Paper Title: Slow decay rate of correlations induced by long-range extended Dzyaloshinskii-Moriya interactions.





Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute,  Chhatnag Road, Jhunsi, Prayagraj - 211019, India


§ ABSTRACT 
    We examine the impact of long-range  Dzyaloshinskii-Moriya (DM) interaction in the extended XY model on the phase diagram as well as the static and dynamical properties of quantum and classical correlation functions. It is known that in the nearest-neighbor XY model with DM interaction, the transition from the gapless chiral phase to a gapped one occurs when the strengths of the DM interaction and anisotropy coincide. We exhibit that the critical line gets modified with the range of interactions which decay according to power-law.   Specifically, instead of being gapless in the presence of a strong DM interaction,  a gapped region emerges which grows with the increase of the moderate fall-off rate (quasi-long range regime) in the presence of a transverse magnetic field. The gapless chiral phase can also be separated from a gapped one by the decay patterns of quantum mutual information and classical correlation with distant sites of the ground state which are independent of the fall-off rate in the gapless zone. We observe that the corresponding critical lines that depend on the fall-off rate can also be determined from the effective central charge involved in the scaling of entanglement entropy.  We illustrate that in a non-equilibrium setting, the relaxation dynamics of classical correlation,  the decay rate of total correlation, and the growth rate of entanglement entropy can be employed to uncover whether the evolving Hamiltonian and the Hamiltonian corresponding to the initial state are gapped or gapless. 

 Slow decay rate of correlations induced by long-range extended Dzyaloshinskii-Moriya interactions
    Tanoy Kanti Konar, Leela Ganesh Chandra Lakkaraju, Aditi Sen(De)
    
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§ INTRODUCTION


Long-range order across quantum systems is desirable for a variety of quantum technological tasks, as it facilitates the sharing of correlation among distant parts of the system. Historically, such long-range order has been observed in many-body systems at quantum criticality or when the spectrum becomes gapless. In the case of the nearest-neighbor (NN) XY model with nonvanishing anisotropy parameter, it has been demonstrated that the classical correlations decay with the distance between the spins  exponentially (polynomially) when it is away from (at) the criticality.
Beyond NN models, long-range interacting (LR) systems with power-law decay in the range of interactions have been shown to exhibit polynomial decay of correlations, even when the spectrum is not gapless. These systems become an intense topic of research in the last few years, especially in the context of building analog quantum simulator<cit.> and due to their natural occurrence in several physical platforms such as the Rydberg atom arrays <cit.>, dipolar systems <cit.>, polar molecules <cit.>, trapped-ion setups <cit.>, and cold atoms in cavities <cit.>. Moreover, several established results that hold for nearest-neighbor systems came into the light of investigation in these LR systems in terms of Lieb-Robinson bound <cit.>, area law <cit.>, novel phases of matter  <cit.>,  dynamical phases <cit.> and from the perspectives of usefulness in quantum technologies like  quantum metrology <cit.> and quantum computation <cit.>. 








The nearest-neighbor Dzyaloshinskii-Moriya (DM) interaction, on the other hand,  involves an asymmetric exchange of spins caused by spin-orbit coupling <cit.>.

It has been widely explored in a variety of solid-state compounds, including Cu(C_6D_5COO)_2. 3D_2O <cit.>, Yb_4As_3<cit.>, Crl_3 <cit.> etc, exhibiting fascinating magnetic properties, like gapless chiral phase when the DM interaction strength is stronger than the anisotropy parameter in the transverse Ising <cit.>,  XY <cit.> and the gamma model <cit.>. It was also shown that the DM interaction can host a ground state in the gapless phase that contains logarithmic entanglement behavior, similar to the LR interacting systems which can enhance the fidelity in quantum teleportation <cit.>, thermal entanglement <cit.> and performance in quantum engines <cit.>.  



In the non-equilibrium domain, the DM interaction has also been examined in the context of non-equilibrium thermodynamics <cit.>, dynamical quantum phase transitions <cit.>, topological phase transition <cit.> and quantum speed limit <cit.>. 



 In this paper, we explore the long-range extended XY model in the presence of long-range extended DM interactions, which decay according to a power-law.  Specifically, we aim to analyze the responses of both long-range and DM interactions in equilibrium and non-equilibrium physics. In the former scenario, we present the phase-classification utilizing conventional order parameters, the law for the decreasing slopes of classical and quantum correlations, and the scaling of entanglement entropy (EE) with the corresponding central charge. In contrast to the NN model <cit.>, we show that given a fixed fall-off rates, there exists a gapped region in which the DM interaction is stronger than the anisotropy parameter. However, we establish that whenever the system is gapless, it is chiral. 
 Notably, in the static case, when the ground state belongs to the chiral gapless region, the classical correlation and quantum mutual information decay with constant decreasing exponent while this is not the case in the gapped regime.   In this model, we observe that the effective central charge associated with EE exhibits interesting behaviors: in the presence of LR and DM interactions, the central charge acts nonlinearly with multiple kinks that are absent from the extended Ising model without DM interactions.

  In comparison to the equilibrium scenario, the dynamical phases could not be efficiently characterized as they depend on the  initial state, evolving operator,  time and system parameters. 
   In order to distinguish between the gapless and gapped phases, we resort to a recently proposed relaxation dynamics of two-point classical correlations <cit.> and entanglement entropy of the evolved state in the transient regimes, as well as mutual information and classical correlation in the steady state domains
  (cf. <cit.> for other dynamical quantities). In our case, the initial state is  prepared by tuning the parameters of the Hamiltonian which have either gapless or gapped energy spectra while  evolution happens when the range of interactions is suddenly quenched, resulting in a gapless or gapped Hamiltonian. Specifically, in the transient  regime,  we show that the long-range DM interactions modify the exponent in the relaxation time of the two-point classical correlation between modes  and establish a generic  scaling law for the growth of entanglement entropy over time. Further, we  observe that  the behavior of mutual information and classical correlation between two arbitrary sites in the steady state differs depending on whether the Hamiltonians corresponding to the initial and dynamical states are in the same (gapped or gapless) phase or in different phases. 

  
  

  
  

 













 

The paper is organized as follows. In Sec. <ref>, we introduce the long-range extended XY model with an additional long-range DM term and find the energy spectra and eigenvectors. While 

we identify the conventional phases in Sec. <ref>, we indicate the decay of classical and quantum correlations in Sec. <ref>. The behavior of  entanglement entropy and the corresponding central charge with respect to system parameters are studied in Sec. <ref>. We then move on to the investigation of features in dynamical states, namely the relaxation dynamics of classical correlations, and the rate of entropy growth in the transient regime in Secs. <ref> and  <ref> respectively while we examine the decay of mutual information with distance between the spins  in the steady state  in Sec. <ref>. The results are summarized  in Sec. <ref>.



§ EMERGENT GAPPED PHASE WITH STRONG DM INTERACTIONS: PHASE DIAGRAM



The  critical lines  for the long-range interacting XY spin models in the presence of Coloumb-like interaction, described by a parameter α and for the nearest-neighbor transverse XY model with DM interaction  are known in the literature <cit.>. However,  the effects of long-range DM interactions has never been addressed before. In this situation, a possible query can be  – can the introduction of long-range DM interactions change the critical lines and phases of the LR Ising models? We answer this question affirmatively by first presenting the analytical prescription of this model for obtaining energy spectra and the eigenvectors, thereby leading to the modified phase-portrait of the extended long-range XY model <cit.> having long-range extended Dzyaloshinskii-Moriya interaction both in the thermodynamic limit and for finite systems.  In particular, we report the gapless chiral phase and its dependence on the fall-off rate, α. In addition, we identify appropriate order parameters that can reveal α-dependence in the modified magentic phase diagram. From the decay patterns of classical correlation including Landau parameters and quantum correlations in terms of quantum mutual information and block entanglement entropy, we again separate the gapless region from the gapped one in this model. 



 §.§ Energy spectrum of the long-range XY and DM interactions


The Hamiltonian, describing long-range extended XY model having long-range extended DM interaction in the presence of transverse magnetic field, reads as


    H   =   ∑_j=1^N∑_r=1^N/2 -J_r^'/𝒩 [1+γ/4σ_j^xℤ_r^zσ_j+r^x+1-γ/4σ_j^yℤ_r^zσ_j+r^y
          +D'/4(σ_j^xℤ_r^zσ_j+r^y-σ_j^yℤ_r^zσ_j+r^x) ]-h'/2∑_j=1^Nσ_j^z,

where ℤ_r^z = ∏_l=j+1^j+r-1σ_l^z, with ℤ_1^z=𝕀
and σ^k(k=1,2,3) is the Pauli matrix, J_r'=J/r^α with α being the strength of power-law decay of the model, γ is the anisotropy parameter, D is the strength of the DM interaction, 𝒩=∑_r=1^N/21/r^α is called the Kac-scaling factor <cit.> which ensures extensivity of the energy in the case of finite-size systems α<1 in the finite-size limit and h' is the strength of the external magnetic field. To make the analysis dimensionless, we redefine the magnetic field as h=h'/J and DM interaction strength as  D=D'/J. We consider the periodic boundary condition (PBC), i.e., σ_N+1≡σ_1.  This model can be solved analytically by mapping spins into free fermions under the following Jordan-Wigner transformation <cit.>:

    σ^x_n    =  ( c_n + c_n^†)
     ∏_m<n(1-2 c^†_m c_m) 
    σ^y_n    =i( c_n - c_n^†)
     ∏_m<n(1-2 c^†_m c_m) 
    and  σ^z_n   =1-2 c^†_n  c_n,

where c_m^†(c_m) is creation (annihilation) operator of spinless fermions and they follow fermionic commutator algebra. The corresponding free fermionic version of the Hamiltonian takes the form 

    H   =   ∑_n∑_rJ_r/2 ((1+iD)c_n^† c_n+r+γ c_n^† c_n+r^† + h.c )
          +h(c_n^† c_n-1/2),

where J_r=1/r^α. One can observe that the Hamiltonian which is quadratic in fermionic operators have long-range interaction both in hopping and superconducting terms. In addition, an extra hoping term emerges due to the DM interaction which also induces a complex phase. It is worthwhile to mention here that in the absence of DM interaction for γ=0, the model is 𝕌(1) symmetric while it is ℤ_2 symmetric with γ 0, D=0. The introduction of  DM interaction leads to a competition between these two symmetries.   In the short-range (SR) models, it is known that the conformal sector changes due to the shift of the symmetry sector <cit.>, responsible for a non-trivial phenomenon. 


Due to the presence of translational invariance in the system, momentum is a good quantum number. Hence, we perform a Fourier transform of the form, c_n=1/√(N)∑ e^-iϕ_kc_k where ϕ_k=2π k/N ∀ k∈[-N/2,N/2] and the corresponding Hamiltonian in the momentum basis can be written as H=k⊕ H_k with 

    H_k    =   ∑_k>0- [∑_r(Re(J_k^α)+DIm(J_k^α))c_k^† c_k
           +(Re(J_k^α)-DIm(J_k^α))c_-k^† c_-k
          -iγIm(J_k^α) (c_k^† c_-k^†- c_-k c_k)]
           +h(c_k^† c_k+c_-k^† c_-k)-h,

and J_k^α = ∑_r = 1^N/2J_r e^iϕ_kr. This Hamiltonian can be diagonalized by applying the Bogoliubov transformation,

    [    c_k; c_-k^† ]=[    u_k -v_k^*;    v_k  u_k^* ][    τ_k; τ_-k^† ],

where u_k=cosθ_k, v_k=isinθ_k and θ_k is called Bogoliubov angle, given by cosθ_k=-h+Re(J_k^α)-λ/√(2(λ^2-(Re(J_k^α)-h)^2)) and λ=√((h-Re(J_k^α))^2+(γ Im(J_k^α))^2). The eigenvalue corresponding to each momentum, k, is given as

    ϵ_± k=-D Im(J_k^α)  ±λ.

Note first that for D 0, although the Bogoliubov basis, (u_k, v_k)^T, are independent of DM interactions,  ϵ_kϵ_-k,  making the analysis non-trivial.

Further, when D=0, this model shows criticality at h=1 and at h=-1+2^1-α for α>1  corresponding to the both ends of the Brillouin zone, i.e., ϕ_k=0 and ϕ_k=π respectively.  In the case of α<1, the energy of the system diverges in the thermodynamic limit, hence, we consider large but finite system size <cit.>. With the introduction of DM, new gapless region emerges, which changes with α for D>γ. Such a region is known for the  nearest-neighbour XY model when D>γ <cit.>  and  gapless to gapped transition occurs at D= γ. However, we now establish that even when D>γ, long-range interactions can induce a gapped phase instead of a gapless one, which implies the modification of the critical lines in presence of LR interactions.






















 §.§ The gapless chiral phase depending on LR interactions


Apart from quantum phase transitions at zero temperature in which the energy gap vanishes, the LR Ising models possess another three distinctive regimes with respect to power-law fall-off rate α, namely, non-local (0 < α < 1), quasi-local (1 < α < 2) and local (α > 2) according to different scaling law of correlation length <cit.>. It is important to note here that in the absence of DM interaction, gap-closing never occurs by tuning the parameter α for h>0 except h_c=1. Further, we know that α≫ 2, the system with NN interactions becomes gapless for D>γ depending on h and the chiral phase emerges along with paramagnetic and ferromagnetic phases. Due to the long-range DM interactions, a competition between the range of interactions, α and the strength of DM interactions surfaces which becomes responsible for the change in condition for the gapless  to gapped transition. More specifically, for moderate values of α(<2), D>γ does not guarantee chiral gapless phase. We find the following in case of LR model. 




