holds for all t = (2m+1)L/v_0 due the 2L periodicity of G^±_ΦΦ(x).
[Moreover, for ξ^±_2(x) = ξ^∓_1(-x), one has |⟨Φ_1| Φ_1⟩| = |⟨Φ_2| Φ_2⟩| when v(x) is even.]
Since (x) = v(x) for all x ∈ [-L/2, L/2], this proves the claim that one-particle PWT is exhibited for all even inhomogeneous CFTs.

As a last remark, one can unfold in a different way to take the spin/statistics of the primary fields into account, so that, e.g., one gets anti-periodic BCs for fermions
on the system of size 2L, and twisted BCs for general anyonic fields.
This can be used to obtain simple explicit formulas for the two-point correlations of any primary fields.
However, since we are interested in recombinations of waves traveling in different directions, it is necessary to identify so as to always have periodic BCs.
In any case, for our purposes, all we needed was the spacetime dependence of the correlations, which are fixed on general grounds in CFT, circumventing any need for explicit formulas.






§ PART B: SPECTRAL PROPERTIES OF INHOMOGENEOUS TLLS





In this part, we first show that reflection symmetry is a necessary condition for PWT for any theory; here under the assumption that the Hamiltonian has pure point spectrum.
Second, we give the details for how to use SL theory to diagonalize the inhomogeneous TLL Hamiltonian (<ref>) and to formulate the PWT requirements on the spectrum.
Third, we review some details on Weyl's law applied to SL operators and, fourth, extend the previous discussion to TLL Hamiltonians with a mass term.
Fifth, we comment on the Liouville normal form used to state the results for the inverse SL problem in the main text.






 §.§ Necessity of reflection symmetry





Before considering TLL theory, we note that all models studied in the main text have pure point spectrum [as long as the integral defining 1/v_0 in Eq. (<ref>) converges].
Thus, we first consider a general Hamiltonian H with pure point spectrum and show that for such a system to exhibit PST/PWT, it is necessary that H commutes with the parity operator , which maps x ↦ -x.
In other words, we will show that reflection symmetry is a necessary condition.

To prove the statement, let E_n denote the eigenenergies of H with the corresponding energy eigenstates |n⟩.
On general grounds, the latter form a complete orthonormal basis for the Hilbert space  of the theory.
Let |Φ_1⟩ be an arbitrary state in  and let |Φ_2⟩≡ |Φ_1⟩.
(We recall that  is a unitary operator satisfying ^2 = 1, meaning that ^-1 = ^† =.)
As usual, define F(t) ≡⟨Φ_2| ^- H t |Φ_1⟩ =  ⟨Φ_1| ^- H t |Φ_1⟩.
For general PST/PWT, we require that there exists T > 0 such that |F(T)| = |⟨Φ_1|Φ_1⟩| for any state |Φ_1⟩.
In particular, this must hold for any energy eigenstate |n⟩, implying |⟨ n||n⟩| = 1.
If |n⟩| is not an eigenstate of , it is straightforward to show that this equality cannot hold.
Consequently, all energy eigenstates must also be eigenstates of , which implies that [, H] = 0.






 §.§ Inhomogeneous TLLs and SL theory





We now turn to inhomogeneous TLLs studied in the main text.
Inserting the expansions (<ref>) into Eq. (<ref>) yields

    H_iTLL
    = 1/2π [πΠ_0]^2
      + 1/2π∑_n = 1^∞( [πΠ_n]^2 + λ_n φ_n^2 )^2 ,

where φ_0 is absent (when E_0 = 0), no different from the standard case with constant K(x) [and v(x)].
For n ≥ 1, let

    φ_n
    = √(π/2)1/λ_n^1/4( a_n + a^†_n),
        Π_n
    = -λ_n^1/4/√(2π)( a_n - a^†_n)

in terms of bosonic creation and annihilation operators a^†_n and a_n, satisfying [ a_n, a^†_n'] = δ_n,n', as in the main text.
Inserting Eq. (<ref>) into Eq. (<ref>) yields Eqs. (<ref>)–(<ref>).

Using the above, one can compute the time evolution of any observable in the algebra of operators generated by the fields φ(x) and Π(x).
For instance, for φ(x, t) ≡^ H_iTLL tφ(x) ^- H_iTLL t and Π(x, t) defined similarly, we obtain

    φ(x, t)
    = ∑_n = 1^∞√(π/2E_n)( a_n^- E_nt + a^†_n^ E_nt)
      u_n(x),
        Π(x, t)
    = ∑_n = 1^∞ (-) √(E_n/2π)( a_n^- E_nt - a^†_n^ E_nt)
      w(x) u_n(x),

where we omitted the zero modes (as they have to be treated separately when E_0 = 0) and used ^ H_iTLL t a_n^- H_iTLL t = a_n^- E_nt and ^ H_iTLL t a^†_n^- H_iTLL t = a^†_n^ E_nt.
Generally, any observable (x, t) in this algebra will contain contributions of the form a^†_n^ E_n t u_n(x) and a_n^- E_n t u_n(x).
Now, in order for PWT to be exhibited, Eq. (<ref>) must hold for any state, which means that it holds true without expectation values, i.e., (x, 0) must equal (-x, T) at some later time T > 0 as an operator identity.
Taking = φ and using the linear independence of the generators a_n and a^†_n, Eq. (<ref>) yields

    u_n(x) = u_n(-x) ^± E_n T  ∀ n = 0, 1, …

for a common T > 0, where we included the trivial case n = 0 [recall that u_0(x) = const and E_0 = 0 when q(x) = 0 in Eq. (<ref>)].
Since u_n(x) = u_n(-x), Eq. (<ref>) implies that each u_n(x) must be an eigenfunction of the parity operator , whose eigenvalues are +1 or -1 (since ^2 = 1).
It follows that all eigenfunctions must have definite parity, and the only possibility for PWT to hold is that the corresponding eigenenergies satisfy E_n T = π m_n for integers m_n for each n, with m_n even (odd) if u_n(x) has even (odd) parity.

Finally, it is straightforward to show the following statement:
The eigenfunctions u_n(x) of a regular SL operator  have definite parity if and only if [, ] = 0.
For our operator in Eqs. (<ref>)–(<ref>), this means that both v(x) and K(x) must be even functions (using that they must be positive to exclude the possibility that both are instead odd).

Given the above, together with Properties <ref>–<ref> for regular SL operators stated in the main text, we have derived our main result on the spectral requirements for an inhomogeneous TLL to exhibit PWT.

Moreover, as stated in the main text, reversing the logic, namely, given a spectrum that satisfy the spectral properties for PWT to be exhibited, our formulation of PWT for inhomogeneous TLLs as an inverse SL problem follows.






 §.§ Weyl's law for SL operators





Weyl's law applied to SL operators, see, e.g., <cit.>, states that the number N(λ) of eigenvalues below a given λ > 0 asymptotically behaves as

    N(λ)/√(λ)∼1/π∫_-L/2^L/2 x √(w(x)/p(x))

for large λ.
This holds under certain technical conditions on the coefficients w(x), p(x) in Eq. (<ref>) [which are satisfied assuming our v(x), K(x) are positive and smooth functions].
Using Eqs. (<ref>) and (<ref>) as well as Property <ref> in the main text, it follows that

    n/E_n∼1/π∫_-L/2^L/2 x/v(x)
    = L/π v_0

for large n, where in the last step we used Eq. (<ref>).
Thus, assuming that the integral defining 1/v_0 converges, then this gives the stated asymptotic behavior, E_n = π v_0 n/L + o(n), in the main text.
(Another possibility would be if the integral diverges, in which case E_n∼const× n^γ for γ < 1, but this is inconsistent with that the eigenenergies must be proportional to integers m_n with a common proportionality constant in order for the theory to exhibit PWT.)






 §.§ “Massive” inhomogeneous TLLs





For massless TLL bosons, as discussed above and in the main text, given a spectrum satisfying the PWT requirements, we saw that K(x) can be found uniquely up to rescaling and changing coordinates.

We now extend to SL operators (<ref>) with q(x) ≠ 0, corresponding to an inhomogeneous (possibly imaginary) mass M(x) = √(-q(x)/v_0^2 v(x)) by adding v_0^2 M(x)^2 φ(x)^2 inside the big parentheses in Eq. (<ref>):

    H_iTLL≡∫_-L/2^L/2 x/2π v(x)
      (
          [πΠ(x)]^2/K(x)
          + K(x)[∂_x φ(x)]^2
          + v_0^2 M(x)^2 φ(x)^2
        )
       ,

Indeed, it is straightforward to extend the previous discussion to such models.
For instance, the PWT requirements on the spectrum are the same, except the spectrum does not start at zero:
For q(x) ≠ 0, a constant u_0(x) no longer solves Eq. (<ref>), thus E_0 > 0.
Instead, we have a “massive” inhomogeneous TLL, for which one can show:


A “massive” inhomogeneous TLL exhibits PWT if and only if v(x), K(x), q(x) are even
 and all eigenvalues are of the form E_n = πm̃_n/T for certain integers m̃_n.


Here, these integers m̃_n = m_n + in terms of the integers m_n for the massless case in the main text, where we allow for an overall shift by a finite even integer .
[The latter must be even so that m̃_n is even (odd) if n is even (odd), as required due to Property <ref> in the main text.]
Moreover, given such a spectrum, the results in <cit.> still imply that q̂(y) is uniquely determined by the spectrum, but this can no longer uniquely determine K(x) since the first term in Eq. (<ref>) is now non-zero in general.

As an example, we revisit the case with v(x) = v √(1 - (2x/L)^2) and K(x) = K [ 1 - (2x/L)^2 ]^α for α > -1/2 and constants v, K > 0 encountered in the main text, whose eigenfunctions were given by Gegenbauer polynomials C^(α)_n(2x/L), but now we include a fine-tuned potential q(x) = - (2vα/L)^2 w(x), where we recall that w(x) = K(x)/v(x).
This fine-tuned potential allows us to complete the square so that the eigenvalues in Eq. (<ref>) are λ_n = (π v_0/L) (n + α)^2  with v_0 = 2v/π, meaning that E_n = π (n + α) v_0/L.
Consequently, any c = α = 0, 2, 4, … works, yielding a discrete family of models with fine-tuned inhomogeneous mass that exhibit PWT.
We stress that the mass in this example satisfies M(x) ∝ 1/L and therefore vanishes in the thermodynamic limit.
Thus, one should not expect hallmarks of massive theories, such as exponential decay of two-point correlations with respect to the spatial separation.






 §.§ Liouville normal form and compactified free bosons





It is curious to note the following “shortcut” to writing our massless bosons (<ref>) in Liouville normal form:
Recall that an inhomogeneous TLL is a theory of compactified free bosons with a position-dependent compactification radius R(x) in curved spacetime given by the metric (h_μν) = (v(x)^2/v_0^2, -1) for coordinates (x^0, x^1) ≡ (v_0t, x); see, e.g., <cit.>.
Given a Luttinger parameter profile K(x), the compactification radius is R(x) = √(2α') K(x) (α' has dimension length squared and is commonly set equal to 2).
In terms of the bosonic field φ(x) (with values taken modulo 2π), the action corresponding to the Hamiltonian (<ref>) is

    S
    = 1/4πα'∫^2 x √(-h)
      R(x)^2 (∂_μφ) (∂^μφ),

where we recall that h ≡(h_μν).
This can be viewed as a non-exactly marginal deformation of a theory with constant compactification radius; cf. <cit.>.
The corresponding action for X(x) ≡ R(x) φ(x) becomes

    S
    = 1/4πα'∫^2 x √(-h)[ (∂_μ X) (∂^μ X) - q̂ X^2/v(x)^2 ]

with q̂ = q̂(y) in Eq. (<ref>) for y = y(x) given by Eq. (<ref>), which agrees with Eq. (<ref>) for (y) = X(x)/√(2α').
I.e., as a theory for X(x), even when q(x) = 0, the position-dependent K(x) corresponds a (possibly imaginary) inhomogeneous mass √(q̂(y))/v(x).
This also extends to the case

    S 
    = 1/2π∫^2 x √(-h)[ K(x) (∂_μφ) (∂^μφ) - v_0^2 M(x)^2 φ^2 ]

with a non-zero mass M(x) = √(-q(x)/v_0^2 v(x)) for the bosonic field φ(x), corresponding to the Hamiltonian (<ref>), the only difference being that the first term in Eq. (<ref>) is then no longer zero.






§ PART C: CORRELATION FUNCTIONS IN INHOMOGENEOUS TLLS





Consider the inhomogeneous TLL theory with Hamiltonian H_iTLL in Eq. (<ref>).
Using the eigenfunctions obtained from SL theory, we can compute any ground-state two-point correlation functions for the fields φ(x) and ϑ(x), where the latter is defined by

    ∂_x ϑ(x)
    ≡πΠ(x).

Indeed, using the expansions (<ref>), that a_n annihilates the ground state |Ω⟩ of H_iTLL, as well as the commutation relations [ a_n, a^†_n'] = δ_n,n', it is straightforward to show that

    ⟨Ω| φ(x,t) φ(x',t') |Ω⟩    = ∑_n = 1^∞π/2E_n u_n(x)u_n(x') ^- E_n (t-t'), 
    ⟨Ω| φ(x,t) ϑ(x',t') |Ω⟩    = ∑_n = 1^∞π/2 u_n(x)U_n(x') ^- E_n (t-t'), 
    ⟨Ω| ϑ(x,t) ϑ(x',t') |Ω⟩    = ∑_n = 1^∞π E_n/2 U_n(x)U_n(x') ^- E_n (t-t'),

where we introduced U_n(x) defined by

    ∂_x U_n(x)
    ≡
    w(x) u_n(x).

Similar correlation functions can be computed for other objects, including vertex operators (i.e., normal-ordered exponentials of the bosonic fields) <cit.>.
It is also straightforward to compute Green's functions; see, e.g., <cit.>, the only difference with the present case being that one must account for the non-trivial weight function w(x) that was simply a constant in <cit.>.

As an important example, consider a theory with K(x) = K > 0 constant but v(x) > 0 general [and q(x) = 0].
We recall that the SL operator in this case is = - v(x) ∂_x v(x) ∂_x = -v_0^2 ∂_y^2 using Eq. (<ref>) with eigenenergies E_n = π n v_0/L and eigenfunctions u_n > 0(x) = √(2 v_0/L)cos( nπ [L + 2y]/2L ) and u_0(x) = √(v_0/L).
Then, one can show that

    ⟨Ω| φ(x,t) φ(x',t') |Ω⟩
    
    = 1/4   (
      logexp( -π/2L [f(x)-f(x')-v_0(t-t')] )
      /
        2 sin( π/2L [f(x)-f(x')-v_0(t-t')+0^+] )
      
    + logexp( -π/2L [f(x)+f(x')+L-v_0(t-t')] )
      /
        2 sin( π/2L [f(x)+f(x')+L-v_0(t-t')+0^+] )
      . 
          + .
      log
        -exp( π/2L [f(x)+f(x')+L+v_0(t-t')] )
      /
        2 sin( π/2L [f(x)+f(x')+L+v_0(t-t')-0^+] )
      
    + log
        -exp( π/2L [f(x)-f(x')+v_0(t-t')] )
      /
        2 sin( π/2L [f(x)-f(x')+v_0(t-t')-0^+] )
      ),

where we included the necessary regularizations in order to interpret coincident spacetime points and used the identity ∑_n = 1^∞1/n^-ξ n = - log( 1 - ^-ξ) for any ξ such that ξ > 0.
Eq. (<ref>) is the analytical expression for what was plotted in Fig. <ref> (assuming K = 1).
Moreover, for delta-like waves, the terms appearing in log F(t) given by Eq. (<ref>) for the free-fermion case (<ref>) closely resemble those in Eq. (<ref>).
Indeed, the former agrees with 4 ⟨Ω| φ(x,t) φ(x',t') |Ω⟩ to leading order, and one can furthermore show that the agreement between the two ways of computing (CFT methods versus bosonization) is exact (as it should be) by evaluating and summing all combinations of two-point correlation functions of the vertex operators describing the right/left-moving free fermionic fields.





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 Paper Title: A concrete construction of a topological operator in factorization algebras.








Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502 Japan masashi.kawahira@yukawa.kyoto-u.ac.jp














§ ABSTRACT 
Factorization algebras play a central role in the formulation of quantum field theories given by Kevin Costello and his collaborators. In this paper, we propose a concrete construction of a topological operator in their formulation. 
We focus on a shift symmetry of a one-dimensional massless scalar theory. And we give some discussions of ℤ-gauging of it, i.e., compact scalar theory and its θ-vacuum.


A12, A13, B05, B34, B39

A concrete construction of a topological operator in factorization algebras
    Masashi Kawahira
    August 5, 2024
===========================================================================







§ INTRODUCTION

Quantum field theory is a central topic in theoretical physics. This is valid to describe various kinds of phenomena and also useful to generate some mathematical conjectures.
In spite of such effectiveness, we have not understood the complete mathematical formulation of quantum field theories.

In the last ten years, Kevin Costello and his collaborators have developed a new formulation which is based on factorization algebras<cit.><cit.>.
This formulation works well in perturbative field theories, conformal field theories and topological quantum filed theories. These days, some people are working on formulations of non-perturbative aspects of quantum field theories <cit.><cit.>. 

In such a context, non-perturbative Noether theorem is also proposed by <cit.> which is motivated by recent developments of generalized symmetries <cit.><cit.><cit.>.
The dogma of generalized symmetries is 

    Symmetry= Topological operator.

A topological operator with 𝒰(U) some support U is invariant under homotopic transformations of U.

A question arises: what does this invariant mean ? 
We will show that 𝒰(U_1) and 𝒰(U_2) are in the same equivalence class of Batalin-Vilkovisky cohomology when U_1 and U_2 are homotopic.
We will propose that this is the definition of the topological operators in the formulation by Costello and his collaborators.

In this paper, we focus only on the case of one-dimensional free massless scalar theory.
This theory has a shift symmetry Φ↦Φ+α where Φ is a field and α is a real number.
We give a concrete construction of the topological operator representing this shift ℝ-symmetry.
A virtue of the topological operator is that we can perform gauging of the subsymmetry ℤ⊂ℝ. 
By the ℤ-gauging, we can obtain a compact scalar theory. It is an interesting theory because this theory has instanton effects. 
There is a work about compact U(1) p-form gauge theory<cit.> motivated by S-duality. 

The case of p=0 is essentially the same as this paper in terms of using ℤ-gauging. However, our construction is more concrete, and we will also discuss the relation to generalized symmetry notions and θ vacuum.

This is organized as follows. 
In section 2, we will review the formulation by Costello and his collaborators.
In section 3, we will propose a concrete construction of the topological operator.
In section 4, we will see the θ-degree and the formulation.
Section 5 is devoted for conclusion and discussion.



§ REVIEW OF FACTORIZATION ALGEBRA OF FREE REAL SCALAR THEORY



 §.§ Observable space

In this section, we will define an observable 𝒪. Let M be a d-dimensional manifold and real scalar fields Φ are in C^∞ (M). 
𝒪 must have two properties:

    
  * 𝒪 is a functional, i.e., 𝒪 is a map from a field Φ to a number 𝒪(Φ).
    
  * 𝒪 has a concept of locality. In other words, 𝒪 has support U⊂ M.

Here U is an open subset of M.
If the shape of U is like a d-dimensional open ball B^d, 𝒪_U is a point operator with a UV cut-off. The case of B^d-1× S^1, it is a loop operator.

Motivated by the above two properties, we will find an example of observables: linear observable.

    𝒪_ linear:Φ↦∫_M fΦ,  
        f∈ C_ c^∞(U,ℂ)

where f is a ℂ-function with compact support in U. The integral ∫_M fΦ refer to only the Φ on U, so in this sense 𝒪_ linear has locality.
We identify 𝒪_ linear with f, then let define linear observable space on U as

    Obs_ linear(U):=C_ c^∞(U,ℂ).


A more general observable can be regarded as a polynomial of functions with compact supports:
𝒪=c+f+f_1*f_2+⋯,
where * is a formal symmetric product, c∈ℂ and f,f_1,f_2,⋯∈ C_ c^∞(U,ℂ). And 𝒪 acts Φ as

    𝒪:Φ↦
        c
        +∫_M fΦ
        +∫_M f_1Φ∫_M f_2Φ
        +⋯

[colframe=red,colback=red!3!]

Observable space Obs(U) is defined as 

    Obs(U):= Sym(C_ c^∞(U,ℂ))
 and the elements are called observables on U.

 
 


 §.§ Classical derived observable space

Roughly speaking, derived means that we will add a concept of degree to the observable space Obs(U).
This formulation is originally given by Batalin and Vilkovisky<cit.><cit.>.