    For a given value of the magnetic field h (both in h≥ 0 and h<0), a gapped to gapless transition occurs when  D>γ depending upon the value of α instead of D=γ, known for the NN model.



    The proof is done by analyzing the dispersion relation in Eq. (<ref>). A system is said to be

gapped when the expression, Δ <cit.>, defined as

    Δ=max{min{ϵ_k},0}>0;  ∀ϕ_k∈ [-π, π],
 
otherwise, the spectrum is gapless (with Δ = 0). The exact point, where the gap-closing takes place, depends on the parameters of the Hamiltonian, and the corresponding momentum, k_F, called the Fermi point, can be obtained as a solution of ϵ_kk=0. However, due to the presence of long-range order, it is cumbersome to find the Fermi point analytically. By differentiating numerically, we observe that the gapless phase never occurs when D<γ while for D>γ, the gapless phase occurs although it is not ubiquitous, and changes with the variation of h,D,γ and α. In particular, when h<0, with the decrease of α, the system is gapped even when D>γ and two distinctive gapped regions emerge when 1≤α<2 and 0<α<1 which depend on the strength of D, thereby inducing (destroying) gapped (gapless) regions. However, for α>2, D > γ guarantees gapless phases and the gapless to gapped transition occurs at D=γ. On the other hand, we find, that when 1≤α<2, strong DM interactions are required to obtain the gapless phase and the desired asymmetric interaction strength for obtaining the gapless phase increases with the decrease of α (see Fig. <ref>). Note, further, that the maximum D necessary for the gapless phase does not happen exactly at α=1, but at a nearby point which can be due to finite size analysis.  



It is now evident that the interplay of α and D along with γ and h can change the critical lines of the phase diagram. Therefore, we now examine the phases in the (h,γ)- and (α, D)-planes.






α-dependent chiral phase. In the nearest-neighbor  XY model with DM interactions, the gapless phase gives rise to a chiral order that measures local current flow between the nearest neighbor sites. It can be established, by computing <cit.> the order parameter,

    𝒞ℋ= 1/4N∑_i=1^N σ^x_iσ^y_i+1-σ^y_iσ^x_i+1

between any two neighboring sites, i and i+1 in the ground state which describes the helical alignment of spins or chiral order in the z-direction. 


In the nearest-neighbor case, i.e., for the NN XY model with D > γ and suitable h values, it was shown that 𝒞ℋ > 0 when Δ = 0. 

In the LR model, we find that for a fixed h, α <2 and D > γ, whenever Δ = 0, i.e., the system is gapless, it possesses chiral order with order parameter 𝒞ℋ > 0 (as shown in Fig. <ref>). Interestingly, however, there exists region in which for D> γ, Δ>0 (as shown in the above proposition), and the corresponding chiral order parameter vanishes, i.e., 𝒞ℋ =0. 








Ferromagnetic and paramagnetic phases. We now calculate the long-range magnetic order parameter, C^xx_N/2 =  σ^x_1σ^x_N/2. When C^xx_N/2 0, the system is said to be in the ferromagnetic-x phase (FM_x) while paramagnetic when it vanishes. On one hand, we show that the FM_x phase occurs when the system is gapped and 0<h<h_c^1=1, 1<α<2, and D < γ (see Fig. <ref>). On the other hand, when h is negative, the system becomes ferromagnetic, i.e., C^xx_N/2 > 0  when h > h_c^2=-1+2^1-α with 1<α<2. When the parameters  are neither chiral nor ferromagnetic, the system is in a paramagnetic phase.




  













 §.§ Constant decaying exponent of correlations with DM interactions



From the above analysis, it is evident that the transition from the chiral gapless phase to the other magnetic gapped phase heavily depends on the fall-off rate α. Let us now investigate how the scaling of classical correlations and quantum mutual information between two spins (a measure of total correlations containing both quantum and classical correlations components) <cit.> can recognize the α-dependent phases discussed above. 

The classical correlation (CC) between two sites, separated by a distance R,  can be represented as C^xx_i,i+R = ⟨σ_i^x σ_i+R^x ⟩ while the quantum mutual information is defined as 
ℐ_i:i+R=S(ρ_i)+S(ρ_i+R)-S(ρ_i, i+R), where S(σ) = -(σlog_2σ) indicates the von-Neumann entropy of the state σ, ρ_i and ρ_i+R are the reduced density matrices of the joint state ρ_i,i+R. As we impose periodic boundary condition, the site index i can be ignored and both the classical and total correlation can be studied as C^xx_R and ℐ_R, where R = {1,2,…N/2}. In typical one-dimensional (1D) quantum spin models, the CC decays exponentially when it is away from criticality, whereas, at the criticality, the polynomial decay with R is observed. However, the extended Ising model deviates from this norm by showing different scaling laws near the critical points and away from it <cit.>. Moreover, in the long-range Kitaev chain (see Eq. (<ref>) with D = 0), quantum mutual information (QMI) persists between two distant segments of the chain, provided the system is in the non-local regime, i.e., α < 1, thereby  predicting the correlation between two distant regions of the system  <cit.>.

In our study, we expect to have a notable scaling law due to the presence of long-range and additional DM interactions which is indeed the case. Firstly, we observe that the trends of C_R^xx and ℐ_R with R is qualitatively different when the system is in the chiral phase and when it is not. Secondly, a counter-intuitive observation is that if one tunes the parameters 

 D, γ, h and α in such a way that 𝒞ℋ 0, the decay exponents of mutual information and classical correlation become constant as depicted in Fig. <ref>. Specifically, we observe that irrespective of the details of the power-law exponent and the strength of the magnetic field, the decay exponent of CC and QMI falls off as 

    ℐ_R    ∝ R^-1  and
    
        C^xx_R    ∝ R^-0.45,

which are independent of α provided the ground state belongs to the gapless chiral region (see Fig. <ref> and Table <ref>). This behavior of ℐ_R and C^xx_R highlights a more complex and highly correlated patterns between α and D within the gapless region. It indicates that although the range of interactions can control the critical lines in the (D,γ)-plane, differentiating gapless and gapped phases, the decay rate of CC and QMI in the gapless phase remains unaltered  and the chiral phase favors the sharing of both quantum and classical correlations between distant sites. More importantly, such α-dependent scaling behavior holds both for quantum and classical correlations (see Fig. <ref>).  

On the other hand,  when the system is in the non-chiral (which we refer to as achiral) phase, the ground state displays the decay exponent of both classical and quantum correlations which vary with α, i.e., depending on the internal descriptions of the model (see Fig. <ref> and Table <ref>). 

Through these observations, we elucidate the nuanced role of long-range DM interactions in modulating correlation properties across both chiral and achiral regions leading to the following proposition.


    In the chiral region of the LR XY model with DM interactions, the scaling exponents of CC and QMI are constant (independent of the fall-off rate, α) while their exponents depend on α in the achiral region.











 §.§ Amendment in scaling of entanglement entropy and central charge owing to long-range DM interactions



In a many-body system, an insight of universality classes can be obtained from the generic features of the ground state. A prominent physical quantity that serves the purpose is the block entanglement entropy  of block-size l <cit.>. In particular, after partitioning the N-party ground state into two blocks, containing l and N-l sites (with l ≪ N), we compute the block entanglement of the ground state as S_l = S(ρ_l) with ρ_l being the reduced density matrix of the N-party ground state.  The l-block entropy can be calculated from the two-point correlation functions between modes, m and n, given as

    C_mn=⟨Ψ|c_m^† c_n | Ψ⟩,    F_mn=⟨Ψ|c_m^† c_n^† |Ψ⟩,

where |Ψ⟩ is the ground state and the corresponding correlation matrix, consisting of correlation functions, can be written as

    ℂ_l=[ 𝕀-C   F; F^†   C ].

If λ_ls are the eigenvalues of ℂ_l, the von-Neumann entropy of |Ψ⟩ in the bipartition l:N-l is given as S_l=-∑_l=1^2Nλ_llog_2λ_l.

If a system is gapped which typically happens away from the criticality and when the range of interactions is relatively local, i.e., the norm of the Hamiltonian does not increase as the system size increases, S_l follows the area law, which implies S_l^NC∝ l^d-1, where d is the spatial dimension of the system and NC stands for “not at criticality" while it deviates from the area law at criticality <cit.>. Hence from the scaling behavior of S_l, one can detect the transition from a gapped to a gapless phase which is interesting since a simple scalar quantity can describe the essential properties of the Hamiltonian instead of a complete microscopic description. 

On the other hand, when the system becomes gapless, representing the critical point, e.g., in the transverse NN XY model,  S_l scales logarithmically, i.e., for a translationally invariant spin chain, at critical points, S_l^C= c/3log_2[N/πsinπ l/N]+a <cit.>, where c is the conformal or effective central charge in the conformal field theory and a is a non-universal constant. Further, it was shown that the central charge carries the signatures of an underlying symmetry. For example, c takes value 1/2 for the transverse-field Ising-like Hamiltonian at the critical point for which the Hamiltonian adheres to the ℤ_2-symmetry, while with the addition of asymmetric nearest-neighbor DM interaction, the Hamiltonian becomes  gapless region having c=1 and gapped with c=1/2. Hence EE turns out to be a powerful tool to separate a gapped phase from a gapless one.


It was also recently shown that if one considers long-range interactions instead of short-range ones <cit.>, where magnetic criticality depends on the value of α, EE depends upon α and the free-fermionic version of long-range Ising spin shows fractal entanglement apart from the volume and area law <cit.> and the scaling of entropy at critical points may not follow S_l^C. In this case, we can define an effective conformal charge, c_eff, to describe the universal properties of the system.

We are interested in finding how the scaling of S_l in the LR Ising Hamiltonian gets altered in the presence of DM interactions. Specifically, we will show the revision of a central charge c_eff, which occurs in S_l^C. Before presenting the effects of DM interaction on the central charge, we observe that at criticality, i.e., h_c^1 = 1 of the extended Ising model, c_eff monotonically increases when 1 < α < 2 and saturates to 1/2 at α≥ 2, (which is the scaling known for the NN model) as shown in Fig. <ref>(a). When α < 2, c_eff varies non-linearly with α which points out that due to the presence of long-range interaction, conformal symmetry breaks down. 

On the other hand, the introduction of long-range  DM interaction makes more modifications in c_eff –  (i) the saturation value changes to 1 when α→α_c = 2, with α_c being the point which mimics the NN transverse Ising model with NN DM interactions; (ii) for α<α_c, the conformal symmetry breaks down whereas when α > α_c,  c_eff→ 1 (see Fig. <ref>(b)). 
Our study reveals that even with the introduction of long-range asymmetric interaction, the conformal symmetry is not restored for all the values of α although c_eff indicating the conformal invariance begins for smaller values of α_c<2 which is found in the absence of DM interactions. This behavior can again be attributed to the fact that when the DM interaction is present, the system becomes gapless upon varying the power-law decaying factor α.











§ DECAY OF DYNAMICAL CORRELATION: UNCOVERING CHIRAL AND ACHIRAL PHASES



Let us now study the trends of physical quantities including quantum correlations in the time-evolved state in the transient and the steady state regimes, starting from the product state or the ground state which evolves according to the LR Hamiltonian with DM interactions. Our aim is to demonstrate that the scaling of correlations with time or with distant sites can indicate the transition from the chiral phase to the achiral ones. 



 §.§ Relaxation behavior of dynamical correlations






Let us first present the method to capture the relaxation of dynamical correlation <cit.> which is shown to comprehend distinct phases in equilibrium. The initial state and the evolution of the system in the momentum basis can be represented by the Bogoliubov–de Gennes (BdG) equation, given as


    ψ_k(0)=[u_k(0), v_k(0)]^T  and  ψ_k(t)=e^-iH_ktψ_k(0).

The corresponding dynamical correlation (DC) is defined as C_mn(t)=c_m^† c_nψ(t) which is a fermionic correlation between two modes. The relaxation of this correlation can be measured by taking the difference between correlation in arbitrary time and time in which the system reaches the steady state, given by δ C_mn(t)=C_mn(t)-C_mn(∞), describing the decay of the correlation function with time. Let the Bogoliubov angle of the initial and quenched Hamiltonian be η_k and η̃_k respectively and the difference between the angels is given as α_k=η_k-η̃_k. The relaxation of DC takes the form <cit.>

    δ C_mn(t) = 
       k∈gapped∫sin 2η̃_ksin 2α_k cos(ϵ_k+ϵ_-k)tcos[k(n-m)]= 
        
       k∉gapless∫sin 2η̃_ksin 2α_k cos(ϵ_k+ϵ_-k)tcos[k(n-m)],

where the initial state belongs to the gapped  and gapless regions in Eqs. (<ref>) and (<ref>) respectively. In our case, we have chosen m-n=1, but our result holds for any arbitrary m  and n.

When δ C_mn(t) ∼ t^-χ, χ indicates the dynamical relaxation exponent characterized by the distinctive values of the system parameters.

In the case of the nearest-neighbor XY model, it was shown that the exponent depends on the phase being commensurate or incommensurate. Specifically, in the commensurate phase, the derivative of the dispersion relation with respect to the momentum vanishes at the extreme ends of the Brillouin zone, while in the incommensurate phase, the derivative vanishes inside the Brillouin zone. If the initial state is a product state, i.e., the ground state of the model when the magnetic field goes to infinity and is then quenched with the Hamiltonian corresponding to the commensurate phase, we find χ = 3/2 while χ = 1/2 when quenched with the incommensurate phase <cit.>. 

In the NN XY model with DM interaction, if the initial state belongs to the gapless phase, the exponent changes to χ = 1 if the quenching Hamiltonian is in the commensurate phase, which indicates the presence of DM interaction. In the case of quenching by a Hamiltonian in the incommensurate phase, the exponent is either χ = 1/2 or χ = 1 depending on the parameters of the initial state <cit.>. On the other hand, considering the LR Kitaev chain, it was found that the scaling law behaves similarly to the SR ones when the initial and post-quench Hamiltonian are non-critical although the exponent changes with the parameters chosen from an equilibrium phase transition <cit.>. 

The model considered in this work possesses both LR DM interactions and LR interactions in the xy-plane and hence the interplay between α and D as mentioned in the static scenario can influence the scaling of δ C_mn(t), resulting to different χ. For investigations, possible three situations arise in the non-local regime (α < 1) - (ni) both pre- and post-quench Hamiltonian are in the chiral phase; (nii) only pre-quench Hamiltonian belongs to the chiral phase; (niii) only post-quench Hamiltonian is chosen from the chiral phase. All three cases can also be considered when α is chosen from a quasi-local regime, i.e., when α > 1 which we refer to as (qi) - (qiii). Eg., we find that χ∼ 1.5 for the case of (niii) and when (qii) χ∼ 0.3 (see Fig. <ref>). 

This shows that the exponent in DC can distinguish between the gapped and the gapless phases  of the post- and pre-quench Hamiltonians  (see Fig. <ref>). In addition, the robustness of correlation can be defined as a quantity 1/χ such that if the value of χ is small, the system does not reach the steady state correlation value for a significantly long time. This reinforces the above result that the state from the gapless phase contains a strong correlation both in terms of space and time. 



 §.§ Slow decay of total correlation and two-point correlation with LR DM interaction



We now focus on six scenarios ((ni) - (niii) and (qi) - (qiii)) to study the decay pattern of total correlation and classical correlation with R in a steady state limit, represented by ℐ_R(∞) and 𝒞^xx_R(∞) respectively. Even though the evolution is unitary, as we are studying the properties of a subsystem, a steady state can be reached in the dynamics. When the quenching Hamiltonian is in the gapped phase ((nii)), the steady state maintains both classical and quantum correlations between distant spins, and  persists even at large distance R (see Figs. <ref>(b), (c), (f) and (g)). Conversely, a gapless quenching Hamiltonian ((ni) and (niii)) results in a steady state where classical correlations are present only between nearby spins as depicted in Figs. <ref> (a), (d), (e) and (h). Notice that the decaying natures of ℐ_ℛ(∞) and C^xx_R(∞) are independent of the initial state. Remarkably, the behavior of total and classical correlations remains similar when the distance between the spins is small. 
















 §.§ Effects of long-range in dynamics of EE



In a final attempt to distinguish the gapless and gapped phases under dynamics, we investigate the
growth of entropy with time. The initial states are prepared as the ground states of gapped as well as from the gapless regions and EE, S_l(ρ(t)) ≡ S_t for a fixed l is computed after quenching the system to the gapped or gapless regions by changing the parameters α for a given γ, h and D. Let us denote the initial and final fall-off rates be α_i and α_q respectively. Firstly, we notice that the rate of growth is shown to be linear in time in the case of the XY model with NN <cit.> and long-range interacting model <cit.>, with the initial state being a product state. The trends of S_t with time depend on both α_i and α_q, thereby indicating its dependence on pre- and post-quench Hamiltonian. 

  Secondly, the rate of increment of EE with time is faster if the initial state is in a gapless region and the EE value gets saturated around time t ∼ l/2, as observed in the case of the nearest-neighbor Ising model quenched from a non-critical regime to a critical one <cit.>. However, if the initial and final states are chosen from a gapless region, the increment rate is slow and the saturation is reached faster around t=l/2 compared to a gapped case (see Fig. <ref>(d) and (e)). Also, the scaling of entropy increment until it reaches saturation can be approximated as

    S_t^±∝ a_1t/α_i ±α_q+a_2,

where a_1, and a_2 are constants (see Fig. <ref>).

When both the initial and the final Hamiltonian are gapped or one of the pre- or post-quench Hamiltonian is gapped, the saturation of EE requires much larger time than that obtained with  the Hamiltonians corresponding to the initial and final states being gapless. 








§ CONCLUSION



Quantum long-range systems are intriguing due to their non-local properties, often displaying counter-intuitive phenomena compared to a short-range Hamiltonian. These models are pervasive in physically realizable systems such as trapped ions and atomic, molecular, and optical setups.   Therefore, comprehending their universal features is essential for the advancement of quantum technologies and condensed matter physics.





We found that the long-range extended Dzyaloshinskii-Moriya (DM) interactions with the extended XY model have unique consequences on the critical behavior of the system with a variation of the fall-off rate, α, in which the range of interactions between the sites decays. Specifically,  even with a comparable or more DM interactions than the anisotropy parameter, we proved that the system remains gapped depending on α, which  varies with the direction of the transverse magnetic field although the gapless phase still possesses chiral order.  
By analyzing classical and total correlations between two arbitrary sites of the ground state in the presence of DM interactions, we observed that both  of them decay with the distance between the sites and the decay exponent is independent of α in the gapless region while it depends on the system parameters in the gapped regime and  decays faster in this case than the gapless zone.  Additionally, we exhibited   the departures from the universal trend in the block entanglement entropy in presence of long-range DM interactions which could successfully capture the interplay between long-range and DM interactions.





This study investigated how dynamical  correlations change in time when the energy spectra of the initial and the evolving Hamiltonian are gapped and gapless, driven by  α for a fixed strengths of DM interactions, and magnetic fields. 

The scaling of both classical correlations and mutual information in the  steady states reveals that the sharing of correlations is favorable when the evolving Hamiltonian is gapped.  Moreover, we found that the growth of entanglement entropy with time  is faster in the gapless regime than in the gapped phase and it saturates quickly when both the pre- and post-quench Hamiltonians are gapless.   On the one hand, these findings contribute to the understanding of fundamental aspects of quantum phase transitions, while on the other, sharability of correlations and entanglement  of the ground state as well as their dynamical patterns can be vital for establishing quantum links amongst quantum computers.



We acknowledge the support from the Interdisciplinary Cyber Physical Systems (ICPS) program of the Department of Science and Technology (DST), India, Grant No.: DST/ICPS/QuST/Theme- 1/2019/23. We  acknowledge the use of https://github.com/titaschanda/QIClibQIClib – a modern C++ library for general purpose quantum information processing and quantum computing (<https://titaschanda.github.io/QIClib>), and the cluster computing facility at the Harish-Chandra Research Institute. This research was supported in part by the INFOSYS scholarship for senior students.







 

§ CORRELATION FUNCTIONS




The two-point correlation functions are the fundamental constituents of our study. In the context of the ground state or the thermal state, the two-site correlation function between sites separated by a distance of R can usually be expressed as a determinant of an R × R Toeplitz matrix <cit.>.
However, as the Hamiltonian includes asymmetric interaction, the bipartite reduced state obtained from the N-party ground  state after tracing out N-2 parties includes the correlators like C^xy≡⟨σ^x_i σ^y_i+R⟩ = tr (σ^x ⊗σ^y ρ_R) and C^yx along with  C^ii (i=x,y,z).  Thus for the corresponding density matrix of the two distant parties of the spin chain and the time-evolved state, we need to deal directly with the Pfaffians in order to evaluate the correlations as a function of time as well as distance since there is a formation of chiral phase due to DM interaction. 

Using the Pfaffian formalism, we first write the spin correlation functions as 








    
    .
    [ C^lm_i,i+R = ⟨σ_i^l σ_i+R^m ⟩= c(l,m)pf  |  . I^lm_1,2                                                      …                                             I^lm_1,R-1                                                 J^lm_1                                                 F^lm_1                                               G^lm_1,2                                                    c .                                                      …                                               G^lm_1,r;                                                                                                             …                                                      …                                                      …                                                      …                                                      …                                                    c .                                                      …                                                      …;                                                                                                                                                         I^lm_R-2,R-1                                               J^lm_R-2                                               F^lm_R-2                                             G^lm_R-2,2                                                    c .                                                      …                                             G^lm_R-2,R;                                                                                                                                                                                                                    J^lm_R-1                                               F^lm_R-1                                             G^lm_R-1,2                                                    c .                                                      …                                             G^lm_R-1,R;                                                                                                                                                                                                                                                                               E^lm                                                 D^lm_2                                                    c .                                                      …                                                 D^lm_R;                                                                                                                                                                                                                                                                                                                                    K^lm_2                                                    c .                                                      …                                                 K^lm_R;                                                                                                                                                                                                                                                                                                                                                                                         H^lm_2,3                                                      …                                               H^lm_2,R;                                                                                                                                                                                                                                                                                                                                                                                                                                                       …                                                      …;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H^lm_R-1,R ] |,






where c(x,x)=c(y,y)=(-1)^R(R+1)/2, 

    I^xx_μ,ν      =  ⟨ A_l+μ(t) A_l+ν(t)⟩, 
    
    J^xx_μ,        =I^xx_μ,R, 
     
    H^xx_μ,ν      = ⟨ B_l+μ-1(t) B_l+ν-1(t)⟩, 
    
    K^xx_ν      = H^xx_1,ν, 
     
    G^xx_μ,ν      = ⟨ A_l+μ(t) B_l+ν-1(t)⟩, 
     
    F^xx_μ      =G^xx_μ,1,  
    
    E^xx      =G^xx_R,1,  
    
    D^xx_ν      =G^xx_R,ν,



    I^yy_μ,ν      =  ⟨ A_l+μ-1(t) A_l+ν-1(t)⟩, 
    
    J^yy_μ      =I^yy_μ,R, 
     
    H^yy_μ,ν      = ⟨ B_l+μ(t) B_l+ν(t)⟩, 
    
    K^yy_ν      = H^yy_1,ν, 
     
    G^yy_μ,ν      = ⟨ A_l+μ-1(t) B_l+ν(t)⟩, 
     
    F^yy_μ      =G^yy_μ,1,  
    
    E^yy      =G^yy_R,1,  
    
    D^yy_ν      =G^yy_R,ν,

with s(x,y)=s(y,x)= -i (-1)^R(R-1)/2, we have

    I^xy_μ,ν      =  ⟨ A_l+μ(t) A_l+ν(t)⟩, 
    
    G^xy_μ,ν      = ⟨ A_l+μ(t) B_l+ν(t)⟩, 
    
    J^xy_μ      = G^xy_μ,0, 
     
    F^xy_μ      =G^xy_μ,1,  
    
    H^xy_μ,ν      = ⟨ B_l+μ(t) B_l+ν(t)⟩, 
    
    E^xy      =H^xy_0,1,  
    
    D^xy_ν      = H^xy_0,ν, 
      
    K^xy_ν      =H^xy_1,ν,

and

    I^yx_μ,ν      =  ⟨ A_l+μ-1(t) A_l+ν-1(t)⟩, 
    
    G^yx_μ,ν      = ⟨ A_l+μ-1(t) B_l+ν-1(t)⟩, 
    
    J^yx_μ      = I^yx_μ,R, 
     
    F^yx_μ      =I^yx_μ,R+1,  
    
    E^yx      =I^yx_r,r+1,  
    
    D^yx_ν      = G^yx_R,ν,  
    
    K^yx_ν      =G^yx_R+1,ν, 
     
    H^yx_μ,ν      = ⟨ B_l+μ-1(t) B_l+ν-1(t)⟩.


Each of the elements in the Pfaffian can be constructed from the expectation values of one of the operators, A_l A_l+R, B_l B_l+R and A_l B_l+R with

A_i=c_i^†+c_i,   B_i=c_i^†-c_i.





When recasted in the same Fourier basis as taken while diagonalization, the operators for the k^th momentum have the matrix form

    (A_l A_l+R)_k =  [[          cos(kR)        -isin(kR)                0                0;        -isin(kR)          cos(kR)                0                0;                0                0 cos(kR)-isin(kR)                0;                0                0                0 cos(kR)+isin(kR) ]],


    (B_l B_l+R)_k =   [[          -cos(kR)           sin(kR)                 0                 0;          -sin(kR)          -cos(kR)                 0                 0;                 0                 0 -cos(kR)+isin(kR)                 0;                 0                 0                 0 -cos(kR)-isin(kR) ]],



    (A_l B_l+R)_k =   [[   cos(kR)  isin(kR)         0         0; -isin(kR)  -cos(kR)         0         0;         0         0         0         0;         0         0         0         0 ]].
 
The single-site transverse magnetization in the same Fourier basis is given by

    σ_z^k = [[ -1  0  0  0;  0  1  0  0;  0  0  0  0;  0  0  0  0 ]].






The corresponding expectation value of the operators for each value of k, say O^k, with respect to the time-evolved  state is computed as ⟨ O ⟩ = ∑_k=1^N/2Tr(ρ_β^k(t) O^k).