    C_ c^∞(U)^0:=(C_ c^∞(U,ℂ),0),
       C_ c^∞(U)^-1:=(C_ c^∞(U,ℂ),-1)

The first one is a degree-0 linear observable, and the second one is a degree-(-1) linear observable which corresponds to anti-field in physics literature.
For simplicity, we denote (f,0)∈ C_ c^∞(U)^0 and (f,-1)∈ C_ c^∞(U)^-1 as f and f^⋆.
The symmetric product * is defined for them as

    a*b=(-1)^|a||b|b*a.

[colframe=red,colback=red!3!]

    Classical derived observable space Obs^ cl(U) is defined as
    
    Obs^ cl(U)
            :=
            ( Sym(C_ c^∞(U)^-1⊕ C_ c^∞(U)^0),Δ_ BV^ cl)

    where Δ_ BV^ cl is a classical Batalin-Vilkovisky operator which is defined in Definition <ref>.
    

By (<ref>), we can rewrite classical derived observable space Obs^ cl(U).

    Obs^ cl(U)  
      =(⋯   ⊕(⋀^2 C_ c^∞(U)^-1*  Sym(C_ c^∞(U)^0))
       ⊕(C_ c^∞(U)^-1*  Sym(C_ c^∞(U)^0))
       ⊕ Sym(C_ c^∞(U)^0)
      ,Δ_ BV^ cl).

This is a decomposition about the degree.
The first line is degree (-2), second one is degree (-1), and third one is degree 0. Actually Δ_ BV^ cl is defined as a map from degree (-n) to degree (-n+1):

    Δ_ BV^ cl:
        ⋀^n C_ c^∞(U)^-1*  Sym(C_ c^∞(U)^0)
        →⋀^n-1 C_ c^∞(U)^-1*  Sym(C_ c^∞(U)^0).

The concrete form of Δ_ BV^ cl depends on what theory we want. In this paper, we are interested in free theory, so we will defined as follows.
[colframe=red,colback=red!3!]

     Classical Batalin-Vilkovisky operator Δ_ BV^ cl is a map C_ c^∞(U)^-1∋ f^⋆↦ -(-Δ+m^2)f∈ C_ c^∞(U)^0. Here Δ is a Laplacian of M. And we define it with Leibniz rule, then we have
     
    Δ_ BV^ cl(f_1^⋆*⋯ *f_n^⋆* P)
           =∑_i=1^n f_1^⋆*⋯*f_i-1^⋆* (-1)^i-1(Δ_ BV^ cl f_i^⋆) *f_i+1^⋆*⋯*f_n^⋆*P
       =∑_i=1^n f_1^⋆*⋯* f_i^⋆ *⋯*f_n^⋆*(-1)^i-1(Δ_ BV^ cl f_i^⋆)*P

    where f_i^⋆∈ C_ c^∞(U)^-1, P∈ Sym(C_ c^∞(U)^0).
    And the (-1)^i-1 comes from the rule that Δ_ BV^ cl and f_i^⋆ are anti-commutative, since Δ_ BV^ cl has a degree +1.
    


By some calculations, we have 

    
  * Leibniz rule : Δ_ BV^ cl(A*B)
    =
    Δ_ BV^ cl(A)*B
    +
    (-1)^|A|A*Δ_ BV^ cl(B),
    
  * Nilpotency : (Δ_ BV^ cl)^2=0.

Nilpotency means that

    Obs^ cl(U)= 
       
       (⋯⋀^2 C_ c^∞(U)^-1*  Sym(C_ c^∞(U)^0)
       C_ c^∞(U)^-1*  Sym(C_ c^∞(U)^0)
        Sym(C_ c^∞(U)^0))

is a chain complex. This is called classical Batalin–Vilkovisky complex.
The cohomology of classical Batalin–Vilkovisky complex
 is called classical Batalin–Vilkovisky cohomology.
 
    H^*( Obs^ cl(U))

Leibniz rule means that H^*( Obs^ cl(U)) has a product *, because the product of A+Δ_ BV^ cl(⋯) and B+Δ_ BV^ cl(⋯) can be rewritten as A*B+Δ_ BV^ cl(⋯).
 [To check this, let us think about (A+Δ_ BV^ clÃ)*(B+Δ_ BV^ clB̃) where A,B satisfy Δ_ BV^ clA=0, Δ_ BV^ clB=0. 
   
    (A+Δ_ BV^ clÃ)*(B+Δ_ BV^ clB̃) 
            =
         A*B+(Δ_ BV^ clÃ)*B+A*(Δ_ BV^ clB̃)+(Δ_ BV^ clÃ)*(Δ_ BV^ clB̃)
       =
         A*B  
         +(Δ_ BV^ clÃ)*B+(-1)^|Ã|Ã*(Δ_ BV^ clB)  
         +A*(Δ_ BV^ clB̃)
       +(-1)^|A|(Δ_ BV^ clA)*B̃  
         +(Δ_ BV^ clÃ)*(Δ_ BV^ clB̃)
         +(-1)^|Ã|Ã*(Δ_ BV^ cl)^2B̃
       =
         A*B  +Δ_ BV^ cl(Ã*B)  +(-1)^|A|Δ_ BV^ cl(A*B̃) +Δ_ BV^ cl(Ã*(Δ_ BV^ clB̃)) 
       =A*B+Δ_ BV^ cl(⋯)
]

   On the ather hand, quantum Batalin-Vilkovisky operator Δ_ BV^ q does NOT have chain rule, as we discuss later, then quantum cohomology H^*( Obs^ q(U)) does NOT have product *.
 

Let us consider the physical meanings of the cohomology H^0( Obs^ cl(U)).
We take a 0-degree observables 𝒪_1,𝒪_2, assume that these are same in the cohomology, i.e., ∃ X  s.t.

    𝒪_2-𝒪_1
        =
        Δ_ BV^ clX

Let Φ_ cl be a solution of the equation of motion -(-Δ+m^2)Φ=0[If you want a theory with some interactions, we need to add some terms to Δ_ BV^ cl.]. We have

    𝒪_2(Φ_ cl)-𝒪_1(Φ_ cl)
           =
        Δ_ BV^ clX(Φ_ cl)
       =
        0.

So we can see H^0( Obs^ cl(U)) as the on-shell evaluation of observables. However [𝒪_1](=[𝒪_2]) is not a number. Then we define a map called state ⟨-⟩.
[colframe=red,colback=red!3!]

    A state ⟨-⟩ is a smooth map:
    
    ⟨-⟩:
            H^0( Obs^ cl(M))
            →ℂ



In some situation, we can calculate H^*( Obs^ cl(M)) explicitly.
[colframe=blue,colback=blue!3!]

    If M is compact and m^2>0, then
    
    H^n( Obs^ cl(M))
            =
            {[            ℂ        (n=0);            0 ( otherwise) ].



[colframe=blue,colback=blue!3!]

    If M=ℝ, then for any value of m
    
    H^n( Obs^ cl(ℝ))
            =
            {[       ℂ[q,p]        (n=0);            0 ( otherwise) ].

    Here q,p has degree 0.


Theorem <ref> tells us that the state ⟨-⟩ can be given as isomorphism, i.e., the state ⟨-⟩ is canonically determined. 
On the other hand, the case of Theorem <ref>, there is room to choose the state ⟨-⟩. Why is there such a difference ? In massive case, we have a reasonable interpretation for this difference.

In the case of non-compact manifold M_ non, the boundary condition is necessary to compute the path integral.
If not, the scalar field Φ might diverge on ∂M_ non.
So the path integral calculation fails.
In order to compute the value of the path integral, we must choose a boundary condition.
Reflecting it, we must choose a map ⟨-⟩:H^0( Obs^ cl(M_ non))→ℂ.

On the other hand, in the case of compact manifold M_ com, the scalar field Φ never diverge on ∂ M_ com.
So we can decide the value of the path integral.
Similarly, we find the state ⟨-⟩ canonically.

The above discussion is valid for massive case.
However, in massless case, even for a compact manifold, H^0( Obs^ cl(M_ com)) is not isomorphic to ℂ. In other words, we can not decide the expectation value naturally, though there is no Φ's divergence on the boundary ∂ M_ com.
In order to understand it, we have to see another factor to interrupt the path integral computation: IR divergence. 
We will propose a treatment of IR divergence in <cit.>.



 §.§ Quantum derived observable space

[colframe=red,colback=red!3!]

    Quantum derived observable space Obs^ q(U) is defined as
    
    Obs^ q(U)
            :=
            ( Sym(C_ c^∞(U)^-1⊕ C_ c^∞(U)^0⊕ℂħ),Δ_ BV^ q=Δ_ BV^ cl+ ħδ)

    where Δ_ BV^ q=Δ_ BV^ cl+ ħδ is a quantum Batalin-Vilkovisky operator and δ is defined in Definition <ref>.
    

Note that C_ c^∞(U)*ℂ=C_ c^∞(U), we can rewrite it as the formal series of ħ:

    Obs^ q(U)= 
     (⋯   ⊕(⋀^2 C_ c^∞(U)^-1* Sym(C_ c^∞(U)^0)[ħ])
       ⊕(C_ c^∞(U)^-1* Sym(C_ c^∞(U)^0)[ħ])
       ⊕ Sym(C_ c^∞(U)^0)[ħ],Δ_ BV^ q).

δ is a quantum correction and will be defined as a map from -n-degree to -n+1-degree:

    δ:
        ⋀^n C_ c^∞(U)^-1*  Sym(C_ c^∞(U)^0)
        →⋀^n-1 C_ c^∞(U)^-1*  Sym(C_ c^∞(U)^0)

[colframe=red,colback=red!3!]

The definition of δ is a bit complicated. So we take three steps.

(1) For f^⋆*g∈ C_ c^∞(U)^-1* C_ c^∞(U)^0, the quantum correction δ is defined as

    [ C_ c^∞(U)^-1* C_ c^∞(U)^0                        δ⟶       ℝ⊂ Sym(C_ c^∞(U)^0);                       90∈                                                 90∈;                     f^⋆*g                         ⟼           ∫_U Q(x)g(x) dx ]

so δ has degree +1.

(2) For f^⋆*P∈ C_ c^∞(U)^-1* Sym(C_ c^∞(U)^0), it is sufficient to think the n-th term g_1*⋯*g_n of P.

    [      C_ c^∞(U)^-1* Sym(C_ c^∞(U)^0)                                  δ⟶                    Sym(C_ c^∞(U)^0);                                 90∈                                                                     90∈;                       f^⋆*g_1*⋯*g_n                                   ⟼ ∑_i=1^n g_1*⋯*g_i*⋯*g_n *δ(f^⋆*g_i) ]


(3) For f_1^⋆*⋯*f_m^⋆*P∈⋀^m C_ c^∞(U)^-1* Sym(C_ c^∞(U)^0)

    [                 ⋀^m C_ c^∞(U)^-1* Sym(C_ c^∞(U)^0)                                                 δ⟶               ⋀^m-1 C_ c^∞(U)^-1* Sym(C_ c^∞(U)^0);                                                90∈                                                                                                   90∈;                                    f_1^⋆*⋯*f_m^⋆*P                                                  ⟼ ∑_i=1^m f_1^⋆*⋯*f_i^⋆*⋯*f_m^⋆*(-1)^i-1δ(f_i^⋆*P) . ]



δ is nilpotent[To check it, we use δ(f^⋆*δ(f̃^⋆*P))=δ(f̃^⋆*δ(f^⋆*P)).] and anti-commute with Δ_ BV^ cl
[We can understand that it is because Δ_ BV^ cl and δ have degrees +1.]: δ^2=0, Δ_ BV^ clδ=-δΔ_ BV^ cl.
Then we have

    (Δ_ BV^ q)^2
          =(Δ_ BV^ cl+ħδ)(δ_ BV^ cl+ħδ)
       
    
          =0.

This means that the following one is a chain complex. This is called quantum Batalin–Vilkovisky complex.

    Obs^ q(U)= (⋯   ⋀^2 C_ c^∞(U)^-1*  Sym(C_ c^∞(U)^0)[ħ]
       C_ c^∞(U)^-1*  Sym(C_ c^∞(U)^0)[ħ]
        Sym(C_ c^∞(U)^0)[ħ])


Obs^ q(U) does not have a Leibniz rule. Instead, we have the following theorem.

[colframe=blue,colback=blue!3!]

For A,B∈ Obs^ q(U) with some degree |A|,|B|,

    Δ_ BV^ q(A*B)=(Δ_ BV^ qA)*B+(-1)^|A|A*(Δ_ BV^ qB)+(-1)^|A|ħ{A,B} .



Here {-,-} is the anti-bracket which is defined just below.[This definition is motivated by the usual anti-bracket in physics literature:

    {a,b}:=∑_i=1^N [(a∂/∂ x^i)(∂/∂x^⋆^ib)-(a∂/∂x^⋆^i)(∂/∂ x^ib)]

where x^i is bosonic and x^⋆^i is fermionic.] 
By this theorem, H^*( Obs^ q(U)) does NOT have a product * in contrast to H^*( Obs^ cl(U)). 
[colframe=red,colback=red!3!]

        For f^⋆∈ C^∞_ c(U)^-1, g∈ C^∞_ c(U)^0,
    we define anti-bracket:
    
    {f^⋆,g}:=∫_U Q(x)g(x) dx∈ℝ⊂ Sym(C^∞_ c(U)^0).

    For any A,B∈ Obs^ cl,q(U) with some definite degrees, we define anti-bracket which has the following properties:
    
        
  * {A,B}=0 if A,B∈ C^∞_ c(U)^0 or A,B∈ C^∞_ c(U)^-1
        
  * {A,B}=-(-1)^(|A|+1)(|B|+1){B,A}
        
  * {A,B*C}={A,B}*C+(-1)^(|A|+1)|B|B*{A,C}
    



However in special situation we can define a product * for quantum cohomology !
Let A and B have compact support in U_1 and U_2 respectively.
And assume U_1 and U_2 are disjoint open subsets of U. Then

    {A,B}_U=0.

The subscription U denotes that this is computed in U. 
This is because the computation of { A,B}_U reduces to the integrals

    ∫_U
        (U_1 -supported function)
        ×
        (U_2 -supported function)
        =0

(<ref>) means that we have Leibniz rule for such A,B, thus we can define a product * for them.

[colframe=blue,colback=blue!3!]

Let U_1 and U_2 be disjoint open subsets of U.
The quantum Batalin–Vilkovisky cohomology has a product. 

    [ H^*( Obs^ q(U_1))× H^*( Obs^ q(U_2))                                    ⟶                      H^*( Obs^ q(U));                                  90∈                                                                       90∈;                    ([A]_U_1,[B]_U_2)                                    ⟼              [A]_U_1*[B]_U_2=[A*B]_U ]

Here subscripts of [ · ]_- denotes that it is in H^*( Obs^ q(-)).
This product is the same as the “operator product" in physics literature.



The cohomology is formal series of ħ, so the definition of a state changes a little.
[colframe=red,colback=red!3!]

    A state ⟨-⟩ is a smooth map:
    
    ⟨-⟩:
            H^0( Obs^ q(M))
            →ℂ[ħ]






 §.§ Weyl algebra in one-dimensional system

In the case of M=ℝ, the quantum cohomology forms Weyl algebra (the canonical commutation relation).
[colframe=blue,colback=blue!3!]

    If M=ℝ, then for any value of m
    
    H^n( Obs^ q(ℝ))
            =
            {[  Wey algebra        (n=0);            0 ( otherwise) ].




The aim of this section is to explain what the generators of Weyl algebra are.
Roughly speaking, the generators of Weyl algebra are ħ,[Q],[P].[The definition of Q and P is written in Appendix <ref>.]
However, it seems strange, because  Q,P have the same support I=(-1/2,1/2), thus we can not define the products of them.

In order to take the products, we will define modified position observable 𝒬_t and modified momentum observable 𝒫_t.
[colframe=red,colback=red!3!]

    Modified position and momentum  observable 𝒬_t and 𝒫_t are in C_ c^∞(I_t), I_t=(-1/2+t,1/2+t). The definition of them is

    𝒬_t(x):=ϕ_p(t)P_t(x)-ϕ̇_p(t)Q_t(x),
      
    𝒫_t(x):=ϕ_q(t)P_t(x)-ϕ̇_q(t)Q_t(x)

where ϕ_q and ϕ_p are defined in Appendix, and Q_t and P_t are defined as

    Q_t(x):=Q(x-t), P_t(x):=P(x-t).



Modified observables have the following special property: 
[colframe=blue,colback=blue!3!]


    ∂/∂ t𝒬_t(·)
           =
        -Δ_ BV^ q(ϕ_p(t)Q^⋆_t(·)),
    ∂/∂ t𝒫_t(·)
           =
        -Δ_ BV^ q(ϕ_q(t)Q^⋆_t(·)).

Remind that Q=𝒬_t=0 and P=𝒫_t=0, then for any t

    [Q]=[𝒬_t],  
        [P]=[𝒫_t].





    ∂/∂ t𝒬_t(x) 
           =ϕ̇_p(t)P_t(x)+ϕ_p(t)∂/∂ tP_t(x)-ϕ̈_p(t)Q_t(x)-ϕ̇_p(t)∂/∂ tQ_t(x) 
       =-ϕ̇_p(t)∂/∂ xQ(x-t)-ϕ_p(t)∂/∂ t∂/∂ xQ(x-t)-ϕ̈_p(t)Q(x-t)-ϕ̇_p(t)∂/∂ tQ(x-t) 
       =-ϕ_p(t)∂/∂ t∂/∂ xQ(x-t)-ϕ̈_p(t)Q(x-t) 
       =-ϕ_p(t)∂/∂ t∂/∂ xQ(x-t)-m^2ϕ_p(t)Q(x-t) 
       =ϕ_p(t)(∂/∂ x)^2Q(x-t)-m^2ϕ_p(t)Q(x-t) 
       =[(∂/∂ x)^2-m^2]ϕ_p(t)Q(x-t) 
       =-Δ_ BV^ q(ϕ_p(t)Q^⋆_t(x))

By the similar calculations, we also obtain

    ∂/∂ t𝒫_t(x)=-Δ_ BV^ q(ϕ_q(t)Q^⋆_t(x))



The support of 𝒬_t and 𝒫_t is I_t=(-1/2+t,1/2+t). 
Therefore we can define products of them like

    [𝒬_t]_I_t*[𝒫_s]_I_s   or  
        [𝒬_t*𝒫_s]
          
        (|t-s|>1)

where I_t and I_s denote being in H^*( Obs^ q(I_t)) and H^*( Obs^ q(I_s)).
Note that this product is independent for the choice t and s because of Theorem <ref>.

[colframe=blue,colback=blue!3!]

    We have the canonical commutation relation:
    
    [𝒬_t*𝒫_s]
            -
            [𝒫_t*𝒬_s]
            =ħ[1]
              
             (s-t>1).

    [𝒬_∙] and [𝒫_∙] are the generator. Then  H^0( Obs^ q(ℝ)) is Weyl algebra.





§ CONSTRUCTION OF TOPOLOGICAL OPERATOR



 §.§ Shift symmetry

In case of m=0, we have a shift symmetry

    Φ↦Φ+α  
        (α∈ℝ)

for equation of motion ΔΦ=0.
Reflecting it, P_t must be conserved.
This is true because 𝒫_t is conserved and in massless case

    𝒫_t(x)
        =
        ϕ_q(t)P_t(x)-ϕ̇_q(t)Q_t(x)
        =
        P_t(x)

In other words, P_t is the Noether charge.



 §.§ Construction of topological operator


First of all, we will review the usual construction of the topological operators.
Let q̂ and p̂ be the generators of Wely algebra, i.e., satisfy [q̂,p̂]=ħ.
In physics literature, the topological operator of the shift symmetry is
[We have no i in front of the αp̂, because Weyl algebra is [q̂,p̂]=ħ.]

    V̂_α
        :=exp(αp̂)=∑_i=0^∞1/n!(αp̂)^n.

Obviously we have

    q̂V̂_α
         =V̂_α(q̂+αħ).


So the naive construction of the topological operator in Batalin-Vilkovisky formalism is

    [𝒱_α]
           :=
        ∑_n=0^∞1/n! α^n
        [P_s_0*P_s_1*⋯*P_s_n],
       (s_1-s_0>1, s_2-s_1>1,⋯,s_n-s_n-1>1).

However there is a problem.
The support of 𝒱_α is infinitely wide.
This is not good as the topological operator because we can not define the action.

So we define the topological operator in another way.
[colframe=red,colback=red!3!]