Once we find all the single-site magnetizations m_j^α ∀ α={x,y,z} at j=i and j=i+R and all possible two-site correlation functions C_i,i+R^l,m ∀  l,m={x,y,z}  from the Pfaffians described above, we can construct the two-site reduced density matrix between sites i and I+R as we know any two party density matrix for the given Hamiltonian is given as


    ρ_ij=1/4 (𝕀_4 + m^z(σ^z_j+σ^z_i) + ∑_k,l C^kl_ijσ^k_i ⊗σ^l_j),

where k,l∈{x,y}.








 Paper Title: Substrate stabilization of Jahn--Teller distortion in a single molecule.





moritz.frankerl@dipc.org
Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany
Donostia International Physics Center (DIPC), E-20018 Donostia-San Sebastián, Spain

Institute for Experimental Physics, University of Regensburg, 93040 Regensburg, Germany
Department of Physical Chemistry, University of Innsbruck, 6020 Innsbruck, Austria

Donostia International Physics Center (DIPC), E-20018 Donostia-San Sebastián, Spain
IKERBASQUE, Basque Foundation for Science, E-48013, Bilbao, Spain

jascha.repp@ur.de
Institute for Experimental Physics, University of Regensburg, 93040 Regensburg, Germany

andrea.donarini@ur.de
Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany


§ ABSTRACT 
    Charge-state transitions of a single Cu-phthalocyanine molecule adsorbed on an insulating layer of NaCl on Cu(111) are probed by means of alternate charging scanning tunneling microscopy. Real-space imaging of the electronic transitions reveals the Jahn–Teller distortion occurring upon formation of the first and second anionic charge states. The experimental findings are rationalized by a theoretical many-body model which highlights the crucial role played by the substrate. The latter enhances and stabilizes the intrinsic Jahn–Teller distortion of the negatively charged molecule hosting a degenerate pair of single-particle frontier orbitals. Consequently, two excess electrons are found to occupy, in the ground state, the same localized orbital, despite a larger Coulomb repulsion than the one for the competing delocalized electronic configuration. Control over the charging sequence by varying the applied bias voltage is also predicted.        


Substrate stabilization of Jahn–Teller distortion in a single molecule
    Andrea Donarini
    August 5, 2024
======================================================================



Introduction Scanning tunneling microscopy (STM) is an excellent tool for investigating the electronic structure of single molecules on surfaces. By combining atomic-scale imaging with spectroscopy of electronic states, STM enables a comprehensive understanding of molecular-scale systems. Ultrathin insulating films of 2-3 monolayers (ML) have been used to separate the molecule from the metallic substrate and thereby inhibit their electronic hybridization, providing access to the (near-) pristine electronic structure of the molecule <cit.>. This enabled the observation of vibrational side bands <cit.>, coherent electron-nuclear coupling <cit.>, many-body correlations <cit.>, charge-state lifetimes <cit.>, and coherent spin control <cit.> on a single molecule. Furthermore, this facilitated the demonstration of single-molecule fluorescence <cit.> permitting vibronic spectroscopy with submolecular resolution <cit.>, single-molecule electrofluorochromism <cit.>, energy transfer in molecular dimers <cit.>, Anti-Kasha emissions <cit.>, switching of the excitonic state of a single molecule <cit.>, and bistability between π-diradical open- and closed-shell states <cit.>.

By employing thicker insulating films (>20 ML), electron exchange with the substrate can be completely inhibited, enabling the stabilization of multiple charge states in  single-molecule investigations <cit.>. This approach requires atomic force microscopy (AFM) for the detection of electron exchange with the tip. It enabled probing and quantifying of single-electron transfer between molecules <cit.>, the reorganization energy upon single-molecule charging <cit.>, excited molecular states <cit.>, as well as their lifetime <cit.>.
Further, based on this approach the imaging of charge-state transitions in single molecules with angstrom resolution <cit.> has been developed. This technique, dubbed single-electron alternate charging scanning tunneling microscopy (AC-STM), is based on driving a single electron to tunnel between the conductive AFM tip and a single molecule deposited on a non-conductive substrate. Charging events are controlled by short voltage pulses (V_ac) of opposite polarity that are superimposed on the DC voltage (V_ dc) across the AFM junction, inducing alternating charging of the molecule. While several charge state transitions can be accessed by gating through V_ dc, the polarity of the voltage pulses of V_ac provides selectivity in probing a specific direction of the charge-state transition <cit.>.


 

Polar insulators strongly stabilize excess charges in adsorbed atoms and molecules <cit.>, facilitating orbital localization through self-trapping of polarons <cit.>, revealing that the insulator does not merely electronically isolate the adsorbed molecule, but plays an active role in determining its electronic properties.  Whereas the reorganization energy is a well studied manifestation of this effect, little is known about how the large polarizability of the environment influences other electronic properties, as, for example, the occupation sequence of the molecular orbitals.

Here we study the electronic states in a charged  Cu-phthalocyanine (CuPc) molecule as a model system that is known to exhibit a Jahn–Teller (JT) distortion. We employ AC-STM imaging to reveal the energetic sequence of orbital occupation for the JT distorted system. Injection of a second electron results in the occupation of the same formerly-degenerate orbital. Our theoretical analysis reveals that this sequence is not merely a molecular property but it is critically governed by the polarizable NaCl substrate, strongly enhancing the JT splitting. Moreover, we predict that the delocalized configuration is accessible at higher bias voltages where it dominates the electronic transition rate due to the state multiplicity.


Experiments We employ AC-STM [<ref>(a)], cf. SI for details, to image the spatial distribution of charge-state transitions for the neutral, anionic and dianionic CuPc species. The map of the 0 → 1^- transition [<ref>(c)] exhibits a 4-fold rotational symmetry, similar to the anionic resonance observed by STM on ultrathin NaCl layers (2 ML)  <cit.>. Toggling of the molecule yields, on average, the superposition of two images rotated of about 32 degrees with respect to each other <cit.>). As reported previously <cit.>, for the reverse transition (1^- → 0)[<ref>(d)] the symmetry reduces from 4- to 2-fold due to the JT distortion, which favors the occupation of one single LUMO <cit.>. 

In the dianionic ground state the second excess electron can either occupy the other (empty) LUMO or further populate the already occupied one. The actual preference results from a competition between JT distortion and Coulomb repulsion, the latter being stronger for two electrons occupying the same orbital.
The AC-STM images of the 1^- → 2^- and 2^- → 1^- transitions, displayed respectively in <ref>(e) and (f), reveal that the second excess electron is occupying the same orbital as the first one.

Model We model the molecule in the AC-STM with the Hamiltonian


    [] = [mol] + [tip] + [tun]+ [as] + [vib] + [e-ph] ,


where the minimal electronic description of the molecule (mol), metallic tip (tip) and tunnelling coupling (tun) is complemented by a small environmental asymmetry (as), the set of molecular and substrate vibrational modes (vib), and by their coupling to the electronic degrees of freedom (e-ph). We write the molecular Hamiltonian in the Fock space of the four frontier orbitals, which are the singly occupied molecular orbital (SOMO, S), the highest occupied molecular orbital (HOMO, H) and the two degenerate lowest unoccupied molecular orbitals (LUMOs, L_xz and L_yz) <cit.>.
The latter two are shown in <ref>(b).
The single-particle energies, ε_S = -12.0, ε_H = -11.7, ε_L_xz/L_yz = -10.7, are renormalized by a crystal field correction of Δ = 2.44 and all symmetry-allowed interaction matrix elements V_ijkl of the Coulomb interaction are retained. The tip is described as a non-interacting Fermi sea with harmonic lateral confinement <cit.> 
[tip] = ∑_kσ( ε_k + c V_b ) [kσ]c^†[kσ]c, where the tip states ε_k are shifted by the applied bias V_b subject to a lever arm c caused by the voltage drop in the insulating layer, [kσ]c destroys an electron in the tip with momentum k and spin σ. 
The tunneling is expressed as 
[tun] = ∑_ikσ t^tip_ikσ[kσ]c^†[iσ]d + H.c., only considering tunneling processes between tip and molecule since molecule-substrate tunneling is inhibited. The tunneling amplitudes t^tip_ikσ are modelled to be proportional to the overlap between the molecule and the tip wave functions and thereby contain the dependence on tip position. The tip wave functions are composed by a Gaussian component stemming from the lateral confinement and an evanescent part along the direction perpendicular to the molecular plane <cit.>.  We incorporate an environmental asymmetry in our model via [as] = -δ_ as ([xz]n-[yz]n), with δ_ as = 4k_BT = 2.




The vibronic Hamiltonian is a collection of harmonic oscillators [vib] = ∑_ζħω_ζ ([ζ]a^†[ζ]a + 1/2) containing molecular and salt modes. Based on symmetry considerations, the vibrational modes of the system fall into three categories, namely, (i) those that do not interact with charges in the molecule, (ii) those that affect both LUMOs the same way, and (iii) those that interact differently with each of the LUMOs. The C_4v symmetry of the combined salt and molecular system facilitates this distinction of vibronic modes. We do not consider the modes falling into the first category. The modes constituting the second category lead to a stabilization of charge states in the molecule. These are the ones transforming as the A_1 irreducible representation and we call them symmetric (S) modes from now on since they do not reduce the symmetry of the system. They are only sensitive to the global charge on the LUMOs, thus


    [e-ph]^ S = ∑_α∈{A_1} g_α ([xz]n  + [yz]n) ( [α]a^† + [α]a) .


The modes in the third category affect the charge distribution within a specific charge state. We refer to them as antisymmetric (AS) modes  since they lower the symmetry of the molecule.
They belong to the B_1 and B_2 irreducible representation. However, due to energetic considerations (cf. SI) we completely neglect the B_2 modes. The antisymmetric (B_1) modes couple to the imbalance between the populations of the  L_xz and L_yz orbitals, i.e.


    [e-ph]^ AS = ∑_β∈{B_1} g_β ([xz]n - [yz]n)( [β]a^† + [β]a) .


We calculated the molecular mode energies as well as the corresponding electron-phonon couplings g_α/β from first principles using  <cit.>, thus obtaining 14 symmetric (A_1)  and 14 antisymmetric (B_1) modes. The coupled salt modes can be combined to obtain just a single symmetric (A_1) and a single antisymmetric (B_1) mode. We take the corresponding electron-phonon couplings as free parameters in our theory and set them to g^salt_A_1 = 68 and g^salt_B_1 = 40, assuming for the energy the one of the transversal optical phonon mode ħω_TO = 20 <cit.>. A microscopic derivation of this minimal model yielding couplings constants of comparable strength is given in the SI. We show a comparison between one symmetric molecular mode and the symmetric salt mode in <ref>(a) and (c). The antisymmetric case is displayed in <ref>(b) and (d).

The influence of the symmetric modes on the many-body spectrum is readily obtained by a canonical Lang-Firsov transformation, leading to a renormalization of the single-particle energies ϵ̃_i = ϵ_i - ∑_αg_α^2/ħω_α as well as of the direct Coulomb interaction terms Ṽ_ijji = V_ijji - 2∑_αg_α^2/ħω_α and i,j ∈{ L_xz, L_yz}. Thus, the energy renormalization induced by the symmetric salt mode explains the stabilization of charged molecules and atoms adsorbed on polar dielectrics due to the Franck-Condon blockade <cit.>.     

For antisymmetric modes, the exchange and pair hopping between LUMOs shows an intricate interplay with the electron-phonon coupling. In this case, the Lang-Firsov transformation does not eliminate the coupling between the electronic and the mechanical degrees of freedom, which persists in the transformed exchange and pair-hopping terms. We identify, instead, a single JT active mode and resort for our analysis into a semi-classical adiabatic approximation. To this end, we introduce the canonical displacement operators

    [m]Q = ∑_β∈ B_1√(ħω_β) A_mβ( [β]a^†+[β]a),

where the first row of the transformation matrix reads A_1β = 1/g√(g_β^2/ħω_β) and yields the JT active mode [AS]Q:= [1]Q. The JT distortion is thus described by the operator 

    [JT](Q_ AS) = Q_ AS^2/4 +  g ([xz]n -  [yz]n) Q_ AS

being Q_ AS the classical displacement of the JT active mode, with the effective coupling g= √(∑_βg_β^2/ħω_β). 
On this basis, we simulate the experimentally investigated many-body eigenstates and energies by diagonalizing the effective Hamiltonian 


    [eff] = [ mol]H̃ + [as] + [JT] (Q_ AS) ,


along the coordinate Q_ AS of the JT-active antisymmetric mode, with [ mol]H̃ containing the Lang-Firsov renormalization due to the symmetric modes.




Electronic structure Some of the eigenenergies of [eff] are the adiabatic potential energy surfaces (APES) plotted in <ref>. The lowest parabola in <ref>(d) is the APES of the neutral state. This neutral ground state defines the reference configuration (Q_ AS = 0). The anionic states with an excess electron in one of the LUMO levels are orbitally degenerate for Q_ AS = 0. Their energy is lowered upon deforming along Q_ AS. Thus, the anionic ground state has one extra electron occupying the L_xz (L_yz) orbital for Q_ AS < 0 (Q_ AS > 0). The small environmental asymmetry δ_ as determines here that the global minimum has the excess electron in L_xz. The singlet-triplet splitting due to SOMO-LUMO exchange is not resolved on this energy scale. So far, the energy spectrum reflects the well-known JT effect. 
The richer structure of the dianionic APESs corresponds to two qualitatively different distributions of the excess electrons.
For one excess electron in each of the L_xz and L_yz orbitals, the coupling to the JT mode vanishes. Thus, the lowest energy of these states (with several different spin multiplicities) is found at Q_ AS = 0 in the upper part of <ref>(d). On the other hand, states with both excess electrons in the same former LUMO strongly couple to the JT-mode. Their ground state is found at Q_ AS≠ 0, analogously to the anionic configuration, but with an even larger JT deformation and reorganization energy. Pair hopping between the real-valued orbitals induces the avoided crossing at Q_ AS = 0, characteristic of the pseudo JT effect <cit.>.

Many-body transition rates A qualitative understanding of the measured charge-transitions is obtained by combining the electronic structure with few other constraints imposed by the experimental set-up. Specifically, the slow tip oscillations synchronized with the bias pulses allow for the system thermalization between consecutive tunnelling events and the large read-out tip-molecule distance yields single electron tunnelling. Moreover, the spectral broadening induced by the symmetric modes is much larger than the environmental asymmetry, which in turns exceeds the thermal broadening. Those premises imply, for example, that the neutral to anionic transition is 4-fold symmetric; it resembles the (incoherent) superposition of the L_xz and L_yz orbitals (with additional toggling), as it involves the spectrally unresolved states associated to both local minima with Q_ AS >0 and Q_ AS < 0 [<ref>(c) and <ref>(b)]. In contrast, environmental asymmetry ensures, for the thermalized anion, the predominant occupation of the state with Q_ AS<0; consequently the anionic to neutral transition acquires the 2-fold rotational symmetry of the  L_xz orbital [<ref>(d) and <ref>(c)]. Mere energetic arguments would predict also for the transition between the anion and the dianion the participation of the states corresponding to both the deformed local minima in the dianionic APES. Single electron tunnelling, though, forbids the contribution of the state with Q_ AS >0 and double occupation of the L_yz orbital, and ultimately yields a many-body transition resembling again L_xz [<ref>(e) and <ref>(g)]. The same appearance is expected for the transition from the dianionic to the anionic ground state [<ref>(h)] although quantum fluctuations in the configuration of the dianion do not completely exclude small contributions from the L_yz orbital, cf. <ref>(f) and the SI for the expression of the dianionic ground state as a function of the deformation coordinate.           

Electron tunnelling implies displacements of both the symmetric and antisymmetric modes. Their interplay is visualized in <ref>(e). The large electron-phonon coupling to the symmetric modes strongly suppresses the transitions between ground states along the effective symmetric coordinate Q_ S [black arrow in <ref>(e)]. On the contrary, w.r.t. the antisymmetric modes transitions between ground states are possible. This phenomenon, visualized for the neutral to anionic transition holds for all other considered transition between states with different charges.

The theoretical transition maps in <ref> and <ref> are calculated with Fermi's golden rule. The transition rate from an initial thermalized state towards a vibrationally excited state with neighbouring charge reads




    R^N→ N+1 = ∑_{n_α},f∑_i,j,σ∏_α∈{A_1}Γ_iσ,jσ(r_ tip)
    	P_λ_α({n_α})f^+(E_f - E_g - e  c V_b)
      ⟨ N+1,f  | [iσ]d^† | N,g ⟩⟨ N,g | [jσ]d | N+1,f ⟩ ,
    	
      
    	
       R^N+1→ N = ∑_{n_α},f∑_i,j,σ∏_α∈{A_1}Γ_iσ,jσ(r_ tip)
    	P_λ_α({n_α})f^-(E_g - E_f - e   c V_b)
      ⟨ N+1,g | [iσ]d^† | N,f ⟩⟨ N,f | [jσ]d | N+1,g ⟩ ,



where the single-particle rate matrix is given by Γ_iσ,jσ(r_tip) = 2π/ħ∑_k t_k,iσ^* t_k,jσδ(E_f - E_g - ϵ_k).
The symmetric modes' Franck-Condon factors yield the Poisson distributions P_λ_α({n_α}), which depend on the Huang-Rhys factors λ_α = g_α/ħω_α and {n_α} is the set of the final state's vibrational excitations w.r.t. all modes α∈{A_1}. Due to the large electron-phonon coupling λ_ salt≈ 5, the salt contribution converges to a Gaussian and it is the main source of spectral broadening <cit.>.   
The Fermi function f^+ (and related f^- = 1-f^+) with the tip temperature T = 6 and chemical potential μ=-4.65 is calculated at the difference between the molecular many-body energies shifted by the fraction c = 0.6 of sample bias V_b dropping between the molecule and the tip. The energy of the final state E_f = E_f^e + ∑_αħω_α n_α includes the electronic potential energy E_f^e (the local minima of the APES in <ref>) and the excitation energies of the symmetric modes, while E_g is the ground state energy of the initial state. Finally, <ref> includes the transition matrix elements between different charge states. To this end, we denote the initial state with |N,g⟩ (|N+1,g⟩) and the final state with |N+1,f⟩ (|N,f⟩). The sum over f extends to all possible states accessible from the designated initial state via a single electron tunnelling event. 
The tunneling rates in <ref> are depicted in <ref> as a function of the tip position and show a strong similarity to the experimental results, with also a remarkable agreement between all the experimental and theoretical bias voltages for the different transitions. 

We notice, in this respect, that the 1^- to 2^- transition denoted with a green arrow in <ref> is calculated, in accordance to the experiment, at the onset of the anion to dianion transition (V_b = 2.75). 




At higher biases we expect to open also transitions to the excited dianionic states, as sketched by the red arrows in <ref>(d) and (f). 
The bias dependence of the transition rate is  shown in <ref>. We evaluate <ref> in the tip positions visualized by the black and teal crosses in the insets. They select tunnelling events involving a single orbital, respectively L_xz for r_1 and L_yz for r_2. The form of the dianionic ground state determines the lower bias onset for the rate in position r_1. At a higher bias voltages both rates saturate, with the one at r_2 being twice as large as the one at r_1: the unoccupied orbital offers two spin channels as compered to the single one for an orbital already occupied by one electron. 
The predicted change of the full rate map is shown in the insets of <ref>.
The comparison to <ref>(g) and <ref> shows the emergence of the orthogonal LUMO (L_yz in our model) with an overall stronger signal.

Control over the landscape of the dianionic APES should also be obtained by changing the underlying salt. It is experimentally known that RbI, NaCl, and LiF have increasing influence onto the spectral broadening of molecular electronic states <cit.> and thus on the reorganization energy of the symmetric modes. We hypothesize that a similar trend also characterizes the antisymmetric modes, with a possible reversal of the orbital occupation sequence of CuPc on RbI with respect to the one on NaCl and LiF.


Conclusions We have shown how the electron-phonon coupling between a molecule and the underlying salt substrate determines the specific charge configuration within a molecule going beyond mere charge stabilization. We extend to this respect existing observations of charge localization in oligophenylene <cit.> and pentacene <cit.> to systems with JT distortion. The cooperative mechanism between salt substrate and molecule should generally appear in molecules and substrates sharing the same point group. This approach could pave the way to control charge configurations in molecules depending on the used substrate, with potential relevance in the study of intermolecular electron transfer, surface chemical reactivity and charge sensing. We also envisage the interplay between JT distortion of neighbouring molecules for the realization of cellular automata based on molecular arrays in which the local information is stored in the charge configuration of a single molecule.

Acknowledgement
M.F., L.L.P., J.R. and A.D. acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within Project-ID 314695032 - SFB 1277 (project B02). J.R. gratefully acknowledges funding from the ERC Synergy Grant MolDAM (no. 951519) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through RE2669/6-2. T.F. and M.F. acknowledge support from the Spanish MCIN/AEI/ 10.13039/501100011033 (PID2020-115406GB-I00), the Basque Department of Education (PIBA-2023-1-0021), and the Euskampus Transnational Common Laboratory QuantumChemPhys.













































§ EXPERIMENTAL TECHNIQUE

Experiments were carried out with a low-temperature scanning tunneling and atomic force microscope equipped with a qPlus tuning fork <cit.> (resonance frequency f_0 ≈ 29.1 kHz, spring constant k = 1.8 kN m^-1, quality factor Q ≈ 3 × 10^4) in ultrahigh vacuum (p = 2 × 10^-10 mbar) and at a temperature of 6.2 K. As substrate we used Cu(111) single crystal covered with >20ML of NaCl. The molecules were deposited onto the sample kept at ≈ 7 K inside the microscope. Bias voltages are given as sample bias with respect to the tip. Positive constant-height offsets Δ z correspond to a distance decrease with respect to the AFM Δ f set point above the clean NaCl. For AC-STM experiments, the ac voltage pulses were produced by an arbitrary waveform generator (Agilent 33522A) and fed to the microscope head through semirigid coaxial high frequency cables.


§ CALCULATING MOLECULAR MODE PARAMETERS


The dependence of the electronic wave functions of the molecule on the nuclear positions is obtained by expanding the electronic Hamiltonian around the reference configuration R_0 up to first order in the displacement coordinate q. By definition, the first derivative of the ground state energy for the reference electronic state vanishes. This is not the case, though, for the each component of the electronic Hamiltonian. 
In terms of the normal modes coordinates, we thus obtain

    [e-ph] = ∑_l,α(∂[e]/∂q_l,α)_q_l,α=0∑_ζv^ζ_l,α√(ħ/2M_lω_ζ)( [ζ]a^† + [ζ]a),

with M_l the mass of the atom l and v^ζ the displacement vector of the normal mode ζ with frequency ω_ζ.
The molecular Hamiltonian  [mol] presented in the main text corresponds to [e](r,R_0). The matrix element describing the electron phonon coupling strength is given by

    g_ij^ζ = ∑_lα⟨ i|∂[e]/∂q_lα| j⟩v^ζ_lα√(ħ/2M_lω_ζ) .

These matrix elements can be evaluated by the INELASTICA python package <cit.>.
To evaluate them we first obtained the molecular geometry with the SIESTA DFT implementation <cit.>.
We employed a Single-Zeta-Pseudo (SZP) basis set and used the GGA-PBE exchange functional.
For details about the calculation of the mode energies ħω_ζ and electron phonon couplings g_ij^ζ we refer to the paper by Frederiksen et al. <cit.>.



§ SEPARATION OF SYMMETRIC AND ANTISYMMETRIC MODES


In this section, we aim to demonstrate the contrasting effects of symmetric and antisymmetric modes. We find that the symmetric modes solely lead to an energy renormalization depending on the total number of charges on the molecule. Conversely, the coupling to the antisymmetric modes varies across different orbital occupations, thereby exerting an influence on the charge distribution of the molecular ground state.

Our discussion begins with the component of the Hamiltonian that encompasses the vibronic degrees of freedom as well as the electron-phonon coupling

    [vib] + [e-ph] = ∑_ζħω_ζ([ζ]a^†[ζ]a + 1/2) + 
    		∑_ζσ∑_i,j = L± g^ζ_ij[iσ]d^†[jσ]d( [ζ]a^† + [ζ]a).

The sum is taken over all contributing modes ζ, the spin σ, and the degenerate LUMOs i and j, as we assume that any excess charge on the molecule to be confined solely to the LUMOs, thus restricting the electron-phonon coupling to these two orbitals.

The electron phonon coupling strength is proportional to the matrix elements with respect to the derivative of the electronic Hamiltonian, see <ref>,

    g_ij^ζ∝⟨ i|∂[e]/∂q_lα| j⟩.

The derivative ∂[e]/∂q_lα is an irreducible tensor operator which transforms in the same way as the respective normal mode ζ.
The wave functions i and j transform according to the E_g irreducible representation of the D_4h point group. Therefore, for the matrix element in <ref> to be nonzero we can only use modes ζ such that their irreducible representation Γ^ζ is contained in the symmetric product

    [ E_g⊗ E_g ]_+ = A_1g⊕ B_1g⊕ B_2g.

The modes transforming according to the A_1g irreducible representation couple to the global excess charge on the LUMOs. We call these the symmetric modes. The associated molecular distortion preserves the symmetry of the system. Therefore, they are not contributing to the JT effect. It is important to consider them, as they give a renormalization to the eigenenergies of the molecule depending on the charge state and they lead to a broadening in the transition rates.

On the other hand, the B_1g and B_2g modes couple to the unbalance between the occupation of the LUMOs. In particular, this imbalance concerns,  for the B_1g modes, the real valued basis L_xz and L_yz shown in Fig. <ref>. Contrary to the totally symmetric A_1g modes, these modes can lower the symmetry of the molecule. Thus, they are responsible for the JT distortion.



 §.§ Energy renormalization due to symmetric modes

In this section, we delve into the impact of the symmetric modes. All relevant parameters are shown in <ref>.
The symmetric modes couple diagonally to the LUMOs, i.e. i=j in <ref>,

    [vib]^A +  [e-ph]^A= ∑_α∈{A_1g}ħω_α([α]a^†[α]a + 1/2) + 
    		∑_α∈{A_1g} g_α∑_iσ[iσ]d^†[iσ]d( [α]a^† + [α]a).