    [𝒰_α,t]
        :=
        ∑_n=0^∞1/n! α^n
        [P_t*P_t*⋯*P_t_n]
        =
        ∑_n=0^∞1/n! α^n
        [P_t^*n]

𝒰_α,t obviously has the finite support I_t.    



Instead of the finite support, we lost the clearness of the following properties.

    
  * topologicalness 
    
    [ d/ dt𝒰_α,t]_I_t=0
 
    
  * action for Q_0 
    
    [Q_0*𝒰_α,t]
            =
            [𝒰_α,s*(Q_0+αħ)],   (t>1,-1>s)


In the next sections, we will prove these properties.



 §.§ Proof of topologicalness

[colframe=blue,colback=blue!3!]

    We have a topological operator 𝒰_α,t. Thus
    
    [ d/ dt𝒰_α,t]_I_t=0.
 
    


    It is enouph to show that
    
    d/ dt[P_t^*n]_I_t=0.
   
    We have
    
    d/ dtP_t^*n=nP_t^*(n-1)* d/ dtP_t,
        d/ dtP_t=-Δ_ BV^ q(ϕ_q(t)Q^⋆_t)=-Δ_ BV^ qQ^⋆_t

    since ϕ_q=1 for the case of m=0.
    Then we obtain
    
    d/ dtP_t^*n=nP^*(n-1)_t*Δ_ BV^ qQ^⋆_t .
   
    It is enough to show that
    
    Δ_ BV^ q(P_t^*(n-1)*Q_t^⋆)=P_t^*(n-1)*Δ_ BV^ qQ_t^⋆.

    However in fact P_t^*(n-1)∈ Sym(C^∞_ c(I_t)^0), Q^⋆_t∈ C_ c^∞(I_t)^-1 then
    
    Δ_ BV^ q(P_t^*(n-1)*Q_t^⋆)=P_t^*(n-1)*Δ_ BV^ qQ_t^⋆+ħδ(P_t^*(n-1)*Q_t^⋆).

    So we need to see that δ(P_t^*(n-1)*Q_t^⋆) vanishes.
    From now, we compute δ(P_t^*(n-1)*Q_t^⋆)
    
    δ(P_t^*(n-1)*Q_t^⋆)
            =(n-1) P_t^*(n-2)∫_ℝP_t(x)Q_t(x) dx

    So We will see that∫_ℝP_t(x)Q_t(x) dx vanishes.
    
    ∫_ℝP_t(x)Q_t(x) dx
             =-∫_ℝQ'(x-t)Q(x-t) dx
       =-∫_ℝQ'(x)Q(x) dx
       =-1/2∫_ℝ∂/∂ x(Q(x)Q(x)) dx

    Q(x) has compact support in I, so ∫_ℝP_t(x)Q_t(x) dx=0.




 §.§ Proof of action for operators

[colframe=blue,colback=blue!3!]

    Q_0 is a charged operator against 𝒰_α,t.
    
    [Q_0*𝒰_α,t]
            =
            [𝒰_α,s*(Q_0+αħ)],
               (t>1,-1>s)




    It is enough to show that
    
    [Q_0*(P_t^*n-P_-t^*n)]=nħ[P_0^*(n-1)]

    for some t>1.
    We have obtained
    
    d/ dtP_t^*n=nΔ_ BV^ q(P_t^*(n-1)*Q_t^⋆)

    in the proof of Theorem <ref>.
    So we have
    
    P_t_2^*n-P_t_1^*n=Δ_ BV^ qH_t_1,t_2(n), 
       H_t_1,t_2(n):=n∫_t_1^t_2 du P_u^*(n-1)*Q_u^⋆.

    By multiplication of Q_0
    
    Q_0*(P_t^*n-P_-t^*n)=Q_0*Δ_ BV^ qH_-t,t(n).

    We will show that the right hand side is nħ P_0^*(n-1) up to the cohomology.

    We know δ(P_u^*(n-1)*Q_u^⋆)=0 from the proof of Theorem <ref>, 
    then we have
    
    Δ_ BV^ q(Q_0*H_-t,t(n)) 
            =Q_0*Δ_ BV^ q(H_-t,t(n))+ħδ(Q_0*H_-t,t(n)),

    so it is enough to show that ħδ(Q_0*H_-t,t) is equivalent to nħ P_0^*(n-1) up to the cohomology.
    
    We compute ħδ(Q_0*H_-t,t(n)).
    
    ħδ(Q_0*H_-t,t(n))
               =
            nħ∫_-t^t du δ(Q_0*P_u^*(n-1)*Q_u^⋆)
       =
            nħ∫_-t^t du (
            δ(Q_0*Q_u^⋆)P_u^*(n-1)
            +
            Q_0*δ(P_u^*(n-1)*Q_u^⋆)
            )
       =
            nħ∫_-t^t du δ(Q_0*Q_u^⋆)P_u^*(n-1)
       =
            nħ∫_-t^t du 
            P_u^*(n-1)∫_ℝ dx Q_0(x)Q_u(x) 
       =
            nħ∫_-t^t du 
            P_u^*(n-1)∫_ℝ dx Q(x)Q(x-u)

    Here we used δ(P_u^*(n-1)*Q_u^⋆)=0.
    Substituting P_u^*(n-1)-P_c^*(n-1)=Δ_ BV^ qH̃_c,u(n-1),
    
    ħδ(Q_0*H_-t,t(n))
               =
            nħ P_c^*(n-1)∫_-t^t du ∫_ℝ dx Q(x)Q(x-u)
            
       +
            nħ∫_-t^t du Δ_ BV^ qH̃_c,u(n-1)∫_ℝ dx Q(x)Q(x-u)

    In the second term of the right hand side, Q(x)Q(x-u) is just a number[These trivially act for the scalar field Φ∈ C^∞(ℝ)], so this term is trivial in cohomology.
    It remains to show that
    
    ∫_-t^t du ∫_ℝ dx Q(x)Q(x-u)=1.

    for some t>1.
    When u is sufficiently large, Q(x)Q(x-u)→ 0.
    So we can take a limit t→∞.
    
    ∫_ℝ du ∫_ℝ dx Q(x)Q(x-u)
            =
            ∫_ℝ du ∫_ℝ dx Q(x)Q(u-x)
            =
            ∫_ℝ du ∫_ℝ dx Q(x)Q(u)
            =1.


[colframe=blue,colback=blue!3!]

    P_0 has no charge against 𝒰_α,t.
    
    [P_0*𝒰_α,t]
            =
            [𝒰_α,s*P_0],
               (t>1,-1>s)




    It is enough to show that
    
    [P_0* (P^*n_t-P_-t^*n)]=0

    for some t>1. We already have

    P_t_2^*n-P_t_1^*n=Δ_ BV^ qH_t_1,t_2,
       H_t_1,t_2:=n∫_t_1^t_2 du P_u^*(n-1)*Q_u^⋆.

By multiplication of P_0

    P_0*(P_t_2^*n-P_t_1^*n)=P_0*Δ_ BV^ qH_t_1,t_2.

We will show the right hand side is zero up to cohomology.

Since

    Δ_ BV^ q(P_0*H_t_1,t_2)
       =
       P_0*Δ_ BV^ q(H_t_1,t_2)
       +
       ħδ(P_0*H_t_1,t_2),

it is enough to show that ħδ(P_0*H_t_1,t_2)=0 up to the cohomology.

We compute ħδ(P_0*H_t_1,t_2).

    ħδ(P_0*H_t_1,t_2)
           =
        ħ∫_-t^t du δ(P_0*P_u^*(n-1)*Q_u^⋆)
       =
        ħ∫_-t^t du (
        δ(P_0*Q_u^⋆)P_u^*(n-1)
        +
        P_0*δ(P_u^*(n-1)*Q_u^⋆)
        )
       =
        ħ∫_-t^t du δ(P_0*Q_u^⋆)P_u^*(n-1)
       =
        ħ∫_-t^t du 
        P_u^*(n-1)∫_ℝ dx P_0(x)Q_u(x)
        
       =
        ħ∫_-t^t du 
        P_u^*(n-1)∫_ℝ dx Q'(x)Q(x-u)

Here we used δ(P_u^*(n-1)*Q_u^⋆)=0. 
Substituting P_u^*(n-1)-P_c^*(n-1)=Δ_ BV^ qH_c,u,

    ħδ(P_0*H_t_1,t_2)
           =
        P_c^*(n-1)ħ∫_-t^t du ∫_ℝ dx Q'(x)Q(x-u)
       +
        ħ∫_-t^t du 
        (Δ_ BV^ qH_c,u)∫_ℝ dx Q'(x)Q(x-u)

    In the second term of the right hand side, Q'(x)Q(x-u) is just a number, so this term is trivial in cohomology.
    It remains to show that
    
    ∫_-t^t du ∫_ℝ dx Q'(x)Q(x-u)=0.

    for some t>1.
    When u is sufficiently large, Q'(x)Q(x-u)→ 0.
    So we can take a limit t→∞.
    
    ∫_ℝ du ∫_ℝ dx Q'(x)Q(x-u)
            =
            ∫_ℝ du ∫_ℝ dx Q'(x)Q(u-x)
            =
            ∫_ℝ du ∫_ℝ dx Q'(x)Q(u)
            =0.





§ GAUGING AND COMPACT SCALAR



 §.§ ℤ-gauging

Topological operators have the advantage over Noether charge, thus we can define discrete gauging. This is because topological operators describe finite group transformations, while Noether charges only do infinitesimal transformations. 
So now we have group action like

    ℝ↷
        H^*( Obs^ q(ℝ)).

Subgroup ℤ⊂ℝ also acts as

    ℤ↷
        H^*( Obs^ q(ℝ)).

So we can define[As (pre)factorization algebra, it is better to consider finite interval: I H^*( Obs^ q(I))/ℤ.]

    H^*( Obs^ q(ℝ))/ℤ

The generators are

    ħ,  [ Exp(± iQ)],  
        [P]

where Exp(∙):=∑_n1/n!∙ ^*n. 
They satisfy

    [ Exp(± iQ)*P_-t]
        -
        [ Exp(± iQ)*P_t]
        =[ Exp(± iQ)]  (t>1).

This is the same as [p̂,e^± iq̂]=ħ e^± iq̂ in a usual notation.
[colframe=red,colback=red!3!]

We saw the algebra generated by

    p̂,e^iq̂,e^-iq̂,ħ 1

and satisfying

    [p̂,e^± iq̂]=ħ e^± iq̂.

We call it periodic Weyl algebra.


We can say

    H^0( Obs^ q(ℝ))/ℤ≅ periodic Weyl algebra

similarly to Theorem <ref>.





 §.§ Compact scalar and θ-vacuum

After ℤ gauging, massless scalar theory becomes compact scalar theory as explained in <cit.>,<cit.>.
In compact scalar theory, we have θ-degree. 
We can express θ as θ-term or θ-vacuum.
However to implement θ-term is difficult in this formalism, because it is based on equation of motion and equation of motion does not have θ-term.
On the other hand, implementing θ-vacuum is relatively easy since a vacuum is just a map from the cohomology to ℂ[ħ].

    H^*( Obs^ q(ℝ))/ℤ→ℂ[ħ]


One way to give a state is to think a representation of H^*( Obs^ q(ℝ))/ℤ.
For simplicity, we use e^± iq̂ and p̂ rather than [ Exp(± iQ)] and [P].
Take eigenvector v_p 

    p̂v_p=pv_p.

Then we find that p must be quantized.
To see that, consider a vector e^iq̂v_p. This satisfies

    p̂(e^iq̂v_p)
        =
        (p+ħ)(e^iq̂v_p)

Therefore e^± iq̂ are ladder operators.
The spectrum of p̂ is

    λ_r(p̂)={ħ(n+r) | n∈ℤ,  r∈ℝ}
.
which depend on r. Obviously r+1-spectrum is equivalent to  r-spectrum.
Physicist denote r as θ/2π.
At θ=0 and θ=π, we can see ℤ_2-symmetry of the spectrum : p↦-p.
So the map (<ref>) is given by a trace

    ⟨-⟩_r=θ/2π:X↦∑_p∈λ_r(p̂) (v_p,Xv_p).

We can see θ-dependence in the state.




§ CONCLUSION AND DISCUSSION

We have seen the construction of a topological operator 𝒰_α,t.
The point of the construction is the order: at first taking products *, and next taking a cohomology [·] (we call it the order A). 
If we take reversed order (we call it the order B), we have

    [𝒱_α]
           :=
        ∑_n=0^∞1/n!(iα)^n
        [P]_I_s_0*[P]_I_s_1*⋯*[P]_I_s_n,
       (s_1-s_0>1, s_2-s_1>1,⋯,s_n-s_n-1>1),

and it has infinitely wide support.
The order A vs the order B is the similar to that path integral formalism vs operator formalism.
In path integral formalism, we can formally think 
of multiplications of operators like 𝒪_1(Φ)𝒪_2(Φ)⋯𝒪_n(Φ) because these are just c-numbers.
On the other hand, in operator formalism, in order to give multiplications of operators, we need to get rid of UV divergences like :𝒪_1𝒪_2⋯𝒪_n: since they are q-numbers.
In the order A, observables 𝒪 are just c-number, so we can freely take the products 𝒪_1*𝒪_2*⋯*𝒪_n. This is the same as path integral formalism. However, in the order B, [𝒪] can not be taken products freely, so this corresponds to operator formalism.
In order to take products, the support of [𝒪_1]*[𝒪_2]*⋯*[𝒪_n] need to be wider than the original support. It is a kind of UV divergences.

And we have seen θ-dependence of the state ⟨-⟩ by representation theory.
It is interesting to construct map in another way.
For example, the state of the massive scalar theory on M=ℝ can be given by the embedding

    f:C_ c^∞(ℝ)
        ↪𝒮(ℝ)

where 𝒮(ℝ) is the set of Schwartz functions on ℝ.
By considering Obs_𝒮:= Sym(𝒮(ℝ)^-1→𝒮(ℝ)^0), we obtain H^0( Obs^ q_𝒮(ℝ)))≅ℂ[ħ]. Then f induce the state

    H^0( Obs^ q(ℝ))
        →
        H^0( Obs^ q_𝒮(ℝ))≅ℂ[ħ]

It is attractive to give a state in similar way in the case of compact scalar.
However, in order to achieve this, we need the technique to remove IR divergence because compact scalar is massless. Such a technique is discussed in <cit.>.

And there is another way to consider the compact scalar theory. It is to focus on factorization homology.
The factorization algebra of compact scalar is essentially a functor which assigns periodic Weyl algebra to each open subsets.
So if we consider this system in S^1, the factorization homology is the same as Hochschild homology of periodic Weyl algebra. 
To extract the θ-information from the Hochschild homology is a fascinating work.



§ ACKNOWLEDGMENT

We would like to thank Yuji Ando, Hayato Kanno, Keisuke Konosu, Hiroaki Matsunaga and Jojiro Totsuka-Yoshinaka for the discussions at the early stage of this work.
We would also like to thank Jun Maeda and Yuto Moriwaki for helpful discussions.

The work of MK is supported by Grant-in-Aid for JSPS Fellows No. 22KJ1989.






ptephy


































§ PROOF OF THEOREMS



 §.§ Proof of Theorem <ref>

[colframe=blue,colback=blue!3!]

    If M is compact and m^2>0, then
    
    H^n( Obs^ cl(M))
            =
            {[            ℂ        (n=0);            0 ( otherwise) ].




    
    A:=
            (C_ c^∞(M)^-1C_ c^∞(M)^0)

    This is an isomorphism. Thus H^*(A)=0, then H^0( Sym(A))=ℂ and H^n≥ 1( Sym(A))=0






 §.§ Proof of Theorem <ref>

From now, we will show 
[colframe=blue,colback=blue!3!]

    If M=ℝ and I⊂ M is an interval, then for any value of m
    
    H^n( Obs^ cl(I))
            =
            {[       ℂ[q,p]        (n=0);            0 ( otherwise) ].

    Here q,p has degree 0.


The proof of this is essentially same as Theorem <ref>.

    We show a quasi isomorphism:
    
    A∼ B

    where
    
    A=(C_ c^∞(I)^-1C_ c^∞(I)^0), 
            B=(0→ℝ^2).

    We denote the basis of ℝ^2 as q,p.
    The cohomology of Sym(A) and Sym(B) are H^*( Obs^ cl(I)) and ℝ[q,p] respectively.
    We will see the following commutative diagram.
    
    C_ c^∞(I)^-1[r] [d]_π_1    C_ c^∞(I)^0 [d]_π_2
    
        0 [r]    ℝ^2

    Later, we will define π_2.
    In order to simplify the notation, in case that m>0 we define ϕ_q,ϕ_p∈ C^∞(ℝ) as

    ϕ_q(x)=1/2(e^mx+e^-mx), ϕ_p(x)=1/2m(e^mx-e^-mx).

They form the kernel of -Δ+m^2.
If m=0, we define

    ϕ_q(x)=1, ϕ_p(x)=x.

We note that ϕ_p'(x)=ϕ_q(x).
   We take g∈ C_ c^∞(I)^0, 
   
    π_2(g)
           :=q∫_I  dx g(x)ϕ_q(x)+p∫_I  dx g(x)ϕ_p(x).

   π(g)=0 holds if g=Δ_ BV^ clf^⋆.




 §.§ Definition of Q and P

We define position observable Q∈ C_ c^∞(I) as

    ∫_I Q(x)ϕ_q(x)  dx=1
          and 
        Q(-x)=Q(x).

By the second condition, we obtain ∫_I Q(x)ϕ_p(x)  dx=0.
For Q_t(·):=Q(·-t)∈ C_ c^∞(I_t), and we define momentum observable as

    P_t:=-Q'_t∈ C_ c^∞(I_t)^0.

The reason why we call Q_t,P_t position and momentum is that if Q_t,P_t act on field Φ∈ C^∞(ℝ), we get

    Q_t(Φ)=∫_I_t Q_t(x)Φ(x)  dx, 
        P_t(Φ)=∫_I_t Q_t(x)Φ'(x)  dx.

If we narrow the width of the interval, Q get close to δ-distribution and Q_t(Φ) and P_t(Φ) approaches Φ(t) and Φ'(t).



[colframe=blue,colback=blue!3!]

   [Q_0] and [P_0] are the generators of H^0( Obs^ cl(ℝ))



    We will show the first one. We have already seen H^0( Obs^ cl(ℝ))=ℂ[q,p] in Theorem <ref>.
    Take
    
    Q_0+Δ_ BV^ clX, 
            P_0+Δ_ BV^ clY,

    where X,Y∈ Sym(C_ c^∞(ℝ))*C_ c^∞(ℝ))^-1.
    By π_2 action,
    
    π_2(Q_0+Δ_ BV^ clX)
            =
            π_2(Q_0)=q,
    π_2(P_0+Δ_ BV^ clY)
            =
            π_2(P_0)=p.

    We used ϕ'_p(x)=ϕ_q(x).































































































































 Paper Title: Long-time Asymptotics for the Ablowitz-Ladik system with present of solitons.



Long-time Focusing Ablowitz-Ladik system]Long-time Asymptotics for the Ablowitz-Ladik system with present of solitons




School of Mathematics and Statistics, Fujian Normal University, Fuzhou, PR China

chenms@fjnu.edu.cn




School of Mathematical Sciences,
	Fudan University, Shanghai, PR China
faneg@fudan.edu.cn



School of Mathematical Sciences,
	Fudan University, Shanghai, PR China
wang_zy@fudan.edu.cn




[2020]Primary 54C40, 14E20; Secondary 46E25, 20C20





§ ABSTRACT 
Abstract]In this article, we investigate some problems about the initial-value problem of the focusing Ablowitz-Ladik system, and the initial data belongs to a discrete weighted ℓ^2 space. 
	On the one hand, we have proved the global well-posedness for the initial-value problem.
	Utilizing the fact that the initial data belongs to the discrete weighted ℓ^2 space, we construct a Riemann-Hilbert problem for the initial-value problem.
	By Fredholm theory, the Riemann-Hilbert problem is uniquely solved with the jump contour consisting of three circles that all center around the origin. 
	On the other hand, based on the global well-posedness and the Riemann-Hilbert problem, we allow the solution to have the higher-order solitons, and then analyze the long-time asymptotics for the solution in different sectors on the upper-half plane.