All modes couple equally to the occupation of the L_xz and L_yz orbitals and thus we omit the subscript specifying the LUMO from the coupling strength g_α. The sum over α encompasses all modes transforming according to the A_1g irreducible representation and g_α is the coupling strength to the LUMOs of mode α. We apply the Lang-Firsov transformation to the full Hamiltonian 

    Ĥ̃ = 
      e^Ŝ( [mol] + [tip] + [tun]+ [as] + [vib] + [e-ph] ) e^-Ŝ ,

where the operator 

    Ŝ = ∑_α∈{A_1g}∑_iσλ_α[iσ]n( [α]a^†-[α]a) .

with λ_α = g_αħω_α. This transforms the annihilation and creation operators on the molecule as

    d̃̂̃_iσ = d̂_iσX,   d̃̂̃^†_iσ = d̂^†_iσX^†.

with X = e^-∑_αλ_α ([α]b^† - [α]b).
The phonon operators are shifted by

    ẫ_α = [α]a - λ_α∑_iσ[iσ]n,   ẫ^†_α = [α]a^† - λ_α∑_iσ[iσ]n.

The transformed molecular Hamiltonian thus obtains a renormalization of the single-particle LUMO energies ϵ̃_i  =  ϵ_i - ∑_α∈ A_1gg^2_α/ħω_α .
and of the direct Coulomb interaction between the LUMOs Ṽ_ijji = V_ijji - 2∑_α∈ A_1gg^2_α/ħω_α, with i,j = L_xz , L_yz.
Furthermore, the symmetric electron-phonon coupling term vanishes, leaving only the harmonic vibrations in the Hamiltonian which describes the effect of the symmetric modes.
The renormalization of the single-particle LUMO energies and the corresponding Coulomb interaction terms leads to a stabilization of charged states on the molecule.
The Lang-Firsov transformation of the tunneling Hamiltonian yields the X̂ operator introduced in Eq. (<ref>). Its matrix elements in the harmonic oscillator eigenbasis are the Franck-Condon factors giving the Poisson distribution in the rates of Eqs. (7) of the main text. Ultimately, the X̂ operator and thus the coupling to the A_1g modes is responsible for the broadening of the calculated transition rates.





 §.§ Treatment of antisymmetric modes

The antisymmetric modes are more sensitive to the actual distribution of the charge between the LUMOs, rather then to their global occupation. The electron vibron coupling Hamiltonian, written for the real basis depicted in Fig. <ref>, reads 

    [e-ph]^B = 
    	∑_β∈{B_1g} g_β( [xz]n - [yz]n)
        ( [β]a^† + [β]a)
    + 
    	∑_β∈{B_2g} g_β([xz]d^†[yz]d + [yz]d^†[xz]d)
        ( [β]a^† + [β]a),

where β encompasses all modes transforming according to either B_1g or B_2g.
This Hamiltonian gives rise to the Jahn-Teller (JT) effect, as each deformation favours the occupation of a determined LUMO thus lifting their degeneracy. The more complex form of the electron-phonon coupling combined with the structure of the exchange Hamiltonian makes the Lang-Firsov transformation unuseful. The latter does not decouple in this case  the electronic from the vibronic degrees of freedom. Therefore, we treat the antisymmetric displacements classically and estimate the ground state confirmation by searching for the minima of adiabatic potential energy surfaces (APES).
To this end, we introduce canonical displacements and momenta as follows

    x̂_β = √(ħω_β)( [β]a^†+[β]a) ,  p̂_β = iħ/2√(ħω_β)( [β]a^† - [β]a) .

This transformation allows us to express the Hamiltonian associated to the modes with the B_1g and B_2g symmetry as

    [vib]^B +  [e-ph]^B =  
      ∑_β∈{B_1/2 g}( ω_β^2 p̂^2_β + 1/4x̂_β^2 ) 
      + 
    	∑_β∈{B_1 g}g_β/√(ħω_β)( [xz]n - [yz]n)
        x̂_β
     +
     ∑_β∈{B_2 g}g_β/√(ħω_β)([xz]d^†[yz]d + [yz]d^†[xz]d)
        x̂_β.

We further simplify the problem by ignoring the kinetic terms and by turning x̂_β into the classical coordinate  x_β. At this point we should diagonalize the effective Hamiltonian 


    [eff]({x_β}) = Ĥ̃_ mol + [as] + ∑_βx_β^2/4 +  [e-ph]^B({x_β})

and look for the ground state energy and configuration for each charge sector. Given the presence of 14 B_1g and 14 B_2g modes, determining the energy minimum of this potentials proves impractical. A first screening on the modes can be performed using their reorganization energy Δ E_ reorg≡ħωλ^2. As can be seen from <ref>, the B_1g modes provide a higher reorganization energy than the B_2g modes. Therefore, we neglect the B_2g modes and focus our attention solely on the B_1g modes.



§ MANY-BODY EIGENSTATES




A crucial role for the understanding of the experiments is played by the effective Hamiltonian in Eq. (<ref>) from which the low energy many-body eigenstates of CuPc in its neutral, anionic and dianionic configuration can be extracted. We restrict ourselves to the Fock space generated from the single particle basis of four frontier orbitals: i.e. the SOMO (S), HOMO (H) and the two degenerate LUMOs. In particular, we will only consider, in the following, the neutral anionic and dianionic states with respectively 0,1 and 2 electrons in the LUMOs. This further simplification does not affect significantly the low energy eigenstates of the neutral and anionic molecule, as obtained by comparison with numerical diagonalization performed in the full Fock space associated to the four frontier orbitals. In the dianionic case this approximation excludes a pair of doublets with triple occupation of the LUMOs. The latter are though not relevant to explain the experimental results, as these states cannot be accessed with a single electron tunnelling event from the anionic ground state.  

As the LUMOs are degenerate, they allow for different representations. We show in <ref> the frontier orbitals with the LUMOs in their real valued representation (L_xz, L_yz). However, in the discussion of the many-body eigenstates, we will also use their rotationally invariant representation which is given by <cit.>

    |L+⟩ = 1/√(2)( |L_xz⟩ + i |L_yz⟩),   |L-⟩ = 1/√(2)( |L_xz⟩ - i |L_yz⟩).

The Hamiltonian we discuss is [eff](x_1) = Ĥ̃_ mol + x_1^2/4 + [e-ph]^B(x_1), where, to start, we restrict ourselves to the B_1g mode with the highest reorganization energy. The renormalization due to the symmetric modes only introduces the same shift to all eigenenergies of a given particle number. In practice, we have

    [eff](x_1) =  
    		∑_i (ε̃_i + Δ) [i]n + 1/2∑_ijkl∑_σσ'Ṽ_ijkl[iσ]d^†[kσ']d^†[lσ']d[jσ]d
    		+  x_1^2/4
    	+ ∑_σg_1/√(ħω_1)( [L+σ]d^†[L-σ]d + [L-σ]d^†[L+σ]d) x_1 .

The summation indices i, j, k and l run over all four frontier orbitals, i.e. i,j,k,l = S,H,L_+,L_-. The model comprises the renormalized single particle energies ε̃_i = ε_i - ∑_α g_α^2/ħω_α  with ε_S = -12.0, ε_H = -11.7, ε_L+/L- = -10.7, also shifted by a crystal field correction of Δ = 2.44. The Coulomb parameters concerning the direct interaction are also renormalized by the reorganization energy of the symmetric modes, i.e. Ṽ_ijkl = V_ijkl - 2∑_α g_α^2/ħω_α. Exchange and pair hopping contributions are instead not influenced by the Lang-Firsov transformation. The bare Coulomb parameters, calculated by direct integration from the frontier orbitals <cit.>, are given in <ref>.
The energy of the considered mode is ħω_1 = 185 and the coupling strength g_1 = 72.





 §.§ The neutral molecule


We start our discussion of the eigenstates form the neutral molecule. By diagonalizing <ref> in the reference position R_0, i.e. x_1 = 0, we find a spin degenerate doublet

    |3,D_0^↑⟩ = [S↑]d^†|Ω⟩,   |3, D_0^↓⟩ = [S↓]d^†|Ω⟩,

where |Ω⟩ is the reference state with doubly occupied HOMO but empty SOMO and LUMOs. We observe the characteristic unpaired spin in the SOMO, due to its localization and correspondingly high charging energy. The first-excited neutral energy level lies about 0.8 above the ground state <cit.> and thus we do not consider it in our calculations.  
The lowest eigenvalue of the Hamiltonian <ref> is  depicted in <ref> (a) as  a function of the displacement coordinate x_1 given in units of √(ħω_1) and represents the adiabatic potential energy surface (APES). As the LUMOs remain unoccupied in the neutral ground state, the electron-phonon coupling does not contribute to the potential energy and we only see the harmonic contribution due to the first antisymmetric mode.




 §.§ The anionic molecule

The effective Fock space for the CuPc anion comprises the eight states given by [Sσ]d^†[Liσ']d|Ω⟩ with σ, σ' = ↑, ↓ and i = ±, which are degenerate in the reference configuration as far as it concerns the direct Coulomb interaction. The exchange coupling between the electrons occupying the SOMO and the LUMO splits this eightfold degeneracy. The eigenstates in the reference configuration can thus be organized as two sets of triplets and singlets

    |4,T_0^+1,i⟩ = [S↑]d^†[Li↑]d^†|Ω⟩,
       |4,T_0^0,i⟩ = 1/√(2)(
        [S↑]d^†[Li↓]d^†
        -
        [Li↑]d^†[S↓]d^†)
        |Ω⟩,
       |4,T_0^-1,i⟩ = [S↓]d^†[Li↓]d^†|Ω⟩,
       |4,S_1,i⟩ = 1/√(2)(
        [S↑]d^†[Li↓]d^†
        +
        [Li↑]d^†[S↓]d^†)
        |Ω⟩,

with the triplet orbitally degenerate ground states separated from the singlets by 2 J_SL = 18 meV. These two sets, labeled by i = ±, arise due to the degenerate LUMOs and are mixed by the coupling to the B_1g vibrational mode. First, we discuss the two triplet states with S_z = +1. The eigenstates of the Hamiltonian <ref> when x_1≠ 0 read

    |4,T_0^+1,xz⟩ = 1/√(2)(|4,T_0^+1,+⟩ + |4,T_0^+1,-⟩),  |4,T_0^+1,yz⟩ = i/√(2)(|4,T_0^+1,+⟩ - |4,T_0^+1,-⟩).

We notice that these states are also eigenstates of <ref> if x_1 = 0, thus giving a single set of eigenstates at all deformations, although the corresponding eigenenergies vary as function of x_1, as shown in Fig. <ref>. From the definition of the rotationally invariant representation given in Eq. (<ref>) we can express these anionic eigenstates states in the real valued representation as 

    |4,T_0^+1,xz⟩ = [S↑]d^†[L_xz↑]d^†|Ω⟩,  |4,T_0^+1,yz⟩ = [S↑]d^†[L_yz↑]d^†|Ω⟩.

The same procedure can be applied to the other 4 triplet states. We obtain

    |4,T^0_0,iz⟩ = 1/√(2)(
      [S↑]d^†[L_iz↓]d^†
      -
      [L_iz↑]d^†[S↓]d^†)
      |Ω⟩,
       |4,T^-1_0,iz⟩ = [S↓]d^†[L_iz↓]d^†|Ω⟩,

with i = x/y.
Analogously, for the singlet states we obtain, in the real valued basis,

    |4,S_1,iz⟩ = 1/√(2)( 
     [S↑]d^†[L_iz↓]d^† 
     +
     [L_iz↑]d^†[S↓]d^†) |Ω⟩,

again with i =x/y. In each of the two dimensional subspaces with a given spin S and S_z the Jahn-Teller Hamiltonian in Eq. (<ref>) acquires the canonical form 

    H^ 1-_ JT = x_1^2/4 1_2 + g_1/√(ħω)_1x_1 σ_x

typical of a E ⊗ b_1 Jahn-Teller problem <cit.>.
The sign of the displacement along the x_1 coordinate determines whether the triplet-singlet set with the added electron in the L_xz or in the L_yz orbital experience a decrease or increase in energy. This leads to a preference of one set of triplet-singlet states over the other depending on the direction of distortion. The APES depicted in <ref> (b) illustrates this phenomenon. The plot shows four parabolas representing the potential energy of both triplet-singlet sets.



 §.§ The dianionic molecule



The effective Fock space for the dianionic molecule consists of twelve states, which, in the rotationally invariant basis, are represented as

    [Sσ]d^†[Liσ']d^†[Ljσ”]d^†|Ω⟩,

where i,j = ±, and σ' = σ̅” if i = j due to the Pauli exclusion principle. These states are degenerate for what concerns the direct Coulomb interaction (terms involving U in [mol]) with an energy E_5 = ε̃_S + 2(ε̃_H + ε̃_L) + 5Δ + Ũ_H + Ũ_L + 2(Ũ_SH + Ũ_SL)+ 4Ũ_HL. By considering the exchange and pair hopping interactions (terms involving J in [mol]), this degeneracy is lifted. We first discuss the spectrum and the eigenstates in the reference configuration x_1 = 0. The eigenenergies are given with respect to the degenerate energy.
The scheme of the energy levels is given in <ref> (a). The ground state is a spin quadruplet

    |5,Q_0^3/2⟩ = [S↑]d^†[L+↑]d^†[-↑]d^†|Ω⟩,
       |5,Q_0^1/2⟩ = 1/√(3)( 
      [L+↑]d^†[L-↑]d^†[S↓]d^† 
      -
      [S↑]d^†[L-↑]d^†[L+↓]d^† 
      +
     [S↑]d^†[L+↑]d^†[L-↓]d^†) 
     |Ω⟩,
       |5,Q^-1/2⟩ = 1/√(3)( 
      [S↑]d^†[L+↓]d^†[L-↓]d^†  
      -
      [L+↑]d^†[S↓]d^†[L-↓]d^† 
      +
      [L-↑]d^†[S↓]d^†[L+↓]d^†) 
      |Ω⟩,
       |5,Q_0^-3/2⟩ = [S↓]d^†[L+↓]d^†[L-↓]d^†|Ω⟩.

The energy of this quadruplet is E_Q_0 = E_5 -J^ex_+--2J^ex_ SL. The first excited states form a spin degenerate doublet, with energy  E_D_1 = E_5 - J^ex_+- + J^ex_SL. The corresponding eigenstates are given by

    |5,D_1^↑⟩ = 1/√(6)(
      2 [L+↑]d^†[L-↑]d^†[S↓]d^†
      +
      [S↑]d^†[L-↑]d^†[L+↓]d^†
      -
      [S↑]d^†[L+↑]d^†[L-↓]d^†)
      |Ω⟩ ,
       |5,D_1^↓⟩  =  1/√(6)(
      2 [S↑]d^†[L+↓]d^†[L-↓]d^†
      +
      [L+↑]d^†[S↑]d^†[L-↓]d^†
      -
      [L-↑]d^†[S↓]d^†[L+↓]d^†)
      |Ω⟩.

The following three energy levels are also spin doublets, with the following eigenenergies 

    E_D_2 = E_5 -J_+-^ p - J_ SL^ ex ,    E_D_3 = E_5  + J_+-^ p - J_ SL^ ex ,    E_D_4 = E_5  + J_+-^ ex - J_ SL^ ex .

The corresponding eigenstates are,

    |5,D_2/3^↑⟩  = 
      1/√(2)(
      [S↑]d^†[L+↑]d^†[L+↓]d^†∓[S↑]d^†[L-↑]d^†[L-↓]d^†) |Ω⟩, 
       |5,D_2/3^↓⟩ = 1/√(2)(
      [L+↑]d^†[S↓]d^†[L+↓]d^†∓[L-↑]d^†[S↓]d^†[L-↓]d^†)|Ω⟩,

and, respectively, 

    |5,D_4^↑⟩ = 1/√(2)(
                [S↑]d^†[L+↑]d^†[L-↓]d^† 
                +
                [S↑]d^†[L-↑]d^†[L+↓]d^†)
                |Ω⟩,
       |5,D_4^↓⟩ = 1/√(2)(
                [L+↑]d^†[S↓]d^†[L-↓]d^† 
                +
    		      [L-↑]d^†[S↓]d^†[L+↓]d^†)
                |Ω⟩.




The first eight states corresponding to the lower energies are not affected by the electron vibron coupling to the modes of symmetry B_1g. This is evident in <ref>, where we observe three parabolas - one is fourfold and the other two are twofold degenerate - with a minimum at x_1 = 0. This antisymmetric mode couples instead the eigenstates |5,D^σ_3⟩ and |5,D^σ_4⟩. For each of the two spin components, the Jahn-Teller Hamiltonian projected on the corresponding two dimensional basis reads 

    H^2-_ JT(x_1) = x_1^2/4 1_2 
    - J_+-^ ex - J_+-^ p/2σ_z 
    +2g_1/√(ħω_1)x_1σ_x .

Due to the finite energy splitting in the reference configuration (x_1 = 0) this Hamiltonian describes the pseudo-Jahn-Teller effect <cit.>. The existence of localized minima corresponding to deformed configurations is only guaranteed for a sufficiently large coupling, i.e. g > √(ħω_1 (J_+-^ ex - J_+-^ p))/4, as can be deduced by analysing the two highest eigenvalues of the dianionic effective Hamiltonian:

    E_D_3/4(x_1) = E_5 - J_ SL^ ex + J_+-^ ex + J_+-^ p/2 + x_1^2/4∓√((J_+-^ ex-J_+-^ p/2)^2 + 4g_1^2/ħω_1x_1^2) .

Differently from the anionic case, here also the eigenstates depend on the deformation coordinate x_1. In particular we obtain

    |5,D^σ_3(x_1)⟩   = cosθ(x_1) |5,D_3^σ⟩ - sgn  x_1 sinθ(x_1) |5,D_4^σ⟩ ,  
    |5,D^σ_4(x_1)⟩   =  sgn x_1 sinθ(x_1) |5,D_3^σ⟩+cosθ(x_1) |5,D_4^σ⟩ ,

where the mixing angle is defined as 

    θ(x_1) = 1/2arccos(1 + 16 g_1^2 x_1^2/ħω_1 (J_+-^ ex - J_+-^ p)^2)^-1/2 .

Within the single mode approximation considered here we obtain a mixing angle at equilibrium θ_ eq≈ 0.24π. The larger is the electron phonon coupling g with respect to the vibrational energy ħω_1 and the energy splitting J_+-^ ex - J_+-^ p of the reference configuration, the larger is the absolute value of the equilibrium configuration.  Thus, the mixing angle converges to θ = π/4. In view of the enhancement of the coupling introduced by the other molecular modes and, above all, by the insulating substrate, we expand the equilibrium eigenstate in the vicinity of this limiting case. 
The lowest energy state having the equilibrium configuration with x_1, eq < 0 reads

    |5,D^σ_3(x_1, eq)⟩   = 
    cos(θ_ eq - π4)1/√(2)(|5,D_3^σ⟩ + |5,D_4^σ⟩) 
    - 
    sin(θ_ eq - π4)1/√(2)(|5,D_3^σ⟩ - |5,D_4^σ⟩) 
       =  sgn σ [Sσ]d^†[
    cos(θ_ eq - π4) [L_xz↑]d^†[L_xz↓]d^† 
    +
    sin(θ_ eq - π4) [L_yz↑]d^†[L_yz↓]d^†]
    |Ω⟩
       ≡cos( θ_ eq - π4) |5,D^σ_xx⟩ + sin( θ_ eq - π4) |5,D^σ_yy⟩ .

This state shows a predominant double occupation of the real valued L_xz orbital (|5,D^σ_xx⟩) with a small admixture of the state with a doubly occupied L_yz orbital (|5,D^σ_yy⟩). The roles of L_xz and L_yz are exchanged if x_1, eq > 0. 

The experimental results we showed in Fig. 1 of the main text suggest that the dianionic ground state has both electrons in the same real valued LUMO. This is clearly not the case in our theoretical model if we consider one mode since the ground state is in the reference configuration where the lowest lying states are the quadruplets which have one electron in each LUMO. We can estimate the reorganization energy required to achieve the desired dianionic ground state by first considering that E_D_3 - E_Q_0 = J_+-^ ex + J_+-^ p + J_SL^ ex = 435 meV. The total reorganization energy scales with the square of the electron-phonon coupling. Thus, the one associated to the B_1g modes scales with the square of the electronic unbalance in the real valued LUMOs. It is thus necessary to achieve a reorganization energy per electron due to B_1g modes of least 109 for the state |5,D^σ_3(x_1, eq)⟩ to become the dianionic ground state.



§ COMBINATION OF MULTIPLE MODES WITH THE SAME SYMMETRY




In this section, our focus is on combining multiple modes that belong to the same irreducible representation into a single effective mode. This allows us to analyze the potential energy of the system with contributions from all modes of  the same irreducible representation. The combined system of molecule and salt has C_4v symmetry. The B_1g modes we calculated for the molecule are all in plane modes and thus they transform as B_1 modes with respect to the C_4v point group. Therefore, we can not only combine the molecular modes among themselves but also include salt contributions to them. We begin this procedure by introducing the transformations of the displacements and momenta given by

    Q̂_n = ∑_β∈{B_1}  A_nβx̂_β,   P̂_m = ∑_β∈{B_1}  B_mβp̂_β.

Imposing canonical commutation relations on these new coordinates yields

    [ Q̂_n, P̂_m] = ∑_ββ' A_nβB_mβ'[ x̂_β,p̂_β'] = 
    	i ħ∑_ββ' A_nβB_mβ'δ_ββ' = 
    	iħ∑_α A_nβB_mβ!= iħδ_nm.

We further assume A to be orthogonal and thus deduce B=A. Moreover, we fix the first row of the transformation matrix to be

    A_1β = 1/g√(g_β^2/ħω_β),

with g = √(∑_βg_β^2/ħω_β). This change of coordinate allows us rewrite the Jahn-Teller Hamiltonian as 

    Ĥ_JT({Q_n}) = 1/4∑_n Q_n^2 + g Q_1 (n̂_xz - n̂_yz) ,

in which the electron-phonon coupling only concerns one effective JT mode, with effective coupling strength g.
However, this transformation comes at the cost of a non-diagonal kinetic part in the Hamiltonian. We are still free to construct the matrix transformation A such that all the Jahn-Teller inactive modes remain orthogonal among themselves. We can thus consider them as phonon bath for the JT active mode Q_1 and we neglect them  in our further discussions. This is justified as far as we are not interested in the dynamics of the JT distortion, but rather on the new equilibrium configurations.

Only considering molecular contributions to the effective coupling strength yields a value of g=218√() corresponding to a reorganization energy of Δ E_B_1 = 48. From our previous discussion in <ref> we know that this reorganization energy is not enough for the states with two electrons in one LUMO to become the new ground states. We can also deduce this from the APES in <ref> (a), which is plotted along the effective JT coordinate Q_ AS := Q_1 in units of characteristic quantity

    L^ AS_c = √(1/g√(∑_βħω_β g_β^2)) ,

where β runs over all considered modes. We notice that an effective Huang-Rhys factor can be defined, starting from Eq. (<ref>), for the JT active mode Q_ AS  as half of the ratio between the deformed equilibrium configuration Q_ AS, min of the anion and the characteristic scaling constant L^ AS_c. In other terms, it reads 

    λ^ AS_ eff = g/L^ AS_c = [ (∑_βg_β^2/ħω_β)^3/∑_β'ħω_β' g_β'^2]^1/4 .

 The latter reduces to the common expression λ = g_β/(ħω_β) when the sums are restricted to a single mode. 
 The deformed minima in the dianionic configuration are closer to the dianionic minimum energy but this still is not enough to explain the double charging of the same LUMO.
 However, if we consider also the contributions from the salt modes with the parameters given in the main text, we obtain g = 359√(), yielding a reorganization energy of Δ E_B_1 = 129. We show the corresponding APES along the Q_ AS displacement coordinate in <ref> (b). For the neutral and anionic charge states we observe no qualitative changes. For the dianionic molecule we see instead a qualitative change, as |5,D^σ_xx⟩ becomes the new global ground state.
The next highest lying states are the |5,D^σ_yy⟩ states. These two pairs of states are split by 2δ_ as, due to the  asymmetry which we included in our Hamiltonian. Otherwise, analogously to the anionic pairs of triplet and singlet states, they would be degenerate.
The energy splitting between the dianionic ground states |5,D^σ_xx⟩ and the quadruplet states is, with our choice of parameters, 70. This analysis confirms that the salt substrate in this system has an effect on the charge distribution within a single molecule going beyond mere stabilization of extra charges. We notice, moreover that, even with the contribution of the salt, λ_ eff≈ 1 thus allowing for direct transitions between vibrational ground state with different charges. The classical limit of non-adiabatic vertical transition does not apply to the JT active mode. Analogous considerations can be reserved also to the symmetric (A_1) modes, yielding a significant difference in the effective Huang-Rhys factors
 
    λ_ eff^ S = 2.36 ,  λ_ eff^ AS = 1.11 .






§ MICROSCOPIC MODEL OF THE SALT ELECTRON-PHONON COUPLING



We present in this section a simple microscopic model to estimate the electron-phonon coupling between an excess charge on the molecule and surface phonons of the underlying salt. We consider the bare Coulomb interaction between the excess electrons in the molecule and the salt ions. Comparison with experimental results indicate that this method seems to overestimate the values for the coupling.


 §.§ Linear vibronic coupling values

We consider vibration of the ions in all three space directions, thus accounting for  both the longitudinal and the transversal phonon modes in the salt. For simplicity, we limit ourselves to an Einstein model of independent harmonic oscillators with the energies ħω_L = 32 for the vibrations parallel to the surface and ħω_T = 20 for the ones perpendicular to it. The linear electron-phonon  couplings are calculated from the change of the Coulomb interaction between the LUMOs and the salt ions upon displacing of the latter. Specifically we evaluate

    [ g_l^α]_ij = q_l e^2/4πϵ_0*∫  drϕ^*_i(r)ϕ_j(r) Δ^α_0l/|r-R^0_l-u_l|u_lα_{u_lα}=0.