KEYWORDS: Global wellposedness, Fredholm theory, long-time asymptotics, higher-order soliton


MSC 2020: 35P25, 35Q15, 35B40, 37K40






[
    Zhaoyu Wang
    August 5, 2024
==================







§ INTRODUCTION


The Ablowitz-Ladik systems read:

    / tq_n=q_n+1-2q_n+q_n-1+σ| q_n|^2(q_n+1+q_n-1),  σ=±1

which is the completely integrable discretization of nonlinear Schrödinger equations:

    u_t=u_xx+σu^2u,

and it was first introduced by Ablowtiz and Ladik in the 1970s <cit.>. 
The inverse scattering transform of (<ref>) has been well investegated <cit.>. 
By the connection between Ablowitz-Ladik systems and orthogonal polynomials on the unit circle, Nenciu <cit.> have constructed the Lax pair for the Ablowitz-Ladik systems.
Using methods of algebraic geometry, Miller et al <cit.> studied the finite genus solutions for Ablowtiz-Ladik systems.
The multi-Hamilton structure is constructed by Gekhtman and Nenciu for the Ablowitz-Ladik system in the defocusing cases<cit.>.
In this article, our research focus on the global well-posedness and the Fredholm theory of the initial-value problem of Ablowitz-Ladik system in the focusing case, and based on these, we study the long-time asymptotics. 
The focusing Ablowitz-Ladik system reads, that is, (<ref>) with σ=1:

    / tq_n=q_n+1-2q_n+q_n-1+| q_n|^2(q_n+1+q_n-1).

We constrain the initial potential in the weighted ℓ^2 space: ℓ^2,1.
In addition, we suppose no spectral singularity for the corresponding Riemann-Hilbert (RH) problem, and do not restrict the poles of the RH problem to be simple.
Under these assumption, by analyzing the Laurent's expansion of the solution for RH problem on the discrete spectrum, we equivalently transform it into RH problem <ref> with the jump contour consisting of three circles. 
It is interesting that RH problem <ref> is a pure RH problem without discrete spectrum.
Utilizing the Vanishing Lemma, we develop the Fredholm theory for RH problem <ref>, which ensure the solvability of this RH problem. 

It is well acknowledged that on the complex plane, a matrix-valued RH problem is not always solved<cit.>. 
In 1989, Zhou have developed a Vanishing Lemma for the n× n RH problem whose jump contour is symmetric about the real line ℝ by Fredholm theory<cit.>. 
In Zhou's work, the jump matrix admits skew-Hermitian outside ℝ: v(z)=v(z̅)^†, and he specified it to be positively definite along ℝ: v|_ℝ>0. The Vanishing Lemma ensures the Fredholm alternative to be zero and consequently the solvability of the matrix-valued RH problem.
For some discrete integrable systems, the discrete and continuous spectrum are symmetric about the unit circle, like Ablowitz-Ladik systems<cit.>, Toda system<cit.>, discrete sine-Gordon equation<cit.>. 
For the discrete integrable systems, if ones also develop the vanishing lemma, then it ensures the solvability of the corresponding RH problem, which is the Lemma <ref>'s results.

The long-time asymptotic behavior for the defocusing nonlinear Schrödinger equation, the equation (<ref>) with σ=-1, has been studied thoroughly<cit.>.
In 2018, Borghese et al  have investigated the long-time asymptotics for the focusing nonlinear Schrödinger equation<cit.>.
Utilizing the dbar steepest descent method, they indicate that the leading terms of the asymptotic formula are determined by the solitons and the RH problem at the stationary phase points, and the error is (t^-3/4).
The dbar steepest descent is first introduced when handling the asymptotic analysis for the orthogonal polynomials<cit.>.
Recently, Chen et al have constructed the inverse scattering theory on the discrete weighted l^2 space for the defocusing Ablowitz-Ladik system, i.e. (<ref>) with σ=-1<cit.>, and based on this, they investigate the long-time asymptotic behavior for the initial-value problem of the defocusing Ablowitz-Ladik system on the background of lower regularity<cit.>. 
The results show that according to the long-time asymptotic behavior, the upper half plane (n,t) is divided into three sectors, two of which are fast decaying regions with the rate (t^-1) while the other one is the oscillatory region where the long-time asymptotic behavior is agree with that for the defocusing nonlinear Schrödinger equation<cit.>. 
The results also consist with the work of Stewart in 2023 that shows the long time asymptotic behavior by linear estimates for some small-norm initial potential <cit.>.





This article is organized as follows. 
In Section <ref>, we present the global well-posedness of the focusing Ablowitz-Ladik system in the discrete weight ℓ^2 space: ℓ^2,1. 
In Section <ref>, we introduce the direct scattering and construct RH problem <ref> for the initial-problem.
When the initial potential belongs to ℓ^2,1, it is verified that the reflection coefficient belongs to H^1, and the discrete spectrum are invariant about the mapping: λ↦λ̅^-1. 
In Section <ref>, we investigate the solvability of RH problem <ref>. 
Transforming it into a new RH problem for M̃, we apply Fredholm theory on M̃, and then uniquely solve the RH problem <ref>. 
In Section <ref>, we study the long-time asymptotic in the region I: ξ∈(-1,1).
In this region, the leading term of the asymptotic formula is the solitonic part q_n^𝒵_ξ(t), and the second one with decaying rate (t^-1/2) is determined by the reflection coefficient around stationary phase points, and the remaining part is dominated by (t^-3/4).
The result is consisting with those for nonlinear Schrödinger equation <cit.>.
In Section <ref>, we study the region II: ξ<-1 and region III: ξ>1.
In these regions, the leading term is also the soliton, but the second term in the region I vanishes and the remaining part is dominated by (t^-1).

List some notation exploited in this paper below:

	
  * The right-shifting operator E: Ef(n)=f(n+1), the left-shifting operator E^-1: E^-1f(n)=f(n-1). 
	
  * ℓ^1: {{a_n}_n=-∞^∞:∑_-∞^∞| a_n|<∞}, where a_n_1=∑_-∞^∞| a_n|.
	
  * ℓ^k,1: {{a_n}_n=-∞^∞:∑_-∞^∞(1+n^2)| a_n|^k<∞}, k=1,2, where the norm reads: a_n_k,1=(∑_-∞^∞(1+n^2)| a_n|^k)^1/k.
	
  * H^1: {f(z)∈ L^2(Σ):∂_θ f(e^θ)∈ L^2([0,2π])}, where Σ={e^θ: θ∈[0.2π]} is the unit circle at the origin and f_H^1=f_L^2(Σ)+∂_θ f_L^2(Σ).
	
  * The Pauli matrices:
	
    σ_1=[
    			0   1
    1   0
    		],  σ_2[
    			0   -
       0
    		],  σ_3=[
    			1   0
    0   -1
    		].

	
  * The phase function on λ∈ℂ:
	
    ϕ(λ,n,t)=- t(λ+λ^-1-2)+nlnλ.

	
  * The discrete spectrum 𝒵∪𝒵^-1:
	
    𝒵={λ_1,…,λ_l}.

	And
	
    𝒵_ξ^+=𝒵∩{ϕ>0},  𝒵_ξ=𝒵∩{ϕ=0}.

	




 §.§ Main results


The initial-value problem of the focusing Ablowitz-Ladik system reads:

    / tq_n=q_n+1-2q_n+q_n-1+| q_n|^2(q_n+1+q_n-1),
        q_n(t=0)=q_n(0)∈ℓ^2,1,

where n∈ℤ is the spatial variable, t∈ℝ is the time variable. 
When we take the initial data shown in (<ref>), the solution is globally well-posed, and in RH problem <ref>, the reflection coefficient r(λ) belongs to H^1. 
r(λ)∈ H^1 means that it is 1/2-Hölder continuous on the unit circle. 
In the residue condition of RH problem <ref>, the order of the poles is set to be α_l' for any l'=1,…,l, and we remove the poles by some triangular rational functions, and equivalently transform RH problem <ref> into RH problem <ref> without poles. 
By the Fredholm theory and Vanishing Lemma, we unique solve RH problem <ref>.

Based on the solvability of RH problem <ref>, we construct the long-time asymptotic formulas, and according to it, we divide the upper-half plane (n,t) into three sectors: two soliton regions and one solitonic-oscillatory region.

	The initial-value problem (<ref>) is uniquely solved, and the solution satisfies the following asymptotic properties on the upper-half plane (n,t):
	
		
  * In region I: ξ∈(-1,1), 
		
    q_n(t)   =(∏_l'=1^lλ_l'^2α_l')e^∫_S_1^S_2ln(1+r(s)^2)/2π s s(q^𝒵_ξ_n(t)+t^-1/2q^Osc_n(t)+(t^-3/4)),

		where q^𝒵_ξ is the soliton, and q^Osc is the oscillatory part defined in (<ref>).
		
  * In regions II and III: ξ<-1 and ξ>1, 
		
    q_n(t)∼ q_n^𝒵_ξ(t)+(t^-1).

		
	
	




	In the above theorem, we have presented the long-time asymptotics for the initial-value problem in region I III, and the results for the transition regions is undergoing.


Our analysis exploits the Lax representation:

    EX   =AX,   X/ t=BX,
    
    	A   =z^σ_3+Q,   B=σ_3/((z-z^-1)^2/2-Q(E^-1Q)+z^σ_3Q-
    	(E^-1Q)z^σ_3),

where Q is a skew-symmetry matrix

    Q=Q(n)=[
    		0   q_n
    -q_n   0
    	].

That a smooth function solves the initial-value problem (<ref>) is equivalent to the compatibility condition:

    A/ t=(EB)A-AB,

which is proved to be the focusing Ablowitz-Ladik system. 
Introducing

    c_n(t)=∏_k=n^∞(1+q_k(t)^2),

it is readily seen that c_-∞=lim_n→-∞c_n(t) is a conserved quantity.
By direct computation,

    / t c_-∞   =/ t∏_n=-∞^∞(1+q_n(t)^2)=c_-∞∑_n=-∞^∞/ tq_n(t)q_n(t)+q_n(t)/ tq_n(t)/1+q_n(t)^2

and taking (<ref>) into (<ref>), we obtain that

    / t c_-∞≡ 0,

that is, c_-∞ is independent on t.




§ GLOBAL WELL-POSEDNESS

In this section, we consider the solution of the initial value problem (<ref>). 
We see that if the initial potential q(0) is in a given weighted ℓ^2 space and the conserved quantity c_-∞ is finite, then the solution is globally well-posed, which we shown in Proposition <ref>. 

	When q(0)∈ℓ^2,1 and c_-∞ is finite, the initial value problem (<ref>) admits a unique solution q(t)∈ℓ^2,1 for t∈ℝ, and the solution is Lipschitz continuous about t. 


	By the definition of c_-∞, notice the initial potential satisfies q(0)_∞<c_-∞.
	We discuss the solution of (<ref>) in the Banach space:
	
    B(q(0),c_-∞){a∈ℓ^∞:a-q(0)_∞<c_-∞}.

	Since the right-hand side of (<ref>) is Lipschitz continuous, by a standard argument about ODEs in a Banach space, there exist t_1>0 such that for t∈(-t_1,t_1), (<ref>) admits a unique solution, and t_1 is determined by the conserved quantity c_-∞.
	Moreover, noticing that q(-t_1/2)_∞,q(t_1/2)_∞≤ c_-∞, we can extended the solution to t∈(-t_1/2,3t_1/2)∪(-3t_1/2,t_1/2)=(-3t_1/2,3t_1/2) by investigating the solution of (<ref>) in the ball B(q(-t_1/2),c_-∞) and B(q(-t_1/2),c_-∞), respectively;
	repeating the procedure, we uniquely extended the solution q(t) to t∈ℝ, and q(t) belongs to B(0,c_-∞). 
	
	Since we have proved the solvability of (<ref>) and the uniqueness of solution in B(0,c_-∞), we claim that the solution q(t) does not blow up in ℓ^2,1 for t∈ℝ. 
	By (<ref>), 
	
    / tq_n(t)≲q_n(t)+q_n-1(t)+q_n+1(t),

	i.e.,
	
    / tq_n(t)^2≲q_n(t)^2+q_n-1(t)^2+q_n+1(t)^2.

	Summing up the both sides in the above inequality, we obtain that there exists positive constant C such that
	
    / tq(t)^2_1,1≤ Cq(t)_2,1^2.

	Integrate both sides of (<ref>) on the interval: [0,t] for t>0 (or [t,0] for t<0), it is readily seen that
	
    q(t)_2,1-q(0)_2,1≤ C∫_0^tq(τ)_2,1τ,

	therefore by Gronwell's inequality, q(t)_2,1≤q(0)_2,1e^Ct never blows up.
	Additionally, in the similar way, for any ϵ>0, we obtain that
	
    q(t±ϵ)-q(t)_2,1≲±∫_0^±ϵq(τ)_2,1τ,

	which means that the solution q(t) is Lipschitz continuous in ℓ^2,1 about t.
	This proves the results.





§ RIEMANN-HILBERT PROBLEM






 §.§ Jost solutions and modified Jost solutions


Denote  X^±(z,n,t) as the Jost solution for the Lax pair, and it satisfies:

    X^±(z,n,t)∼ z^nσ_3e^-/2(z-z^-1)^2tσ_3.

Notice that the Jost solution satisfies that

    X^+(z,n,t)≡ c_n(t)^-1,   X^-(z,n,t)≡ c_-∞c_n(t)^-1.

Recalling that c_-∞ is positive, by (<ref>), we see that X^±(z,n,t) are invertible. 
By uniqueness of the fundamental solution, there is a unique 2×2 matrix-valued function S(z) called scattering matrix such that

    X^-(z,n,t)=X^+(z,n,t)S(z).
 
We note that the matrix Q(n) is skew-Hermitian, therefore, by the Lax pair (<ref>), it follows that

    X^±(z,n,t)=σ_2X^±(z̅^-1,n,t)σ_2.

We naturally introduce the modified Jost solutions

    Y^±=Y^±(λ,n,t)=[
    		1   0
    0   z
    	]X^±(z,n,t)z^-nσ_3e^ t/2(z-z^-1)^2σ_3[
    		1   0
    0   z^-1],

λ=z^2, and it follows that Y^±∼ I as n→±∞.
By (<ref>) and (<ref>), we see that

	
    Y^±(λ,n+1,t)   =(I+Q(n,t))λ^σ_3/2Y^±(λ,n,t)λ^-σ_3/2,
    ∂_tY^±(λ,n,t)   =B̃(λ,n,t)Y^±(λ,n,t)+(λ+λ^-1-2)/2Y^±(λ,n,t)σ_3,


where

    B̃=σ_3/(λ+λ^-1-2/2I-Q(E^-1Q)+Q-(E^-1Q)λ^σ_3)

and it follows that Y^± are single-valued functions. 
By (<ref>), it is readily seen that

    λ^-n+1/2σ_3Y^±(λ,n+1,t)λ^n+1/2σ_3-λ^-n/2σ_3Y^±(λ,n,t)λ^n/2σ_3=λ^-(n/2+1)σ_3[
    			0   q_n(t)
    -q_n(t)   0
    		]Y^±(λ,n,t)λ^n/2σ_3,

and then taking summation on both sides of (<ref>), we rewrite Y^±(λ,n,t) in the form of summation 

	
    Y^+(λ,n,t)=I-∑_k=n^∞λ^n-k-2/2σ_3[
    			0   q_n(t)
    -q_n(t)   0
    		]Y^+(λ,k,t)λ^-n-k/2σ_3,
       Y^-(λ,n,t)=I+∑_k=-∞^n-1λ^n-k-2/2σ_3[
    			0   q_n(t)
    -q_n(t)   0
    		]Y^-(λ,k,t)λ^-n-k/2σ_3.


Recalling the assumption that q(0)∈ℓ^2,1∈ℓ^1, as shown in <cit.>, [Y^+_1,Y^-_2] is holomorphic on D_-={|λ|<1} and continuously extended to D_-∪Σ, Σ={|λ|=1}, while [Y^-_1,Y^+_2] is holomorphic on D_+={|λ|>1} and continuously extended to D_+∪Σ.

By (<ref>), taking the WKB expansion, it follows that the modified Jost solutions satisfy asymptotic properties at λ=0 and λ=∞:

	
    [Y^+_1,Y^-_2](λ,n,t)∼[
    			c_n^-1(t)+O(λ)   q_n-1(t)+O(λ)
    c_n^-1(t)q_n(t)λ+O(λ^2)   1+O(λ)
    		],  λ→0,
       [Y^-_1,Y^+_2](λ,n,t)∼[
    			1+O(1/λ)   -c_n^-1(t)q_n(t)1/λ+O(1/λ^2)
    -q_n-1(t)+O(1/λ)   c_n^-1(t)+O(1/λ)
    		],  λ→∞.





 §.§ Scattering coefficients


Introduce scattering coefficients:

    a(λ)=c_n(t)[Y^-_1,Y^+_2](λ,n,t),   b(λ)=e^ϕ(λ,n,t)c_n(t)[Y^+_1,Y^-_1](λ,n,t).

Factually,

    a(λ)=S_1,1(z),   b(λ)=zS_2,1(z),

which follows by (<ref>), (<ref>) and (<ref>).
Since Y^± is single-valued, we see that a(λ), b(λ) are also single-valued functions of λ. 
By the analyticity of Y^±, it follows that a(λ) is holomorphic on D_+ and continuously extended to D_+∪Σ;
in addition, b(λ) is continuous on Σ. 
By the symmetric property (<ref>) and the definition (<ref>), it is readily seen that

    ă(λ)a(λ̅^-1)=c_n(t)[Y^+_1,Y^-_2](λ,n,t),

and that ă(λ) is holomorphic on D_-. 
What is more, utilizing the asymptotic property (<ref>), we obtain that

    lim_λ→∞a(λ)=1,  ă(λ)=1.

The reflection coefficient is defined as

    r(λ)=b(λ)/a(λ),

and we claim that when q(0)∈ℓ^2,1, the corresponding reflection coefficient r belongs to H^1 as shown in Proposition <ref>. 

	Suppose that q(0)∈ℓ^2,1 and no spectral singularity, i.e., a admits no zero on Σ. 
	Let r be the corresponding reflection coefficient as defined in (<ref>).
	We assert that r∈ H^1. 

Before we prove Proposition <ref>, we introduce

    Ỹ=Ỹ^±(λ,n)=λ^-nσ_3/2Y^±(λ,n,0)λ^nσ_3/2,

and it is a little abuse of notation that we simply note q_n=q_n(0) in the proof.
Substituting (<ref>) into (<ref>), we obtain that

    a(λ)=c_n(0)[Ỹ^-_1,Ỹ^+_2](λ,n),   b(λ)=c_n(0)[Ỹ^+_1,Ỹ^-_1](λ,n).

and that Y^± satisfy the following Volterra summation equation:

    Ỹ^±=I+𝒯^±Ỹ^±,

where for any 2×2 matrix function Ỹ=Ỹ(n)

	
    𝒯^-Ỹ(λ,n)=∑_k=-∞^n-1[
    			0   q_k
    -q_k   0
    		]λ^(k+1)σ_3Ỹ(k),
       𝒯^+Ỹ(λ,n)=-∑_k=n^+∞[
    			0   q_k
    -q_k   0
    		]λ^(k+1)σ_3Ỹ(k).


Recalling that q∈ℓ^2,1, we obtain that the Volterra summation equation (<ref>) uniquely solved by

    Y^±=∑_k=0^+∞(𝒯^±)^kI.

In addition, by (<ref>), it is readily seen that for λ∈Σ, λ=λ̅^-1 and 𝒯^± admits the symmetry on the unite circle

    σ_2𝒯^±Ỹσ_2=𝒯^±(σ_2Ỹσ_2).


	Since a have no zeros on the jump contour Σ, if we prove that
	
    a,b∈ H^1,

	then, we obtain the results.
	Moreover, by (<ref>), to prove (<ref>), we only have to verify
	
    Y^±(·,0)∈ H^1.

	
	By direct computation, we obtain that for any m∈ℤ^+,
	
		
    (𝒯^-)^2m-1(λ,n)=(-1)^m∑_-∞<k_2l-1<⋯<k_1<n[
    				0
    ∏_Even jq_k_j∏_Odd jq_k_jλ^A_2m-1],
       (𝒯^-)^2m(λ,n)=(-1)^m∑_-∞<k_2m<⋯<k_1<n[∏_Odd jq_k_j∏_Even jq_k_jλ^1-A_2m
    0
    			],
       (𝒯^+)^2m-1(λ,n)=(-1)^m∑_n≤ k_1≤⋯≤ k_2m-1<+∞[
    				0
    ∏_Even jq_k_j∏_Odd jq_k_jλ^A_2m-1],
       (𝒯^+)^2m(λ,n)=(-1)^m∑_n≤ k_1≤⋯≤ k_2m<+∞[∏_Odd jq_k_j∏_Even jq_k_jλ^1-A_2m
    0
    			],

	
	where A_m=∑_j=1^m(-1)^j-1k_j+1.
	Denoting N(m,λ,n) as the non-zero entry of (𝒯^-)^k(λ,n), we obtain that for any g(λ)∈ L^(Σ)
	
    ∫_ΣN(m,λ,n)g(λ)λ/2π=∑_-∞<k_2m-1<⋯<k_1<n(∏_Even jq_k_j)(∏_Odd jq_k_j)ĝ(A_2m-1)
           ≤∑_-∞<k_2m-1<⋯<k_1<nq_k_1⋯ q_k_2m-1ĝ(A_2m-1)≤q_1^2m-2/(2m-2)!q_2ĝ_2≤(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!g_L^2(Σ),

	where ĝ=ĝ(n) are the Fourier coefficients of g, thus, 
	
    N(m,·,n)_L^2(Σ)≤(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!.

	We also have
	
    ∂_θ N=∑_j=1^2m-1(-1)^j-1N_j,
       N_j=N_j(m,λ,n)=∑_-∞<k_2m-1<⋯<k_1<n(∏_Even j'q_k_j')(∏_Odd j'q_k_j')k_jλ^A_2m-1.

	Using the same technique in (<ref>), we have
	
    ∫_ΣN_j(m,λ,n)g(λ)λ/2π   ≤∑_-∞<k_2m-1<⋯<k_1<n(∏_j' jq_k_j')(k_jq_k_j)ĝ(A_2m-1)
       ≤q_1^2m-1/(2m-2)!q_2,1ĝ_2≤(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!g_L^2(Σ).

	Considering (<ref>) and (<ref>), we obtain that
	
    ∂_θ N(m,·,n)_L^2(Σ)≤(2m-1)(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!.

	By (<ref>) and (<ref>), it follows that
	
    N(m,·,n)_H^1≤2m(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!,

	i.e.,
	
    (𝒯^-)^2m-1(·,n)_H^1≤2m(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!.

	Applying the technique for the estimate in (<ref>), we estimate the others in (<ref>) and obtain that
	
    (𝒯^±)^m(·,n)_H^1≤(m+1)(1+π^2/3)^m-1/2q^m_2,1/(m-1)!.

	Additionally, by (<ref>), it is readily seen that for any n∈ℤ^+,
	
    σ_2(𝒯^±)^mIσ_2=𝒯^±(σ_2(𝒯^±)^m-1Iσ_2)=⋯=σ_2(𝒯^±)^mIσ_2,

	thus, 
	
    (𝒯^±)^m(·,n)=[
    			0   1
    -1   0
    		](𝒯^±)^m(·,n)∈ H^1,

	and
	
    𝒯^±)^m(·,n)_H^1≤(m+1)(1+π^2/3)^m-1/2q^m_2,1/(m-1)!.

	
	By (<ref>), (<ref>) and (<ref>), applying the Dominated Convergence Theorem and taking n=0, we obtain that (<ref>). 
	This proves the results.





 §.§ Discrete spectrum


Under the assumption of no spectral singularity, the holomorphic function a(λ) admits at most  finite zeros the discrete spectrum 𝒵∈ D_+, and denote the zero's order at λ_l' as α_l'>0. 
By the symmetric relation (<ref>), it is clear that ă(λ) admits an α_l'-order zero at each λ=λ̅_l'^-1, l'=1,…,l. 

Since λ_l' is the α_l'-order pole of a(λ), by the first formula in (<ref>), it turns out that there exist constants: β_λ_l',0,β_λ_l',1,…,β_λ_l',α-1, such that for any α=0,…,α_l'-1,

    ∂_λ^αY^-_1(λ_l')=∑_α'=0^α(α
    α'
    	)∂_λ^α-α'Y^+_2(λ_l')(∑_j=0^α'(α'
    j
    	)β_λ_l',j∂_λ^α'-j(e^-ϕ(λ_l'))),

where  (α
α'
) denotes the binomial coefficient and β_λ_l',00. 
Similarly utilizing (<ref>), for λ̅_l'^-1∈ D_-, there exist constants β_λ̅_l'^-1,0,…,β_λ̅_l'^-1,α_l'-1, such that for any fixed α belonging to 0,…,α_l'-1, 

    ∂_λ^αY^-_2(λ̅_l'^-1)=∑_α'=0^α(α
    α'
    	)∂_λ^α-α'Y^+_1(λ̅_l'^-1)(∑_j=0^α'(α'
    α”)β_λ̅_l'^-1,j∂_λ^α'-j(e^ϕ(λ̅_l'^-1))),

especially, β_λ̅_l'^-1,0=-β_λ_l',00. 





 §.§ The reconstruction formula

In this part, we construct the Riemann-Hilbert problem and the reconstruction formula. 
Set the 2×2 matrix-valued function

    M=M(λ,n,t)=(P[Y^+_1,Y^-_2/ă])(λ,n,t),   λ∈ D_-,
    (P[Y^-_1/a,Y^+_2])(λ,n,t),   λ∈ D_+,

where P=P(n,t)=[
	1   0
c_n(t)q_n-1(t)   c_n(t)
] and ă=ă(λ)=a(λ̅^-1).
We claim that M is the unique solution of RH problem <ref>. 
By the analyticity of Y^± and (<ref>), we obtain the analyticity of M.
By the asymptotic behavior (<ref>) of Y^± at the infinity, we derive the normalization of M at λ→∞.
Now, we focus on the residue condition.
Recalling that λ=λ_l' is an α_l'-order zero of a(λ), it is naturally an α_l'-order pole of M(λ,n,t).
If denote λ=λ'_-αM as the coefficient of (λ-λ')^-α in the Laurent's expansion of M at λ=λ', then by the argument of complex analysis, we obtain the formula to compute the Laurent's coefficients:

    λ=λ_l'_-αM=1/(α_l'-α)!∂_λ^α_l'-α[M(λ)(λ-λ_l')^α_l']|_λ=λ_l'.

Considering (<ref>), (<ref>) and (<ref>), we obtain the residue condition (<ref>). In addition, the residue condition (<ref>) is verified similarly. 
The jump condition is a consequence of (<ref>), (<ref>) and (<ref>). 

	Find a 2×2 matrix-valued function M such that
	
		
  * Analyticity: M is holomorphic on ℂ∖Σ.
		
  * Normalization: As λ→∞, M∼ I+(λ^-1).
		
  * Residue condition: On the α_l'-order pole λ=λ_l'∈𝒵, we have
		
			
    λ=λ_l'_-αM=∑_j=0^α_l'-α∂_λ^α_l'-α-jM(λ_l')/(α_l'-α-j)![
    					0   0
    p_λ_l',je^ϕ(λ_l')   0
    				],
 
			where α=0,…,α_l'-1 and p_λ_l',j is a polynomial of (n,t) of order at most j
			
    p_λ_l',j
    				=∑_j'=0^j∑_j”=0^j'∂_λ^j”((·-λ_l')^α_l'/a)|_λ=λ_l'β_λ_l',j'-j”∂_λ^j-j'e^ϕ(λ_l')/(j-j')!(j'-j”)!j”!e^ϕ(λ_l').

			And λ=λ̅_l'^-1 is also a α_l'-order pole of M on D_-, and
			
    λ=λ̅_l'^-1_-αM=∑_j=0^α_l'-α∂_λ^α_l'-α-jM(λ̅_l'^-1)/(α_l'-α-j)![
    					0   p_λ̅_l'^-1,je^-ϕ(λ̅_l'^-1)
    0   0
    				],
 
			
		
		where
		
    p_λ̅_l'^-1,j=∑_j'=0^j∑_j”=0^j'∂_λ^j”((·-λ̅_l'^-1)^α_l'/ă)|_λ=λ̅_l'^-1β_λ̅_l'^-1,j'-j”∂_λ^j-j'e^-ϕ(λ̅_l'^-1)/(j-j')!(j'-j”)!j”!e^-ϕ(λ̅_l'^-1)

		
  * Jump condition: If we set the boundary value of M=M(λ,n,t) on λ∈Σ as M_±=M_±(λ,n,t), then
		
    M_+=M_-V,   V=V(λ,n,t)=[
    				1+| r(λ)|^2   r(λ)e^-ϕ(λ,n,t)
    r(λ)e^ϕ(λ,n,t)   1
    			].

	


By the asymptotic property (<ref>) of Y^± at λ→0 and the definition  (<ref>), we find that the solution of (<ref>) is recovered by the solution of RH problem <ref>, i.e. the reconstruction formula,

    q_n(t)=[M(0,n+1,t)]_1,2.








§ FREDHOLM ALTERNATIVE, PURE RH PROBLEM AND THE SOLVABILITY


In this section, utilizing some triangular matrix function [
	1   0
fe^ϕ   1
] and [
	1   f̆e^-ϕ
0   1
], we remove the poles in the original RH problem and transform it into a new RH problem without singularity.
In the new RH problem, we find that the jump matrix is self-adjoint, and as a consequence, using the Fredholm theorem, we prove that the RH problem is uniquely solved. 



 §.§ Remove the poles

In this section, we construct lower/upper triangular functions [
	1   0
fe^ϕ   1
], [
	1   f̆e^-ϕ
0   1
], such that f, f̆ are rational function and M[
	1   0
fe^ϕ   1
], M[
	1   f̆e^-ϕ
0   1
] are analytic on D̃_+, D̃_- respectively.
The regions are as shown in FIGURE <ref>.

Before proving the results for the region D̃_+ and D̃_- respectively, we consider the pole at each λ_l' or λ̅_l'^-1 belonging to the discrete spectrum as the following proposition.


	There exist polynomials f_λ_l'(λ), f_λ̅_l'^-1(λ) of (λ-λ_l)^-1, (λ-λ̅_l^-1)^-1 with the order not more than α_l', respectively, such that for l'=1,…,l and α=1,…,α_l',
	
    λ=λ_l'_-α(M[
    			1   0
    f_λ_l'e^ϕ   1
    		])≡0,   λ=λ̅_l'^-1_-α(M[
    			1   f_λ̅_l'^-1e^-ϕ
    0   1
    		])≡0.



	Set 
	
    f_λ_l'(λ)=∑_s=1^α_l'f_λ_l',s(λ-λ_l')^-s.

	Rewrite the first equality in (<ref>), 
	
    0≡λ=λ_l'_-α(M[
    			1   0
    f_λ_l'e^ϕ   1
    		])=λ=λ_l'_-αM+λ=λ_l'_-α(M[
    			0   0
    f_λ_l'e^ϕ   0
    		]).

	Recalling that the second column of M is holomorphic on λ∈ D_+, we rewrite the second term on the right-hand side of (<ref>):
	
    λ=λ_l'_-α(M[
    			0   0
    f_λ_l'e^ϕ   0
    		])=∑_j=0^α_l'-α∂_λ^α_l'-α-jM/(α_l'-α-j)![
    			0   0
    ∑_j'=0^jf_λ_l',α_l'-j'∂_λ^j-j'e^ϕ/(j-j')!   0
    		]|_λ=λ_l'.
 
	Substituting (<ref>) and (<ref>) into (<ref>), we obtain that
	
    f_λ_l',s=∑_s'=0^α_l'-s∂_λ^s'((λ-λ_l')^α_l'/a(λ))|_λ=λ_l'β_λ_l',α_l'-s-s'/(α_l'-s-s')!s'!

	solve (<ref>), i.e.
	
    f_λ_l'(λ)=∑_s=1^α_l'∑_s'=0^α_l'-s∂_λ^s'((λ-λ_l')^α_l'/a(λ))|_λ=λ_l'β_λ_l',α_l'-s-s'(λ-λ_l')^-s/(α_l'-s-s')!s'!.

	For the existence of f_λ̅_l'^-1, the proof is parallel.
	We complete the proof. 

As shown in Proposition <ref>, we have proved that the pole of M can be removed by the rational triangular function at each point of the discrete spectrum, which means in RH problem <ref>, the residue condition can be transformed into the jump.
Based on this, we want to do more, and find a rational triangular function on D_+ or D_- to remove the poles at the discrete spectrum uniformly, which is helpful when we prove the unique solvability of the newly obtained RH problem. 
By the theory of generalized Vandermonde matrix, we obtain the rational function on D_+ and D_- as the proposition below.


	There exists a rational function f(λ), f̆(λ) such that M[
		1   0
fe^ϕ   1
	], M[
		1   f̆e^-ϕ
0   1
	] are holomorphic on D_+, D_-∖{0}, respectively. 
	


	Notice that when we take l=1, it is Proposition <ref>; thus, Proposition <ref> is a generalized version of Proposition <ref>. To prove the results, we introduce the generalized Vandermonde matrix <cit.>
	
    V(λ_1,…,λ_l,α_1,…,α_l)=[
    			1   λ_1   ⋯   λ_1^α_1   ⋯   λ_1^τ
    
    			0   1   ⋯   α_1λ_1^α_1-1   ⋯   τλ_1^τ-1
    ⋯   ⋱   ⋱   ⋯   ⋯   ⋯
    
    			0   ⋯   0   1   ⋯   τ!/α_1!λ_1^τ-α_1
    ⋯   ⋯   ⋯   ⋯   ⋯   ⋯
    
    			1   λ_l   ⋯   λ_l^α_l   ⋯   λ_l^τ
    
    			0   1   ⋯   α_lλ_l^α_l-1   ⋯   τλ_l^τ-1
    ⋯   ⋱   ⋱   ⋯   ⋯   ⋯
    
    			0   ⋯   0   1   ⋯   τ!/α_l!λ_l^τ-α_l],

	where τ=∑_l'=1^lα_l'. The generalized Vandermonde matrix is invertible and its determinant is
	
    V(λ_1,…,λ_l,α_1,…,α_l)=∏_j> l(λ_j-λ_k)^α_jα_k.

	
	To construct f(λ), it is sufficient to construct a polynomial g(λ) of λ such that
	
    f(λ)=g(λ)∏_l'=1^lf_λ_l'(λ),   g(λ)=∑_s=0^τ-1g_sλ^s,

	and for any l'=1,…,l, α=1,…,α_l',
	
    λ=λ_l'_-αf(λ_l')=f_λ_l',α.

	Seeing (<ref>), since f is consisting of f_λ_l' and the holomorphic function g, we learn that f admits an α_l'-order pole at λ=λ_l'.
	Calculate the Laurent's coefficients of f: 
	
    λ=λ_l'_-αf(λ_l')   =1/(α_l'-α)!∂_λ^α_l'-α((·-λ_l')^-α_l'f_λ_l'g∏_l” l'f_λ_l”)(λ_l')
       =∑_s=0^α_l'-α1/s!f_λ_l',α+s∂_λ^s(g∏_l” l'f_λ_l”)(λ_l').

	By comparing (<ref>) and (<ref>), it is sufficient to solve the following linear system with τ variables g_0,…,g_τ-1:
	
    ∂_λ^s(g∏_l” l'f_λ_l”)(λ_l')=δ_0,s, 
    		  δ_0,s=
    			1   s=0
    
    			0   s0
    		,   s=0,…,α_l'-1.

	Equivalently, we rewrite (<ref>) in the form of matrix:
	
    blkdiag({J_l'(λ_l')}_l'=1^l) V(λ_1,…,λ_l,α_1,…,α_l)(g_0,…,g_τ-1)^t=(1,α_1-10,…,0,…,1,α_l-10,…,0)^t

	where blkdiag(·) denotes the block diagonal matrix, and J_l' is the α_l'×α_l' matrix function:
	
    J_l'=[∏_l” l'f_λ_l”   0   ⋯   0
    ∂_λ∏_l” l'f_λ_l”   ∏_l” l'f_λ_l”   ⋱   ⋮
    ⋮   ⋱   ⋱   0
    ∂_λ^α_l'∏_l” l'f_λ_l”/α_l'!   ⋯   ∂_λ∏_l” l'f_λ_l”   ∏_l” l'f_λ_l”].

	We default f_λ_l”(λ_l')0 for all l” l', or let f_λ_l” plus a constant to ensure its non-zero property at λ_l';
	therefore, the block diagonal matrix in (<ref>) is invertible;
	in addition, by the fact that the generalized Vandermonde matrix is invertible, it is readily seen that the linear system (<ref>) is uniquely solved.
	We have confirmed the existence of f. 
	
	For the existence of f̆, since M[
		1   0
fe^ϕ   1
	](λ) is holomorphic on λ∈ D_+, it follows that σ_2 M[
			1   0
fe^ϕ   1
		](λ̅^-1)σ_2 is holomorphic on λ∈ D_-.
	By (<ref>) and the symmetry shown in (<ref>), it is readily seen that
	
    σ_2 M[
    					1   0
    fe^ϕ   1
    				](λ̅^-1)σ_2=   σ_2Pσ_2^2e^-ϕσ_3/2σ_2^2[Y^-_1,Y^+_2]σ_2^2diag(a,1)σ_2^2e^ϕσ_3/2σ_2^2[
    					1   0
    fe^ϕ   1
    				]σ_2(λ̅^-1)
    
    			=   σ_2Pσ_2(e^-ϕ/2σ_3[Y^+_1,Y^-_2]diag(1,ă)e^ϕ/2σ_3)(λ)[
    				1   -f(λ̅^-1)e^-ϕ(λ)
    0   1
    			]
    
    			=   (σ_2P̅σ_2 P^-1)M(λ)[
    				1   -f(λ̅^-1)e^-ϕ(λ)
    0   1
    			];

	therefore, by the fact that the matrix function σ_2P̅σ_2 P^-1 is invertible, it follows that when we take f̆(λ)=-f(λ̅^-1), M[
		1   f̆e^-ϕ
0   1
	] is holomorphic on D_-. 
	We complete the proof.




 §.§ The solvability

We have constructed the rational triangular matrix functions in Proposition <ref> to remove poles of M, however since ϕ admits an essential singularity at λ→∞, the newly obtained matrix function is not normalized at the infinity, i.e., M[
	1   0
fe^ϕ   1
] diverges at λ→∞. 
Moreover, because the essential singularity of ϕ at λ=0, M[
	1   f̆e^-ϕ
0   1
] also admits an essential singularity at λ=0. 
To avoid these essential singularities, we choose a circle Σ̃_1 such that its radius is greater than 1 and the region D̃_+ between Σ̃_1 and Σ contains all the discrete spectra in D_+.
Writing Σ̃_2=(Σ̃_1)^-1 and D̃_-=(D̃_+)^-1, by the symmetry of the discrete spectrum, we claim that D̃_- contains all the discrete spectra belonging to D_-.
Taking the transformation:

    M̃=M
    		I    λ∈(D_+∪ D_-)∖(D̃_+∪D̃_-),
    [
    			1   0
    fe^ϕ   1
    		]   λ∈D̃_+,
    [
    			1   f̆e^-ϕ
    0   1
    		]   λ∈D̃_-,

we obtain a matrix-valued function M̃ that is the solution of RH problem <ref>.
In RH problem <ref>, it is easily to find that the jump matrix admits an upper-lower triangular factorization:

    V   =(I-w_-)^-1(I+w_+)
    
    	w_-   =0    on Σ̃_1,
    [
    			0   -f̆e^-ϕ
    0   0
    		]    on Σ̃_2,
    [
    			0   -(r+f)e^-ϕ
    0   0
    		]    on Σ,
    	,
    
    	w_+   =[
    			0   0
    fe^ϕ   0
    		]    on Σ̃_1,
    0    on Σ̃_2,
    [
    			0   0
    (r+f)e^ϕ   0
    		]    on Σ.



	Find a 2×2 matrix-valued function M̃ such that
	
		
  * M̃ is holomorphic on ℂ∖Σ̃, where Σ̃=Σ∪Σ̃_1∪Σ̃_2.
		
  * As λ→∞, M̃∼ I+(λ^-1).
		
  * On the jump contour Σ̃,
		
    M̃_+=M̃_-Ṽ,  Ṽ=[
    					1   0
    fe^ϕ   1
    				]   λ∈Σ̃_1,
    [
    					1   -f̆e^-ϕ
    0   1
    				]   λ∈Σ̃_2,
    [
    					1+f+r^2   (r+f)e^-ϕ
    (f+r)e^ϕ   1
    				]   λ∈Σ.

		
		
	


Introduce the Cauchy-type integral operator:

    C^Γ f(λ)=∫_Γf(ζ)/ζ-λζ/2π,

where Γ consists of finite oriented Jordan curves with finite intersection points.  


	

Denote C_±^Γf as the boundary values of C^Γf. 
It is well known that C_±^Γ are bounded operators on L^2(Γ).
Taking Γ=Σ̃=Σ∪Σ̃_1∪Σ̃_2 and a function f on the jump contour Σ̃,
we define the bounded operator:

    T^Σ̃f=C_-^Σ̃(C_+^Σ̃(fw)w_+)+C_+^Σ̃(C_-^Σ̃(fw)w_-),

where w=w_++w_-. By rationally approximating w_+, w_- in the 1st, 2nd term on the right-hand side of (<ref>), respectively, we obtain that both C_-^Σ̃(C_+^Σ̃(· w)w_+) and C_+^Σ̃(C_-^Σ̃(· w)w_-) are compact operators on L^2(Σ̃). 
Exactly, we simply rewrite (<ref>) as

    T^Σ̃=(C_w)^2,
 
where

    C_wf=C_+^Σ̃(fw_-)+C_-^Σ̃(fw_+),

and then by the compactness of T^Σ̃, -T^Σ̃=(+C_w)(-C_w) is Fredholm. 
In addition, referring to Corollary <ref> that the equation C_wμ=μ admits no non-trivial solution, the Fredholm alternative is zero, and therefore, we obtain that both -T^Σ̃ and -C_w are invertible with the relation

    (-C_w)^-1=(-T^Σ̃)^-1(+C_w).

Thus, RH problem <ref> is uniquely solvable and the solution of RH problem <ref> is

    M̃(λ)=I+∫_Σ̃[(-C_w)^-1Iw](ζ)/ζ-λζ/2π,

which is also well known as Beals-Coifman solution.





	Setting Γ⊂ℂ as a jump contour consisting of finite Jordan contours and the unit circle with at most finite intersections, we assume that the jump contour is invariant under the mapping λ↦λ̅^-1.  
	Suppose a 2×2 matrix-valued function V^Γ=V^Γ(λ) satisfies
	
		
  * V^Γ admits the symmetry on Γ∖Σ:
		
    V^Γ(λ̅^-1)=(V^Γ(λ))^†;

		
  * on λ∈Σ, 
		
    V^Γ(λ)>0;

		
  * V^Γ admits the factorization:
		
    V^Γ=(I-w^Γ_-)^-1(I+w^Γ_+),

		where w^Γ_± are some continuous 2× 2 matrix-valued functions on Γ.
	
	Define the Cauchy-type integral operator for f∈ L^2(Γ):
	
    C_w^Γf=C^Γ_+(fw^Γ_-)+C^Γ_-(fw^Γ_+),

	where
	
    w^Γ=w^Γ_++w^Γ_-,

	then, we claim that there is no non-trivial solution μ∈ L^2(Γ) for the equation:
	
    C_w^Γμ=μ.

	



	Suppose μ is a non-trivial solution. 
	It is readily seen that C^Γ(μ w^Γ)(λ) is analytic outside Γ and satisfies
	
		
  * on λ∈Γ, 
		
    C^Γ_+(μ w^Γ)=C^Γ_-(μ w^Γ)V^Γ,

		
  * as λ→∞, 
		
    C^Γ(μ w^Γ)(λ)∼(λ^-1).

	
	The holomorphism is the direct result of μ, w^Γ∈ L^2(Γ). 
	For (<ref>), it is well-known that when Γ consists of finite Jordan curves,  C^Γ_± are bounded on L^2(Γ) and 
	
    C_+-C_-=,

	therefore, 
	
		
    C^Γ_+(μ w^Γ)=C_w^Γμ+(C^Γ_+-C^Γ_-)(μ w^Γ_+)=μ(I+w^Γ_+),
       C^Γ_-(μ w^Γ)=C_w^Γμ+(C^Γ_–C^Γ_+)(μ w^Γ_-)=μ(I-w^Γ_-),

	
	i.e. we obtain (<ref>). 
	(<ref>) is the direct consequence of the compactness of Γ. 
	
	Now, we investigate the analytic function on ℂ∖Γ:
	
    Φ=C^Γ(μ w^Γ).

	By the above argument, it follows that Ψ=Ψ(λ)=Φ(λ̅^-1)^† is holomorhpic on ℂ∖Γ and satisfies that
	
		
  * on λ∈Γ, 
		
    Ψ_+(λ)=(V^Γ(λ̅^-1)^†)^-1Ψ_-(λ),

		
  * as λ→∞,
		
    Ψ(λ)∼(1),

	
	thus by (<ref>), (<ref>) and (<ref>), it follows that ΦΨ admits no jump over Γ∖Σ, that is, on λ∈Γ∖Σ,
	
    Φ_+(λ)Ψ_+(λ)=Φ_-(λ)(V(λ̅^-1)^†)^-1Ψ_+(λ)=Φ_-(λ)Ψ_-(λ).

	On Σ, Recalling the asymptotic property (<ref>) of Φ, we have that at λ=0,
	
    Ψ(λ)∼(λ),

	therefore, Φ(λ)Ψ(λ)λ^-1 is holomorphic on the unit disc {λ<1}, and by Cauchy's Integral Theorem, the integral
	
    0   =∫_ΣΦ_-(λ)Ψ_-(λ)λ^-1λ/2π=∫_Σ[Φ_-(V^Γ)^†Φ_-^†](λ)λ^-1λ/2π=∫_0^2π[Φ_-(V^Γ)^†Φ_-^†](e^θ)θ/2π.

	However, recalling that V^Γ|_Σ>0, we obtain that the right-hand side of (<ref>) vanishes if and only if Φ(λ) tends to zero almost everywhere on the boundary of {λ<1}, which by (<ref>) is equivalent to that μ is vanishing on the support of w^Γ.
	This controverts to the assumption.
	We have proved the results.



	Set the Cauchy-type integral operator C_w as defined in (<ref>).
	There is no non-trivial solution in L^2(Σ̃) for the equation below
	
    C_wμ=μ.

	



	(<ref>) shows that Ṽ admits a proper factorization.
	And since r∈ H^1 and f is a rational function, by (<ref>) and (<ref>), w_± are 1/2-Hölder continuous on Σ̃.
	By the choice of f and f̆ in Proposition <ref>, Ṽ satisfies that on Σ̃_1∪Σ̃_2,
	
    Ṽ(λ̅^-1)=Ṽ(λ)^†.

	Recalling that Ṽ is self-adjoint on Σ, it follows that
	
    Ṽ|_Σ>0.

	Thus, the results is the consequence of Lemma <ref>.
	We have proved the results.



	We have proved the solvability of RH problem <ref>.
	And by (<ref>) and (<ref>), the solution of the initial-value problem is written by M̃ in the form,
	
    q_n(t)=[M̃(0,n+1,t)]_1,2.

	Especially, under the reflectionless condition, i.e., r≡0, the RH problem is uniquely solved, which is equivalent to the solvability of RH problem <ref> with r≡0, and we denote the solution with the discrete spectral data (𝒵,) as 
	M^(𝒵,). 
	And the soliton solution is
	
    q_n(t|𝒵,)=[M^(𝒵,)(0,n+1,t)]_1,2.

	
	It is readily seen that M^(𝒵,) is unique. 
	In fact, we assume that there is another solution M^(𝒵,)' solves RH problem <ref> for r≡0, then we have that by Proposition <ref>, M^(𝒵,)(M^(𝒵,)')^-1 is entire, and by Liouville's Theorem, M^(𝒵,)(M^(𝒵,)')^-1≡ I on the whole plane, i.e., 
    M^(𝒵,)≡ M^(𝒵,)'.

	



	
	

	
	
	
		
			
			
			
		
	
	
		
		
	
	
		
		
	
	
		
		
	
	
		
		
	
	
		
		
	
	
	




§ LONG-TIME ASYMPTOTICS FOR Ξ<1

Previously, we have presented the direct scattering transform and constructed the RH problem for the initial-value problem that is unique solvable. 
When ξ=n/2t<1, there are two stationary phase points on the jump contour

    S_1=-ξ+√(1-ξ^2),   S_2=-ξ-√(1-ξ^2),

and the Taylor's expansions of ϕ at S_1 and S_2 are

    ϕ(λ,n,t)-ϕ(S_j,n,t)∼   (-1)^j t√(1-ξ^2)S_j^-2(λ-S_j)^2+(λ-ξ^3)
    	,  λ→ S_j.

In this section, we take a series of RH transformations for M. 
Under the condition ξ=n/2t<C_0 for some positive constant C_0<1, it asserts that the asymptotic behavior of q_n(t) is solitonic and oscillatory, that is, as t→+∞.
	
    q_n(t)   =(∏_l'=1^lλ_l'^2α_l')e^∫_S_1^S_2ln(1+r(s)^2)/2π s s(q^𝒵_ξ_n(t)+t^-1/2q^Osc_n(t)+(t^-3/4)),

where q^𝒵_ξ is the solitonic part

    q^𝒵_ξ_n(t)=[M^(𝒵_ξ)]_1,2(0,n+1,t),

and q^Osc is the oscillatory part

    q^Osc_n-1(t)=   -1/√(2)(1-ξ^2)^1/4(T_1^2M^(𝒵_ξ)_1,2(S_1,n,t)γ_2(r(S_1))
       +T_1^-2M^(𝒵_ξ)_1,1(S_1,n,t)γ_1(r(S_1))+T_2^2M^(𝒵_ξ)_1,2(S_2,n,t)γ_1(r(S_2))
    		+T_2^-2M^(𝒵_ξ)_1,1(S_2,n,t)γ_2(r(S_1)))

In addition, M^(𝒵_ξ) is defined as (<ref>), and for j=1,2

    (γ_j(r(S_1)))=(-1)^j((r(S_1))-π/4-(Γ(iν(r(S_1))))),
       (γ_j(r(S_2)))=(-1)^j-1((r(S_2))+π/4+(Γ(iν(r(S_2))))).




At the beginning, we introduce the scalar function that admits a jump on an upper-half arc S_1S_2 of the unit circle Σ,

    T(λ)=T(λ,ξ)=e^∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λζ/2π∏_λ_l∈𝒵_ξ^+(λ-λ_l/λ-λ̅_l^-1)^α_l,  ξ∈(-1,1),

where 𝒵_ξ^+=𝒵∩{Reϕ>0} and the integral is on the arc from S_2 to S_1.
We assert that T(λ) admits properties shown in the following proposition. 

	The function defined by (<ref>) satisfies the following properties:
	
		
  * T(λ) is a meromorphic function on λ∈ℂ∖S_1S_2, and admits α_j-order pole at λ=λ_j.
		
  * For λ∈ℂ∖S_1S_2, T(λ) admits the symmetry
		
    T(λ̅^-1)=T(0)T^-1(λ)

		
  * For λ∈S_1S_2, the boundary value T_±(λ) of T(λ) satisfy
		
    T_+(λ)=T_-(λ)(1+r(λ)^2).

		
  * As λ→∞,
		
    T(λ)∼ 1+(λ^-1).

		
  * On the neighborhood of S_j, T(λ) admits the following asymptotic property,
		
    T(λ)∼∏_λ_l∈𝒵_ξ^+(S_j-λ_l/S_j-λ̅_l^-1)^α_l(λ-S_2/λ-S_1)^ν_je^α_j(S_j)+(λ-S_j^1/2),

		where 
		
    ν_j=ln(1+r(S_j)^2)/2π,  α(λ)=∫_S_2^S_1ln(1+r(s)^2)-ln(1+r(S_j)^2)/s-λ s/2π.

	


	(a) is directly verified by (<ref>). 
	For (b), we have
	
    T(λ)T(λ̅^-1)   =exp(∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λζ/2π+∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λ̅^-1ζ/2π)∏_λ∈𝒵_ξ^+(λ-λ_j/λ-λ̅_j^-1·λ^-1-λ_j^-1/λ^-1-λ̅_j)^α_j
       =exp(∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λζ/2π+∫_S_1^S_2ln(1+r(ζ)^2)/(ζ̅^-1-λ̅^-1)ζ̅^2ζ̅/2π)∏_λ∈𝒵_ξ^+λ_j^-2α_j
       =exp(∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λ(1-λ/ζ)ζ/2π)∏_λ∈𝒵_ξ^+λ_j^-2α_j
       =exp(∫_S_1^S_2ln(1+r(ζ)^2)ζ^-1ζ/2π)∏_λ∈𝒵_ξ^+λ_j^-2α_j=T(0).

	For (c), by Sokhotski-Plemelj formula, we have that for any λ∈S_1S_2,
	
    ∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λ_+ζ/2π-∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λ_-ζ/2π=ln(1+r(λ)^2),

	where λ_± means the ± limits of λ∈S_1S_2. 
	Since r∈ H^1, (d) is the direct result of the definition (<ref>).
	
	The remaining is (e). We refer to a conformal mapping (<ref>) that transforms the unit circle to the straight line, and then discuss it on the real line ℝ. We take
	
    δ(λ)=δ(λ,ξ)=T(λ)∏_λ_l∈𝒵_ξ^+(λ-λ_l/λ-λ̅_l^-1)^-α_l=e^∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λζ/2π.

	Changing the variable on Σ: λ→λ', s→ s',
	
    λ=√(S_1S_2)√(S_1)+√(S_2)λ'/√(S_2)+√(S_1)λ',   s=√(S_1S_2)√(S_1)+√(S_2)s'/√(S_2)+√(S_1)s',

	we obtain that
	
    ∫_S_2^S_1f(s)-f(S_1)/(2π)(s-λ)ds=√(S_2)+√(S_1)λ'/2π∫_-∞^0f(s)-f(S_1) s'/(√(S_2)+√(S_1)s')(s'-λ'),

	where f(s)=ln(1+| r(s)|^2).
	Since r(s)∈ H^1⊂ L^∞(Σ), it naturally follows that
	
    F(s')=f(s)-f(S_1)/√(S_2)+√(S_1)s',   s'≤0,
    
    			0,   s'>0,

	belongs to H^1(ℝ) and F(s'=0)=0;
	therefore, we apply Lemma 23.3 in <cit.> and get that for any λ∈ℂ∖ℝ
	
    |∫_-∞^+∞F(s')/(2π)(s'-λ')ds'-∫_-∞^+∞F(s')/2π s'ds'|≲∥ F∥_H^1|λ'|^1/2,

	i.e.
	
    δ(λ)∼(λ-S_2/λ-S_1)^ν_je^α_j(S_j)+(λ-S_j^1/2),

	which is an equivalence of (<ref>).
	These complete the proof. 



 §.§ Transform to M^(1)

By the property of functions M and T, as usual, we take the RH transformation:

    M^(1)=MT^σ_3.

In the new RH problem below, the jump matrix admits a proper factorization and the exponential functions e^ϕ, e^-ϕ in the residue condition do not blow up.

We assert that (<ref>) solves the following RH problem.

	Find a 2×2 matrix-valued function M^(1) s.t.
	
		
  * M^(1)(λ) is meromorphic on ℂ∖Σ.
		
  * As λ→∞,
		
    M^(1)∼ I+(λ^-1)

		
  * On λ∈Σ, 
		
    M^(1)_+=M^(1)_-V^(1),

		where
		
    V^(1)=[
    					1+r^2   rT^-2e^-ϕ
    rT^2e^ϕ   1
    				]   λ∈Σ-S_1S_2,
    [
    					1   r̅/1+r^2T_+^-2e^-ϕ
    r/1+r^2T_-^2e^ϕ   1+r^2
    				]   λ∈S_1S_2.

		
  * Residue condition: On the poles 𝒵∪𝒵^-1, we have
		
			
    λ=λ_l'_-αM^(1)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)(λ_l')/(α_l'-α-j)![
    						0   p^T_λ_l',je^-ϕ(λ_l')
    0   0
    					]    λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)(λ_l')/(α_l'-α-j)![
    						0   0
    p^T_λ_l',je^ϕ(λ_l')   0
    					]    λ_j∈𝒵∖𝒵_ξ^+,
    				
       λ=λ̅_l'^-1_-αM^(1)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   0
    p^T_λ̅_l'^-1,je^ϕ(λ̅_l'^-1)   0
    					]   λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   p^T_λ̅_l'^-1,je^-ϕ(λ̅_l'^-1)
    0   0
    					]   λ_j∈𝒵∖𝒵_ξ^+,
 
			where p^T_λ_l',j and p^T_λ̅_l'^-1,j are polynomials of (n,t) of order at most j.
		
		
	

That M^(1)=MT^σ_3 satisfies the analyticity, normalization, and jump condition in RH problem <ref> is easily verified by RH problem <ref> and the first three items in Proposition <ref>.
The remaining to check is the Residue condition on 𝒵. 
On the one hand for λ_l'∈𝒵∖𝒵_ξ^+, 
since T is analytic at λ_l', we calculate the Laurent's coefficients of M^(1)_2,

    λ=λ_l'_-αM^(1)_1   =∂_λ^α_l'-α((·-λ_l')^α_l'M_1T)(λ_l')/(α_l'-α)!
       =∑_j=0^α_l'-α∂_λ^α_l'-α-j((·-λ_l')^α_l'M_1)(λ_l')/(α_l'-α-j)!∂_λ^jT(λ_l')/j!
       =∑_j=0^α_l'-α∂_λ^jT(λ_l')/j!λ=λ_l'_-α-jM_1.

By (<ref>) and Leibniz Rule, it follows that

    ∂_λ^α M_2(λ_l')/α!=∑_j=0^α∂_λ^α-j M_2^(1)(λ_l')/(α-j)!∂_λ^jT(λ_l')/j!.

Thus, considering (<ref>), (<ref>), and (<ref>), we obtain that there are polynomials p^T_λ_l',j of (n,t) of order at most j, j=0,…,α_l'-1, such that

    λ=λ_l'_-αM^(1)_1=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)_2(λ_l')/(α_l'-α-j)!p^T_λ_l',je^ϕ(λ_l'),  α=1,…,α_l'.

On the other hand for λ_l'∈𝒵_ξ^+, by the transform (<ref>), we see that at any λ_l'∈𝒵_ξ^+, the pole of M^(1) is on the second column while the first column is holomorphic. 
Calculate the Laurent's coefficients,

    λ=λ_l'_-αM^(1)_2=∑_j=0^α_l'-αλ=λ_l'_-α-jT^-1∂_λ^jM_2(λ_l')/j!.

By Leibniz Rule, We also have that for any natural number α<α_l',

    ∂_λ^α M^(1)_1(λ_l')/α!   =(α!)^-1∂_λ^α[(·-λ_l')^α_l'M_1T/(·-λ_l')^α_l'](λ_l')
       =∑_j=0^α∂_λ^α-j[(·-λ_l')^α_l'M_1](λ_l')/(α-j)!∂_λ^j[(·-λ_l')^-α_l'T]/j!
       =∑_j=0^αλ=λ_l'ℒ_α-j-α_l'M_1∂_λ^j[(·-λ_l')^-α_l'T]/j!.

By (<ref>), (<ref>), and (<ref>), it is readily seen that there are polynomials p^T_λ_l',j of (n,t) of order at most j such that for any λ_l'∈𝒵_ξ^+,

    λ=λ_l'_-αM^(1)_2=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)_1(λ̅_l'^-1)/(α_l'-α-j)!p^T_λ_l',je^-ϕ(λ_l'),  α=1,…,α_l'.

Combining (<ref>) and (<ref>), we obtain the residue condition (<ref>), and we also verify (<ref>) in the similar way. 
Thus, we complete the assertion. 





 §.§ Split the jump contour

In this part, applying the ∂̅-RH transformation, we split the jump contour Σ along the arcs S_1S_2 and Σ∖S_1S_2. 
In the follow, properly choose the jump contour Σ^(2)=Σ^(2)_1∪Σ^(2)_3∪Σ^(2)_4∪Σ^(2)_6as shown in FIGURE <ref> such that it consists of line segments S_j(1+e^±π/4(-ϵ_0,ϵ_0)) and arcs centering at the original.
Positive number ϵ_0 is so small enough that the ⋃_k=1,3,4,6Ω_k is away from the discrete spectrum.

Introduce the continuous functions on ℂ∖{0}

	
    R_1(λ)=r(λ/λ),   R_6(λ)=R_1(λ̅^-1),
       R_3(λ)=(r̅/1+r^2)(λ/λ),   R_4(λ)=R_3(λ̅^-1),


and the 2×2 matrix-valued function

    M^(2)=M^(1)ℛ,  ℛ=
    		I   λ∈Ω_2∪Ω_5,
    [
    			1   0
    -R_1T^2e^ϕ   1
    		]   λ∈Ω_1,
    [
    			1   -R_3T^-2e^-ϕ
    0   1
    		]   λ∈Ω_3,
    [
    			1   0
    R_4T^2e^ϕ   1
    		]   λ∈Ω_4,
    [
    			1   R_6T^-2e^-ϕ
    0   1
    		]   λ∈Ω_6.

Utilizing RH problem <ref> and the transformation (<ref>), ones easily check that M^(2) uniquely solves the following ∂̅-RH problem, where the jump contour Σ^(2) as shown in FIGURE <ref>.

	Find a function such that
	
		
  * M^(2) is continuous on ℂ∖{Σ^(2)∪𝒵∪𝒵^-1} and holomorphic on (Ω_2∪Ω_5)∖{𝒵∪𝒵^-1}.
		
  * As λ→∞, M∼ I+(λ^-1).
		
  * On Σ^(2),
		
    M^(2)_+=M^(2)_-V^(2),

		where
		
    V^(2)=[
    					1   0
    R_1T^2e^ϕ   1
    				]   λ∈Σ^(2)_1,
    [
    					1   R_3T^-2e^-ϕ
    0   1
    				]   λ∈Σ^(2)_3,
    [
    					1   0
    R_4T^2e^ϕ   1
    				]   λ∈Σ^(2)_4,
    [
    					1   R_6T^-2e^-ϕ
    0   1
    				]   λ∈Σ^(2)_6.

		
  * On the poles on 𝒵∪𝒵^-1, 
		
			
    λ=λ_l'_-αM^(2)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(2)(λ_l')/(α_l'-α-j)![
    						0   p^T_λ_l',je^-ϕ(λ_l')
    0   0
    					]    λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(2)(λ_l')/(α_l'-α-j)![
    						0   0
    p^T_λ_l',je^ϕ(λ_l')   0
    					]    λ_j∈𝒵∖𝒵_ξ^+,
    				
       λ=λ̅_l'^-1_-αM^(2)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(2)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   0
    p^T_λ̅_l'^-1,je^ϕ(λ̅_l'^-1)   0
    					]   λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(2)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   p^T_λ̅_l'^-1,je^-ϕ(λ̅_l'^-1)
    0   0
    					]   λ_j∈𝒵∖𝒵_ξ^+.
 
			
		
		
  * ∂̅ condition: On Ω_1∪Ω_3∪Ω_4∪Ω_6,
		
    ∂̅M^(2)=M^(2)∂̅ℛ.

	




Rewrite λ∈ℂ in the polar coordinates: λ=ρ e^θ.
We note that the function R_j are constant along the ray ℝ^+e^θ, thus utilizing the formula ∂̅=e^iθ(∂_ρ+ρ^-1∂_θ)/2, we calculate

	
    ∂̅R_1   =/2ρ^-1e^θ∂_θ(r(e^θ)),
    ∂̅R_3   =/2ρ^-1e^θ∂_θ(r(e^θ))-r(e^θ)^2∂_θ(r(e^θ))/(1+r(e^θ)^2)^2.


Through (<ref>), it is readily seen that the L^1-norm of ∂̅R_j on any circle {λ>0} is well dominated by the norm of r∈ H^1, which is applied to those estimates in Section <ref>.




 §.§ The factorization of the ∂̅-RH problem

In this part, we decompose the solution for ∂̅-RH problem into a product of solutions for an RH problem and a ∂̅ problem. i.e.,

    M^(2)=M^(2)_DM^(2)_RH.



	Find a 2×2 matrix-valued function M^(2)_RH satisfying the ∂̅-RH problem <ref> with ∂̅_λℛ≡ 0.


	Find a 2×2 matrix-valued function M^(2)_D such that
	
		
  * M^(2)_D is continuous on ℂ and analytic on Ω_2∪Ω_5.
		
  * ∂̅ condition: On Ω_1∪Ω_3∪Ω_4∪Ω_6, 
		
    ∂̅M^(2)_D=M^(2)_Dℛ̃,  ℛ̃=M^(2)_RH∂̅ℛ(M^(2)_RH)^-1.

	
	




  §.§.§ Reflection coefficient vanishes


In this part, we discuss the RH problem <ref> when the reflection coefficient vanishing and denote its solution as M^(𝒵)=M^(𝒵)(λ,n,t).

	Find a 2×2 matrix-valued function M^(𝒵) solves RH problem <ref> with r≡0.

For any λ_l'∈𝒵_ξ^+, by the method to prove Proposition <ref>, there are two rational functions f_λ_j^𝒵=f_λ_j^𝒵(λ), f_λ̅_j^-1^𝒵=f_λ̅_j^-1^𝒵(λ) such that M^(𝒵)[
	1   f_λ_l'^ξe^-ϕ
0   1
], M^(𝒵)[
	1   0
f_λ̅_l'^-1^ξe^ϕ   1
] are holomorphic at λ=λ_l', λ=λ̅_l'^-1, respectively.
Similarly, for λ_l'∈𝒵∖(𝒵_ξ^+∪𝒵_ξ), there are rational functions f_λ_j^𝒵=f_λ_j^𝒵(λ), f_λ̅_j^-1^𝒵=f_λ̅_j^-1^𝒵(λ) such that M^(𝒵)[
	1   0
f_λ_l'^ξe^ϕ   1
], M^(𝒵)[
	1   f_λ̅_l'^-1^ξe^-ϕ
0   1
] are holomorphic at λ=λ_l', λ=λ̅_l'^-1, respectively.
By these facts, choosing a small enough positive constant ϵ_1, we construct the matrix-valued function

    M^(3)=M^(𝒵)[
    			1   f_λ_l'^ξe^-ϕ
    0   1
    		],   λ-λ_l'<ϵ_1,  λ_l'∈𝒵_ξ^+,
    [
    			1   0
    f_λ̅_l'^-1^ξe^ϕ   1
    		],   λ-λ̅_l'^-1<ϵ_1,  λ_l'∈𝒵_ξ^+,
    [
    			1   0
    f_λ_l'^ξe^ϕ   1
    		],   λ-λ_l'<ϵ_1,  λ_l'∈𝒵∖(𝒵_ξ^+∪𝒵_ξ),
    [
    			1   f_λ̅_l'^-1^ξe^-ϕ
    0   1
    		],   λ-λ̅_l'^-1<ϵ_1,  λ_l'∈𝒵∖(𝒵_ξ^+∪𝒵_ξ),
    
    		I,   otherwise.

Clearly, by the choice of M^(3), it solves a RH problem without residue condition that the jump condition is over Σ^(3)=⋃_λ∈𝒵∖𝒵_ξ{λ-λ_l'=ϵ_1 or λ-λ̅_l'^-1=ϵ_1}, and seeing the sign table of ϕ in FIGURE <ref>, if ϵ_1 is small enough, there exists a positive constant C such that

    V^(3)-I_L^∞(Σ^(3))≤ e^-Ct.

Explicitly, the RH problem reads as follows.

	Find a 2×2 matrix-valued function M(3), such that
	
		
  * M^(3) is holomorphic on ℂ∖(Σ^(3)∪𝒵_ξ∪𝒵_ξ^-1).
		
  * As λ→∞, M^(3)∼ I+(λ^-1).
		
  * On Σ^(3),
		
    M^(3)_+=M^(3)_-V^(3),

		where
		
    V^(3)=[
    					1   f_λ_l'^ξe^-ϕ
    0   1
    				],   λ-λ_l'=ϵ_1,  λ_l'∈𝒵_ξ^+,
    [
    					1   0
    f_λ̅_l'^-1^ξe^ϕ   1
    				],   λ-λ̅_l'^-1=ϵ_1,  λ_l'∈𝒵_ξ^+,
    [
    					1   0
    f_λ_l'^ξe^ϕ   1
    				],   λ-λ_l'=ϵ_1,  λ_l'∈𝒵∖(𝒵_ξ^+∪𝒵_ξ),
    [
    					1   f_λ̅_l'^-1^ξe^-ϕ
    0   1
    				],   λ-λ̅_l'^-1=ϵ_1,  λ_l'∈𝒵∖(𝒵_ξ^+∪𝒵_ξ).

		
  * The residue condition is as that of RH problem <ref> on 𝒵_ξ if 𝒵_ξ∅.
	


Set 

    M^(𝒵_ξ)=
    		I   𝒵_ξ=∅,
    
    		M^(𝒵_ξ,^T_ξ)   𝒵_ξ∅,

where M^(𝒵_ξ,^T_ξ) is as defined in Remark <ref>, and the discrete spectral data (𝒵_ξ,^T_ξ) is the restriction of (𝒵,^T) on 𝒵_ξ, where (𝒵,^T) is the discrete spectral data of RH problem <ref>.
Recalling the discussion in Remark <ref>, M^(𝒵_ξ) uniquely exists. 
Referring to Proposition <ref>, for any λ_l'∈𝒵_ξ, there is a rational function f_λ_l'=f_λ_l'(λ) such that

    M^(3)[
    		1   0
    f_λ_l'   1
    	],   M^(𝒵_ξ)[
    		1   0
    f_λ_l'   1
    	]

are holomorphic at λ=λ_l', and then it follows that

    M̃^(3)=M^(3)(M^(𝒵_ξ))^-1

is analytic at λ_l'.
Similarly, it is also analytic at λ̅_l'^-1.
Based on the above arguments, it is clear that (<ref>) solves a small-norm RH problem without poles

	Find a 2×2 matrix-valued function M̃^(3) such that
	
		
  * M̃^(3) is holomorphic on Σ^(3).
		
  * As λ→∞, M̃^(3)∼ I+(λ^-1).
		
  * On Σ^(3), M̃^(3)_+=M̃^(3)_-V^(3,R), where V^(3,R)=M^(𝒵_ξ)V^(3)(M^(𝒵_ξ))^-1.
	

By (<ref>) and the boundedness of M^(𝒵_ξ) on Σ^(3), for t≫0, the jump matrix V^(3,R) decays to the identity exponentially,

    V^(3,R)-I_L^∞(Σ^(3))≲ e^-Ct.

Then by the small norm theory for the RH problems <cit.>, it follows that

    M^(3)(λ)=(I+∫_Σ^(3)μ^(3)(s)(V^(3)(s)-I)/s-λ s/2π)M^(𝒵_ξ)(λ),

where μ^(3) is the solution in L^2(Σ^(3)) of 

    (-C_V^(3,R)-I)μ^(3)=I,   C_V^(3,R)-I=C^Σ^(3)_-(·(V^(3,R)-I)).

It is well understood that the projection operator C^Σ^(3)_- is bounded on L^2(Σ^(3)), then it follows that the bound C_V^(3)-I_L^2(Σ^(3)) of the linear operator of C_V^(3)-I is dominated by

    C_V^(3,R)-I_L^2(Σ^(3))≤C^Σ^(3)_-_L^2(Σ^(3))V^(3,R)-I_L^∞(Σ^(3))≲ e^-Ct.

By (<ref>), for t≫0, the solution μ^(3)=(-C_V^(3,R)-I)^-1I for (<ref>) uniquely exists and as t→∞,

    μ^(3)_L^2(Σ^(3))≤(1).

By (<ref>), (<ref>), (<ref>) and (<ref>), it follows that

    M^(𝒵)(0)=(I+(e^-Ct))M^(𝒵_ξ)(0).

and also

    M^(𝒵)(S_j)=(I+(e^-Ct))M^(𝒵_ξ)(S_j),   j=1,2.





  §.§.§ Solvability of M^(2)_RH


In this part, we discuss the solvability of M^(2)_RH. 
We choose small discs U_j as the neighborhood of S_j. 
We rewrite M^(2)_RH in the form

    M^(2)_RH=
    		EM^(𝒵),   λ∈ℂ∖(Σ^(2)∪ U_1∪ U_2),
    
    		EM^(𝒵)M^(2,j)_loc,   λ∈ U_j∖Σ^(2),

where M^(𝒵) is the solution of RH problem <ref>, M^(2,j)_loc is the solution of RH problem <ref>, and E admits the RH problem <ref>.

	Find a 2×2 matrix-valued function M^(2,j)_loc such that
	
		
  * M^(2,j)_loc is analytic on ℂ∖Σ^(2,j).
		
  * M^(2,j)_loc∼ I+(λ^-1) as λ→∞. 
		
  * M^(2,j)_loc admits the jump condition on Σ^(2,j)
		
    (M^(2,j)_loc)_+=(M^(2,j)_loc)_-V^(2,j)_loc,    on Σ^(2,j).

		where
		
			
    V^(2,1)_loc=V^(2,1)_loc(λ)=[
    						1   0
    T_1^2r(S_1)ζ^-2ν_1e^ζ^2/2   1
    					]   λ∈Σ^(2,1)_1,
    [
    						1   T_1^-2r(S_1)/1+r(S_1)^2ζ^2ν_1e^-ζ^2/2
    0   1
    					]   λ∈Σ^(2,1)_2,
    [
    						1   0
    T_1^2r(S_1)/1+r(S_1)^2ζ^-2ν_1e^ζ^2/2   1
    					]   λ∈Σ^(2,1)_3,
    [
    						1   T_1^-2r(S_1)ζ^2ν_1e^-ζ^2/2
    0   1
    					]   λ∈Σ^(2,1)_4,
    				
       V^(2,2)_loc=V^(2,2)_loc(λ)=[
    						1   T_2^-2r(S_2)ζ^-2ν_2e^ζ^2/2
    0   1
    					]   λ∈Σ^(2,2)_1,
    [
    						1   0
    T_2^2r(S_2)/1+r(S_2)^2ζ^2ν_2e^-ζ^2/2   1
    					]   λ∈Σ^(2,2)_2,
    [
    						1   T_2^-2r(S_2)/1+r(S_2)^2ζ^-2ν_2e^ζ^2/2
    0   1
    					]   λ∈Σ^(2,2)_3,
    [
    						1   0
    T_2^2r(S_2)ζ^2ν_2e^-ζ^2/2   1
    					]   λ∈Σ^(2,2)_4,

		
		and 
		
    λ=β_jζ+S_j,  β_j=(-1)^j/√(2)(1-ξ^2)^-1/4t^-1/2S_j,
       T_j=∏_λ_l∈𝒵_ξ^+(S_j-λ_l/S_j-λ̅_l^-1)^α_l(-2√(2)(1-ξ^2)^3/4t^1/2)^(-1)^j-1ν_j( S_j)^(-1)^jν_j/e^ t[1+√(1-ξ^2)+ξ((1+(-1)^j)/4π+(-1)^j-1arcsinξ)]-α_j(S_j).

	


	Find a 2×2 matrix-valued function such that
	
		
  * E is analytic on Σ^E=∂ U_1∪∂ U_2∪(Σ^(2)).
		
  * As λ→∞, E(λ)∼ I+(λ^-1).
		
  * On Σ^E,
		
    E_+=E_-V^E,

		where 
		
    V^E=
    				M^(𝒵)V^(2)(M^(𝒵))^-1,   λ∈Σ^(2)∖(∂ U_1∪∂ U_2),
    
    				M^(𝒵)M^(2,j)_locV^(2)(V^(2,j)_loc)^-1(M^(𝒵)M^(2,j)_loc)^-1,   λ∈Σ^(2)∩ U_j,
    
    				M^(𝒵)M^(2,j)_loc(M^(𝒵))^-1,   λ∈∂ U_1∪∂ U_2.

	

Using (<ref>), we obtain the solution M^(2)_RH of RH problem <ref> by constructing M^(𝒵), M^(2,j)_loc and E one by one. 

To see the solvability of M^(2)_RH, we first investigate the part of M^(𝒵) and M^(2,j)_loc. 
On the one hand, recalling the discussion in Section <ref>, we see that M^(𝒵) is uniquely solved for t≫0.
On the other hand, M^(2,j)_loc can be represented by 

	
    M^(2,1)_loc(λ)=T_1^-σ_3M^(PC)(ζ,r(S_1))T_1^σ_3,
       M^(2,2)_loc(λ)=σ_1T_2^σ_3M^(PC)(ζ,r(S_2))T_2^-σ_3σ_1,


where M^(PC) the well-known as the model RH problem <ref>. 


Based on the solvability of M^(𝒵) and M^(2,j)_loc, we come to the error part of E. 
By (<ref>), (<ref>), (<ref>) and the sign table of ϕ, it is readily seen that for some positive constant ϵ

    V^(2)(λ)-I≤ e^-ϵ t,  λ∈Σ^(2)∖(U_1∪ U_2).

By (<ref>), (<ref>) and Proposition <ref>, we obtain that

    V^(2)(V^(2,j)_loc)^-1-I≲V^(2)-V^(2,j)_loc≲ t^-1/4ζ^1/2e^-ζ^2/2,
    	  λ∈Σ^(2)∩ U_j.

By (<ref>), (<ref>) and (<ref>), it follows that

    (M^(2,j)_loc)^-1-I≲ t^-1/2,  λ∈∂ U_j.

Considering (<ref>), (<ref>) and (<ref>), by the boundedness of M^(𝒵) on Σ^E, we conclude that

    V^E-I≲ t^-1/4,

that is, E is the solution of a small-norm RH problem.
Recalling that C_-^Σ^E is a bounded operation on L^2(Σ^E), by (<ref>), it is readily seen that the Cauchy-type integral operator

    C_V^E-I: C_V^E-If=C_-(f(V^E-I))

is consequently bounded on L^2(Σ^E), and 

    C_V^E-I_L^2(Σ^E)≲ t^-1/4.

(<ref>) indicates that as t→+∞, the operator decays, and for sufficiently large t, (-C_V^E-I)^-1 exists and is bounded on L^2(Σ^E).
By RH problem <ref>, it follows that the solution can be written in the integral form:

    E(λ,n,t)=I+∫_Σ^E[(-C_V^E-I)^-1I](s,n,t)(V^E(s,n,t)-I)/s-λ s/2π.


Now, we focus on the estimate of E(0)=E(0,n,t), and assert that as t→+∞,

    E(0,n,t)-I=   t^-1/2M^(𝒵_ξ)(S_1,n,t)/√(2)(1-ξ^2)^1/4[
    			0   -γ_1(r(S_1))/T_1^2
    γ_2(r(S_1))T_1^2   0
    		](M^(𝒵_ξ)(S_1,n,t))^-1
         +t^-1/2M^(𝒵_ξ)(S_2,n,t)/√(2)(1-ξ^2)^1/4[
    			0   -γ_2(r(S_2))/T_2^2
    γ_1(r(S_2))T_2^2   0
    		](M^(𝒵_ξ)(S_2,n,t))^-1+(t^-3/4).

Taking the value of (<ref>) at λ=0, 

    E(0)=I+∮_∂ U_1∪∂ U_2s^-1[(-C_V^E-I)^-1C_V^E-II](s)(V^E(s)-I) s/2π
         +∮_Σ^(2)s^-1[(-C_V^E-I)^-1I](s)(V^E(s)-I) s/2π+∮_∂ U_1∪∂ U_2s^-1(V^E(s)-I) s/2π.

In (<ref>), the first integral is dominated by (t^-3/4) because of (<ref>) and (<ref>), and using (<ref>) and (<ref>), it follows that the second integral is constrained by (t^-3/4).
Then we rewrite (<ref>) in the asymptotic form that as t→+∞,

    E(0,n,t)=I+∮_∂ U_1∪∂ U_2s^-1(V^E(s,n,t)-I) s/2π+(t^-3/4).

Using (<ref>), (<ref>), (<ref>), (<ref>) and the Residue Theorem, we formulate the integral term in (<ref>), and consequently obtain (<ref>).















	If we set the transformation on ℂ, j=1,2: λ=β_jζ+S_j, where β_j are defined as (<ref>) and S_j are the stationary phase points,
	then when ξ∈(-1,1), we have the following asymptotic properties that as λ→ S_j,
	
    R_1(λ)∼ r(S_j)+(t^-1/4ζ^1/2),   R_3(λ)∼r(S_j)/1+r(S_j)^2+(t^-1/4ζ^1/2),
       T^2(λ)e^ϕ(λ)∼ T_j^2ζ^(-1)^j2ν_je^-(-1)^jζ^2/2+(t^-1/4ζ^1/2e^-(-1)^jζ^2/2),

	


	On the one hand, recalling that r∈ H^1, we have that for λ on the neighborhood of S_j,
	
    R_1(λ)-r(S_j)=r(β_jζ+S_j/β_jζ+S_j)-r(S_j)≲β_jζ+S_j/β_jζ+S_j-S_j^1/2≲β_jζ^1/2,

	and we proved the result for R_1 in (<ref>).
	In addition, when r∈ H^1, it is readily seen that r/1+r^2∈ H^1, therefore, we similarly obtain R_3(λ)'s result in (<ref>) and complete the proof of (<ref>).
	On the other hand, the result (<ref>) is a consequence of (<ref>), (<ref>) and (<ref>).
	This completes the proof.


We have proved that M^(2)_RH is uniquely solved,
then we claim that M^(2)_RH is invertible.
It is readily seen that M^(2)_RH has no jump over ℂ and tends to 1 at the infinity.
In addition, utilizing the technique to prove Proposition <ref>, we can remove the pole of M^(2)_RH at λ=λ_l' by a triangular function with constant determinant, 
that is, the singularities of M^(2)_RH at λ=λ_l' are removable.
In the similar way, its singularities at λ=λ̅_l'^-1 are also removable. 
By Liouville's Theorem, M^(2)_RH is the constant 1 over ℂ, that is, M^(2)_RH is invertible.

Defining the function

    M^(2)_D=M^(2)(M^(2)_RH)^-1.

Since M^(2)_RH is invertible, the definition (<ref>) is well posed,
and comparing ∂̅-RH problem <ref> to RH problem <ref>, we find that M^(2)_D is the solution of ∂̅ problem <ref>. 



  §.§.§ Estimate on the ∂̅ problem

Since we have already proved the existence of M^(2)_RH, in this part, we concentrate on some estimates of M^(2)_D. 

That M^(2)_D solves ∂̅ problem <ref> is equivalent to the integral equation

    M^(2)_D(λ)=I+∬_ℂ[M^(2)_Dℛ̃](s)/s-λ L(s)/π,

where L(s) is the  Lebesgue's measure on the complex plane.
Rewrite (<ref>) as

    (-S)M^(2)_D=I,   Sf(λ)=∬_ℂ[fℛ̃](s)/s-λ L(s)/π.

The linear integral operator S is proved bounded on L^∞(ℂ) in Proposition <ref>, and the bound S_L^2(ℂ) decays as t→∞.
Therefore, (<ref>) is uniquely solved in L^∞(ℂ) for t≫0.


	Define the integral operator S as (<ref>).
	If r∈ H^1, then S is bounded on L^∞(ℂ), and as t→+∞, the bound of S on L^∞(ℂ) is dominated by t^-1/4, that is,
	
    S_L^∞(ℂ)≲ t^-1/4.



	Recalling that ℛ̃ is supported on ⋃_k=1,3,4,6Ω_k and the discrete spectrum of RH problem <ref> is away from ⋃_k=1,3,4,6Ω_k, we have that M^(2)_RH, (M^(2)_RH)^-1, T and T^-1 are all bounded on ⋃_k=1,3,4,6Ω_k, and consequently,
	
    Sf(λ)=∬_ℂ[fℛ̃](s)/s-λ L(s)/π=∬_⋃_k=1,3,4,6Ω_k[fℛ̃](s)/s-λ L(s)/π
         ≲f_∞∑_k=1,3,4,6∬_Ω_k∂̅ℛ(s)/s-λ L(s)/π
          =f_∞∑_k=1,3,4,6∬_Ω_k[∂̅R_ke^(-1)^[k/2]ϕ](s)/s-λ L(s)/π,

	where (-1)^[k/2]=
		1   k=1,4
-1   k=2,6.
	
	For the region Ω_1, we write s∈Ω_1 in the polar coordinates: s=ρ e^iθ, (ρ,θ)∈(1,ρ_0)×(θ_ρ,-π-θ_ρ), where ρ_0=1+ϵ_0e^π/4, and ρ e^θ_ρ, ρ e^-π-θ_ρ are on the upper boundaries of Ω_1: S_1(1+e^π/4(0,ϵ_0)), S_2(1+e^-π/4(0,ϵ_0)), respectively.
	Taking Taylor's expansion of (<ref>) at λ=S_1, we see that as ρ e^θ_ρ→ S_1, 
	
    ϕ(ρ e^θ_ρ)=   ϕ(ρ e^θ_ρ,n,t)
    			∼ -2t√(1-ξ^2)(ρ-1)^2+(ρ-1)^3.

	Utilizing (<ref>), (<ref>) and the boundedness of T on Ω_1, we estimate the integral on Ω_1 for λ0,
	
    ∬_Ω_1[∂̅R_1e^ϕ](s)/s-λ L(s)/π =∫_1^ρ_0∫_-π-θ_ρ^θ_ρρ^-1∂_θ(r(e^θ))e^ϕ(ρ e^θ)/2π(ρ e^θ-λ)ρθρ
         ≤∫_1^ρ_0∫_-3π/2^π/2∂_θ (r(e^θ))e^ϕ(ρ e^θ_ρ)/2πρ e^θ-λθρ
         ≤∫_1^ρ_0ρe^ϕ(ρ e^θ_ρ)/2πr_H^1(∫_-π^πρ e^θ-λ^-2θ)^1/2
         ≲∫_1^ρ_0ρe^ϕ(ρ e^θ_ρ)/2πρ-λ^1/2.

	By (<ref>) and (<ref>), since we have assumed ϵ_0 small enough, it is readily seen that
	
    ∬_Ω_1[∂̅R_1e^ϕ](s)/s-λ L(s)/π≤∫_0^ρ_0ρe^-t√(1-ξ^2)(ρ-1)^2/2πρ-λ^1/2≲ t^-1/4.

	
	For regions Ω_3, Ω_4, Ω_6, letting λ0, it admits the same estimates as (<ref>), i.e., for sufficient small ϵ_0 and k=1,3,4,6,
	
    ∬_Ω_k[∂̅R_ke^(-1)^[k/2]ϕ](s)/s-λ L(s)/π≲ t^-1/4.

	Utilizing (<ref>) and (<ref>), we finally prove the results.


Now, we focus on the estimate of M^(2)_D at λ=0.

    M^(2)_D(0)=I+∬_ℂ[M^(2)_Dℛ̃](s)s^-1 L(s)/π.

Again taking s=ρ e^θ∈Ω_1, by (<ref>) it is easy to check that for any fixed ξ∈(-1,1), there is a positive constant ϵ̃(ξ) such that

    sinθ-sinθ_ρ≤ϵ̃(ξ)(θ-θ_ρ)   θ∈(-π/2,θ_ρ),
    
    			-ϵ̃(ξ)(θ+π+θ_ρ)   θ∈(-π-θ_ρ,-π/2).

Utilizing the techniques applied in Proposition <ref> and the estimate (<ref>), we obtain that the following integral on Ω_1 is dominated by t^-3/4, that is,

    ∬_Ω_1[M^(2)_Dℛ̃](s)s^-1 L(s)/π≲∬_Ω_1∂̅R_1(s)e^ϕ(s)s^-1 L(s)/π
         =∫_1^ρ_0e^ϕ(ρ e^θ_ρ)/2πρρ∫_-π-θ_ρ^θ_ρ∂_θ(r(e^θ))e^ϕ(ρ e^θ)-ϕ(ρ e^θ_ρ)θ
         ≤r_H^1∫_1^ρ_0 e^(ρ e^θ_ρ)/2πρρ(∫_-π/2^θ_ρe^2t(ρ-ρ^-1)(θ-θ_ρ)cosθ_ρθ+∫_-π-θ_ρ^-π/2e^-2t(ρ-ρ^-1)(θ+π+θ_ρ)cosθ_ρθ)^1/2
         ≲∫_1^ρ_0e^-t√(1-ξ^2)(ρ-1)^2/2πρt^-1/2(ρ-ρ^-1)^-1/2ρ≲ t^-3/4.

Utilizing the same technique, the results still hold for Ω_3, Ω_4 and Ω_6:

    ∬_Ω_k[M^(2)_Dℛ̃](s)s^-1 L(s)/π≲ t^-3/4,   k=1,3,4,6.

Substituting (<ref>) into (<ref>), it is readily seen that as t→+∞,

    M^(2)_D(0)∼ I+(t^-3/4).






  §.§.§ The asymptotic formula


By (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we obtain the asymptotic formula (<ref>).



§ LONG-TIME ASYMPTOTICS FOR THE SOLITON REGION

When ξ∈(-1,1), we have defined the scalar function T(λ) as (<ref>), otherwise, we set it as

    T(λ)=T(λ,ξ)=∏_λ_l'(λ-λ_l'/λ-λ̅_l'^-1)^α_l'   ξ≤-1,
    
    		e^-∫_Σln1+r(s)^2/s-λ s/2π∏_λ_l'(λ-λ_l'/λ-λ̅_l'^-1)^α_l'   ξ≥1.

In the following, we discuss the long-time asymptotics in the two sectors: ξ>1 and ξ<-1.




In the sector ξ<-1, the stationary phase points are off the jump contour Σ={λ=1}.
In this vein, set the positive number

    ϵ_2=min{λ_1-1,…,λ_l-1,√(ξ^2-1)+ξ-1}/2.

By the sign table shown in FIGURE <ref> and (<ref>), it is clear that

    Σ_ϵ_2={λ=1+ϵ_2}⊂{ϕ>0},
       Σ_ϵ_2^-1⊂{ϕ<0},

and there is no discrete spectrum in the region between Σ and Σ_ϵ_2.
By (<ref>) and (<ref>), it follows that there is a positive constant C such that the jump matrix define in (<ref>) satisfies

    V^(4)-I_L^∞(ℛ)≲ e^-C t,   t→+∞,

Define

    M^(4)=MT^σ_3[
    			1   0
    -R_1T^2e^ϕ   1
    		]   λ∈Ω_8,
    [
    			1   R_6T^-1e^-ϕ
    0   1
    		]   λ∈Ω_9,
    
    		I   λ∈Ω_7∪Ω_10.

It is clear that M^(4) solves the following ∂̅-RH problem uniquely, and one can refer to its jump contour Σ^(4)=Σ_ϵ_2∪Σ_ϵ_2^-1 in FIGURE <ref>.

	Find a 2×2 matrix-valued function M^(4) such that
	
		
  * M^(4) is continuous on ℂ∖{Σ^(4)∪𝒵∪𝒵^-1} and holomorphic on (Ω_7∪Ω_10)∖{𝒵∪𝒵^-1}.
		
  * As λ→∞, M^(4)∼ I+(λ^-1).
		
  * On Σ^(4),
		
    M^(4)_+=M^(4)_-V^(4),

		where
		
    V^(4)=[
    					1   0
    R_1T^2e^ϕ   1
    				]   λ∈Σ_ϵ_2,
    [
    					1   R_6T^-2e^-ϕ
    0   1
    				]   λ∈Σ_ϵ_2^-1.

		
  * On the poles on 𝒵∪𝒵^-1, 
		
			
    λ=λ_l'_-αM^(4)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(4)(λ_l')/(α_l'-α-j)![
    						0   p^T_λ_l',je^-ϕ(λ_l')
    0   0
    					]    λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(4)(λ_l')/(α_l'-α-j)![
    						0   0
    p^T_λ_l',je^ϕ(λ_l')   0
    					]    λ_j∈𝒵∖𝒵_ξ^+,
    				
       λ=λ̅_l'^-1_-αM^(4)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(4)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   0
    p^T_λ̅_l'^-1,je^ϕ(λ̅_l'^-1)   0
    					]   λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(4)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   p^T_λ̅_l'^-1,je^-ϕ(λ̅_l'^-1)
    0   0
    					]   λ_j∈𝒵∖𝒵_ξ^+.
 
			
		
		
  * ∂̅ condition: On Ω_8∪Ω_9,
		
    ∂̅M^(4)=M^(4)[
    					0   0
    -∂̅R_1T^2e^ϕ   0
    				]   λ∈Ω_8,
    [
    					0   ∂̅R_6T^-1e^-ϕ
    0   0
    				]   λ∈Ω_9.

	


As we have done in the case ξ∈(-1,1), one can factorize the solution M^(4) into a product of solutions of a ∂̅ problem and an RH problem below,
that is, M^(4) admits the following factorization:

    M^(4)=M^(4)_DM^(4)_RH.



	Find a 2×2 matrix-valued function M^(4)_RH solves ∂̅-RH problem <ref> with ∂̅M^(4)_RH≡0 over the complex plane ℂ.


	Find a 2×2 matrix-valued function M^(4)_D=M^(4)_D(λ,n,t) such that
	
		
  * M^(4)_D is continuous on ℂ.
		
  * As λ→∞, M^(4)_D∼ I+(λ^-1).
		
  * On Ω_7,…,Ω_10, it admits the ∂̅ condition:
		
    ∂̅M^(4)_D=M^(4)_Dℛ^(4),  ℛ^(4)=M^(4)_RH[
    					0   0
    -∂̅R_1T^2e^ϕ   0
    				](M^(4)_RH)^-1   λ∈Ω_8,
    [
    					0   ∂̅R_6T^-1e^-ϕ
    0   0
    				](M^(4)_RH)^-1   λ∈Ω_9.

		
	

On the one hand for M^(4)_RH, we claim that M^(4)_RH is well-defined as what ones have done in Section <ref>.
With a little abuse of notation, for ξ>0, denote M^(𝒵,^T) as the solution of RH problem <ref> with V^(4)≡ I, where (𝒵,^T) is the discrete spectral data.
Referring to the techniques of small norm RH problem theory in Section <ref>, it is clear that for some positive constant C,

    M^(𝒵)(0,n,t)=(I+(e^-Ct))M^(𝒵_ξ)(0,n,t),

where M^(𝒵_ξ)=M^(𝒵_ξ)(λ,n,t) is defined for ξ>0 as
	
    M^(𝒵_ξ)=
    			I   𝒵_ξ=∅,
    
    			M^(𝒵_ξ,^T_ξ)   𝒵_ξ∅,
    		  𝒵_ξ=𝒵∩{ϕ=0},

and M^(𝒵_ξ,^T_ξ)=M^(𝒵_ξ,^T_ξ)(λ,n,t) is as defined in Remark <ref> with the discrete spectral data (𝒵_ξ,^T_ξ) that is the restriction of (𝒵,^T) on 𝒵_ξ. 
Utilizing the techniques in Section <ref>, it is clear that M^(4)_RH admits the form

    M^(4)_RH=E^(4)M^(𝒵),

where E^(4) is the solution of the small-norm RH problem:

	Find a 2×2 matrix-valued function E^(4)=E^(4)(λ,n,t) such that
	
		
  * E^(4) is analytic on ℂ∖Σ^(4).
		
  * As λ→∞, E^(4)∼ I+(λ^-1).
		
  * On Σ^(4), M^(4)_+=M^(4)_-V^(4,E), where V^(4,E)=M^𝒵V^(4)(M^𝒵)^-1.
	

By the boundedness of M^𝒵 and (M^𝒵)^-1 on Σ^(4), recalling (<ref>), ones obtain that as t→+∞, 

    V^(4,E)-I_L^∞(Σ^(4))≲ e^-Ct.

Again utilizing the small norm theory for the RH problems, we obtain that RH problem <ref> is uniquely solved and as t→+∞, 

    E^(4)(λ,n,t)∼ I+(e^-Ct).

Substituting (<ref>) into (<ref>) and utilizing (<ref>), it follows that as t→+∞,

    M^(4)_RH(0,n,t)∼ M^(𝒵_ξ)(0,n,t)+(e^-Ct).
 

On the other hand for M^(4)_D, we assert that when t→+∞, M^(4)_D(0,n,t) decays to the identity matrix like that it have done for ξ∈(-1,1), but the decaying rate is (t^-1),

    M^(4)_D(0,n,t)∼ I+(t^-1).

Recalling ∂̅ problem <ref>, rewrite M^(4)_D in the form

    M^(4)_D=(-S^(4))^-1I,   Sf(λ)=∬_ℂ[fℛ^(4)](s)/s-λ L(s)/π.

The integral operator is bounded on L^∞(ℂ) and the bound decays to zero for t≫0, that is, 

    S^(4)_L^∞≲(t^-1/2).

For λ0 and f∈ L^∞(ℂ), by the boundedness of M^(𝒵), (M^(𝒵))^-1 and T on Σ^(4), it is similar to the estimates in (<ref>) and (<ref>) that

    ∬_Ω_8[fℛ^(4)](s)/s-λ s/π   ≲f_L^∞(ℂ)∬_Ω_8∂̅R_1e^ϕ(s)/s-λ L(s)/π
       =f_L^∞(ℂ)∫_1^1+ϵ_2∫_-π^π∂_θ(r(e^θ))e^t((ρ-ρ^-1)sinθ+2ξlnρ)/2πρ e^θ-λθρ
       ≤f_L^∞(ℂ)r_H^1∫_1^1+ϵ_2e^t((ρ-ρ^-1)+2ξlnρ)ρ/2π(∫_-π^πρ e^θ-λ^-2θ)^1/2
       ≲f_L^∞(ℂ)∫_1^1+ϵ_2e^-2t(ρ-1)((-ξ)ln(1+ϵ_2)/ϵ_2-1)/ρ-λ^1/2ρ≲ t^-1/2f_L^∞(ℂ).

and also

    ∬_Ω_8[fℛ^(4)](s)/s-λ s/π≲ t^-1/2f_L^∞(ℂ),

thus, (<ref>) follows. 
Utilizing (<ref>), we obtain that (<ref>) is well-posed, and at λ=0, similarly we have the estimate (<ref>).

Considering (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we have that for ξ<-1

    q_n(t)=q_n^𝒵_ξ(t)+(t^-1),

where q_n^𝒵_ξ(t)=[M^(4)]_1,2(0,n+1,t).
By the same technique, it is readily seen that for ξ>1, the solution q_n(t) also satisfies (<ref>). 







§ PARABOLIC CYLINDER MODEL







Here, describe the parabolic cylinder model that is first introduced by <cit.> and further by <cit.>.
And it has been frequently applied to the literature of long-time asymptotics <cit.>.
Set Σ^(PC)=⋃_k=1^4Σ^PC_k, where Σ^PC_k is the ray e^π/4(2k-1) oriented as shown in FIGURE <ref>.
The regions Ω^PC are the sectors on ζ∈ℂ that is divided by Σ^(PC) and ζ.
Consider the parabolic-cylinder RH problem below. 

	Find a 2×2 matrix-valued function M^(PC)=M^(PC)(ζ,τ) on ζ∈ℂ, such that
	
		
  * M^(PC) is analytic on ℂ∖Σ^(PC).
		
  * As ζ→∞, M^(PC)∼ I+(ζ^-1).
		
  * On Σ^(PC), 
		
    M^(PC)_+=M^(PC)_-V^(PC),  
    			V^(PC)=V^(PC)(ζ,τ)=
    			[
    					1   0
    τζ^-2ν_je^ζ^2/2   1
    				]   ζ∈Σ^PC_1,
    [
    					1   τ/1+τ^2ζ^2ν_je^-ζ^2/2
    0   1
    				]   ζ∈Σ^PC_2,
    [
    					1   0
    τ/1+τ^2ζ^-2ν_je^ζ^2/2   1
    				]   ζ∈Σ^PC_3,
    [
    					1   τζ^2ν_je^-ζ^2/2
    0   1
    				]   ζ∈Σ^PC_4.

		
	

The solution of RH problem <ref> is explicitly solved by:

    M^(PC)(ζ,τ)=Φ^(PC)(ζ,τ)𝒬(ζ,τ)ζ^-νσ_3e^/4ζσ_3,
       Φ^(PC)(ζ,τ)=[
    			e^-3πν/4D_ν(e^-3π/4ζ)   -γ_1(τ)e^π(ν-)/4D_-ν-1(e^-π/4ζ)
    γ_2(τ)e^-3π(ν+)/4D_ν-1(e^-3π/4ζ)   e^πν/4D_-ν(e^-π/4ζ)
    		]   ζ∈{ζ>0},
    [
    			e^πν/4D_ν(e^π/4ζ)   -γ_1(τ)e^-3π(ν-)/4D_-ν-1(e^3π/4ζ)
    γ_2(τ)e^π(ν+)/4D_ν-1(e^π/4ζ)   e^-3πν/4D_-ν(e^3π/4ζ)
    		]   ζ∈{ζ<0},
    	
       𝒬(ζ,τ)=[
    			1   0
    -τ   1
    		]   ζ∈Ω^PC_1,
    [
    			1   -τ̅/1+τ^2
    0   1
    		]   ζ∈Ω^PC_3,
    [
    			1   0
    τ/1+τ^2   1
    		]   ζ∈Ω^PC_4,
    [
    			1   τ̅
    0   1
    		]   ζ∈Ω^PC_6,
    
    		I   ζ∈Ω_2∪Ω_5,

where D_a(λ) is the solution of the parabolic cylinder equation:

    (∂_λ^2+1/2-λ^2/2+a)D_a(λ)=0

and the complex constants γ_1, γ_2 and ν read

    γ_1=γ_1(τ)=√(2π)e^π/4-πν(τ)/2/τΓ(-ν(τ)),  γ_2=γ_2(τ)=√(2π)e^-π/4-πν(τ)/2/τ̅Γ(ν(τ)),  ν=ν(τ)=ln(1+τ^2)/2π.

It is well acknowledged in <cit.> that M^(PC) admits the asymptotic property at ζ→∞,

    M^(PC)(ζ,τ)∼ I+1/ζ[
    		0   -γ_1(τ)
    γ_2(τ)   0
    	]+(ζ^-2).








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