Here l labels an atom in the salt, i,j = L± specify one of the degenerate LUMOs and α = x,y or z represents the spatial direction of the displacement. Furthermore, we have q_i = ± 1 depending on the atom being a chloride (+1) or sodium (-1) atom, e is the elementary charge and ϵ_0 the vacuum permittivity.
The atom's equilibrium position is specified by R^0_l, u_l is the displacement coordinate of the considered atom and we have introduced the zero point fluctuations Δ^x/y_0l = √(ħ/2m_lω_L) and Δ^z_0l = √(ħ/2m_lω_T).
The central copper atom of the molecule, which we define to be at the origin, is centered on a chloride ion and we take 20 layers of salt into account. We checked that the integral in Eq. (<ref>) is converged in the z direction (perpendicular to the salt surface) when taking into account this number of layers. In the x and y direction we consider a square with an area of 400^2, which again suffices for the convergence of the electron phonon couplings. Moreover, we choose a distance of 2.8 between salt and molecule, which is consistent with values found in the literature <cit.>. By calculating the derivative in Eq. (<ref>), we obtain the expression

    [ g_l^α]_ij = q_l e^2/4πϵ_0∫ dxdydz
        ϕ^*_i(x,y,z)ϕ_j(x,y,z)(α-R^0_lα)Δ^α_0l/[(x-R^0_lx) ^2+(y-R^0_ly)^2+(z-R^0_lz)^2]^3/2 ,

which represents the starting point for the numerical calculation of the coupling constants.


 §.§ Numerical evaluation of the integral

The numerical integration of <ref> faces the problem of a divergent integrand when the integration variable coincides with one of the equilibrium positions of the atoms in the salt r = R_l. The integral is though not diverging. To better understand this issue, we expand the product of the wave functions around the diverging point as

    ϕ^*_i(r)ϕ_j(r) =     ϕ^*_i(R_l^0)ϕ_j(R_l^0) + 
    	∑_α(∂/∂ r_αϕ^*_i(r)ϕ_j(r))_ r =  R^0_l( r_α - R^0_lα) + 
       1/2∑_α∑_β(∂^2/∂ r_α∂ r_βϕ^*_i(r)ϕ_j(r))_ r =  R^0_l( r_α - R^0_lα) ( r_β - R^0_lβ) + ...,

where α,β = x,y or z. We now evaluate the integral <ref> for the first few terms of this expansion in the vicinity of a diverging point r =  R_l^0.
Furthermore, we drop all constants since we are only interested in the behavior of the integral when the integrand is diverging and without loss of generality set α = z.
By shifting to cylindrical coordinates with ζ = z-R_l z and ρ = √((x-R_l x)^2 + (y-R_l y)^2) and integrating over a small volume around the diverging point, we obtain, for the constant term in Eq. (<ref>)

    ∫_0^2π  dϕ∫_0^R   dρ∫_-ϵ^ϵ  dζρζ/( ρ^2+ζ^2 )^3/2 = 0.

For the linear component of <ref>, we distinguish two cases. The first case is for α = z, in which it is convenient to choose spherical coordinates with r = √(ρ^2 +ζ^2) and θ = arctan( ρ/ζ). We thus express the integral in the vicinity of the diverging point in these coordinates and obtain 

    -∫_0^2π  dϕ∫_0^π  dθ∫_0^R   dr r cos^2(θ)sin(θ)	,

which is not diverging and vanishes in the limit R → 0. The second case we need to discuss is when α≠ z. We can choose α=x as the case with α = y is completely analogous.
Now, we rotate our coordinate system and obtain ζ' = y-R_l y and ρ' = √((x-R_l x)^2 + (z-R_l z)^2).
With this transformation, the local integral is proportional to

    ∫_0^2π  dϕ∫_0^R   dρ'∫_-ϵ^ϵ  dζ'ρ'^3 cos(ϕ)sin(ϕ)/( ρ'^2+ζ'^2 )^3/2 = 0 .

Analogously, one can prove that also the integrals involving higher orders expansions will be finite and vanish in the limit of infinitesimal integration volume around the point with diverging integrand. Therefore, in the numerical evaluation we can set the integrand at the diverging point to 0 and can just use standard numerical integration schemes. 
The electron-phonon coupling relative to each vibrational mode of a salt ion is a Hermitian 2x2 matrix which can be expanded in terms of the Pauli matrices. We obtain 

    g_l^α = g^α_l,A_1 1_2 + g^α_l,B_1σ^x+g^α_l,B_2σ^y.

as, due to the form of the LUMO orbitals, the σ^z component vanishes exactly. We labelled the different electron-vibron coupling components according to the representations of the corresponding irreducible tensor operator with respect to the C_4v point symmetry group. In fact,


    ∑_σ∑_ij[Liσ]d^†δ_ij[Ljσ]d∈ A_1 , 
      ∑_σ∑_ij[Liσ]d^†σ^x_ij[Ljσ]d∈ B_1 , 
      and  ∑_σ∑_ij[Liσ]d^†σ^y_ij[Ljσ]d∈ B_2 .


So far we have shown how to calculate the electron-phonon couplings for each individual atom in the salt. We are, however, more interested in the collective behavior of the substrate. Thus, despite of the simplified Einstein model, we look for the modes with finite coupling to the excess charge in the molecular LUMOS. Analogously to the scheme proposed in Sec. <ref> we thus combine all modes with the same symmetry. We start with collective transversal modes, whose coordinates read

    x_Γ^T = ∑_l g^z_l,Γ/√(∑_m(g^z_m,Γ)^2)ũ_l z,

being Γ = A_1, B_1, and B_2 the corresponding irreducible representation and ũ_l z = u_l z/Δ^z_0l the displacement of each individual atom in z direction normalized by the respective zero point fluctuation. Analogously, we define for the longitudinal modes the collective mode coordinates

    x_Γ^L = ∑_l g^x_l,Γũ_l x + g^y_l,Γũ_l y/√(∑_m (g_m,Γ^x)^2 + (g_m,Γ^y)^2) .

These coordinate transformations identify six collective salt modes with a finite electron phonon coupling. All the other modes are not influenced by an excess charge on the CuPc molecule and are treated as a dissipative phonon bath.
The coupled modes exhibit respectively the transversal and longitudinal coupling strengths g^T_Γ = √(∑_l (g^z_l,Γ)^2) and g^L_Γ = √(∑_l (g^x_l,Γ)^2+(g^y_l,Γ)^2), with Γ = A_1, B_1 and B_2. Starting from the numerical calculation of the integrals in (<ref>), one obtains the following coupling strengths for the collective modes

    g^T_A_1   = 116,         g^T_B_1 = 20.6,         g^T_B_2 = 2.5
    
    		g^L_A_1   = 92.7,         g^L_B_1 = 11.1,         g^L_B_2 = 11.7.

Similarly to what we observed with the molecular modes, also in the salt the contribution to the electron-phonon coupling of the B_1 mode dominates over the one of the B_2 modes. Therefore, we concentrate our discussion solely on the A_1 and B_1 modes. They yield a total reorganization energy

    Δ  E_reorg^salt = (g^T_A_1)^2+(g^T_B_1)^2/ħω_T+
    	(g^L_A_1)^2+(g^L_B_1)^2/ħω_L = 966.

These values of the reorganization energy due to the salt deformation can be taken as an estimate for their order of magnitude. Their quantitative accuracy is though problematic.

On the one hand, by following the conclusions from the experiments of Fatayer et al. <cit.> we expect that the total reorganization energy upon charging should be in the range of 300≤Δ E_reorg^tot≤500. As the molecular component is Δ E_reorg^mol = 187, a salt reorganization energy in the range 113≤Δ E_reorg^salt≤313 is to be expected. 
This leads us to the conclusion that our model overestimates the total reorganization energy stemming from the salt.

On the other hand, in the molecular reference condition the energy difference between the dianionic ground state (both electrons in different LUMOs) and the localized configuration is ≈200.
Thus the minimum reorganization energy stemming from the antisymmetric modes, in order to obtain the localized configuration as the dianionic state with lowest energy, must be ≈100.

We assume that these problems can be solved by a more realistic treatment of the salt and salt-molecule interaction which includes the dispersion relation of the phonons and a screening of the bare Coulomb interaction between the molecule and the underlying ions. While keeping the form of the Hamiltonian Eq. (<ref>), we have considered the effective couplings as free parameters and explored their effects on the many body transitions. 
We have chosen them to be Δ E_A_1^salt = 231.2 and Δ E_B_1^salt = 80.
Combined with the molecular contributions this choice of parameters gives a total reorganization energy of Δ E_A_1 + Δ E_B_1 = 500.
Thus not overestimating the total reorganization energy while having the localized dianionic configuration as the ground state.






 Paper Title: An Inexpensive, Configurable Two-Tone Electron Spin Resonance Spectrometer.






Department of Physics, Hamilton College, Clinton, NY 13323, USA
cacollet@hamilton.edu

Department of Physics, Muhlenberg College, Allentown, PA 18104, USA


School of Chemistry, The University of Manchester, Manchester M13 9PL, UK

§ ABSTRACT 
Electron spin resonance (ESR) is a powerful tool for characterizing and manipulating spin systems, but commercial ESR spectrometers can be expensive and designed to work in narrow frequency bands. This work presents an inexpensive spectrometer that, when coupled with easy-to-design resonators, enables ESR over a broad frequency range, including at frequencies outside the standard bands. It can operate at either a single frequency or at two frequencies simultaneously. The spectrometer is built from off-the-shelf parts and controlled by a field programmable gate array (FPGA), and new capabilities can be easily added by reconfiguring the FPGA and adding or swapping components. We demonstrate the capabilities of the spectrometer using the molecular nanomagnet Cr_7Mn.

An Inexpensive, Configurable Two-Tone Electron Spin Resonance Spectrometer
    Grigore A. Timco
    August 5, 2024
==========================================================================





§ INTRODUCTION



Electron spin systems are a subject of significant interest for the possibility of forming spin qubits <cit.>.
One of the main ways of manipulating and reading out the states of these spins is electron spin resonance (ESR), where radio-frequency (RF) voltage signals are transduced into oscillating magnetic fields that interact with the spins <cit.>.
The response of the spins to these RF excitations is then detected, with the signal generation and detection being done by an ESR spectrometer <cit.>.
This work presents an ESR spectrometer that is both inexpensive and flexible, making it possible to precisely tune to spins across a wide frequency range from ∼2-15 GHz, either at a single frequency in monochromatic mode, or simultaneously at any two frequencies in that range in two-tone mode.

As currently configured our spectrometer addresses the 2-6 GHz range, needing only to change the RF circulator to swap between 2-4 GHz and 4-6 GHz, but minor adjustments laid out in Appendix <ref> can allow it to address the 6-15 GHz range.
It is also possible to address multiple ranges simultaneously by using different components in two-tone mode as described below.
Our spectrometer excites spin samples by coupling a signal generator, able to produce sinusoidal voltages at a variety of frequencies and powers, with a resonator, which converts the voltage into an oscillating magnetic field applied to the spin sample. 
Any signal from the sample itself is detected using the same resonator, then downconverted using an IQ mixer for readout on an oscilloscope. 
Overall control is provided by a field-programmable gate array (FPGA).

In recent years it has become much easier and more affordable to design ESR spectrometers by coupling high-frequency FPGAs with digital-to-analog converters (DACs) for signal generation, and analog-to-digital converters (ADCs) for detection <cit.>.
Such designs offer many benefits, including very granular control over the frequency and amplitude of the produced signals.
However, their frequency ranges can be somewhat limited by the sampling rate of the electronics, with, for instance, a 5 GSample/s (GSPS) converter needing to work in Nyquist zone 3 to generate a 6 GHz signal,<cit.> potentially resulting in a reduction in power and the presence of unwanted sidebands.
By trading the granular control for a simpler design using affordable off-the-shelf components, we have developed a spectrometer that has only very rudimentary signal shaping capabilities, mostly constrained to square pulses of varying powers, but that can work at much higher frequencies just by swapping out components to match the desired range.
The relative simplicity of the spectrometer also makes it fairly straightforward to configure and extend its functionality, and to understand how each part of the spectrometer works.

We developed this spectrometer in part to allow for both individual and simultaneous control and readout of the spins in a dimer of molecular nanomagnets (MNMs).
MNMs are a class of magnetic materials allowing for chemical engineering of the properties of the quantum spins, with potential quantum computing applications using the molecules as spin qubits <cit.>.
Performing multi-qubit gates with MNMs will involve simultaneous ESR on multiple spin qubits with frequencies separated by multiple GHz <cit.>.
Our lab has recently focused on MNMs engineered to have atomic clock transitions <cit.>, which extend the quantum state lifetime T_2 by making the spins less sensitive to local field fluctuations.
Dimers of these MNMs can be used to perform quantum gates <cit.>, but only with a spectrometer that can drive each spin simultaneously, and our spectrometer makes this possible.

Below we describe in detail the composition and performance of our spectrometer.
Most of the components are commercially available, and we will only briefly discuss their role; a more general discussion of spectrometer design can be found elsewhere <cit.>.




§ SPECTROMETER DESIGN

Using the configuration presented in this work, the spectrometer can produce pulses as short as 20 ns and up to 22 dBm (150 mW) of power at the output to the resonator.
It can switch between continuous-wave (CW) and pulsed signals programmatically, though CW mode is mostly useful for doing preliminary characterization of resonator frequencies, as more sensitive and precise CW measurements can be done with a vector network analyzer.
Our main goal in developing this spectrometer is to perform pulsed experiments, and as such only pulsed signals and results are presented here.

The design of this spectrometer can be broken down into three parts: the source and detection electronics, the resonator portion that directly interacts with the spins, and the FPGA controller.



 §.§ Electronics

Fig. <ref> shows a circuit diagram of the spectrometer electronics. 
Details of the exact off-the-shelf components used can be found in Appendix <ref>.






  §.§.§ Monochromatic Source

An RF source generates one or two CW output signals at a single RF frequency (the carrier frequency ν_c), which is our excitation and detection frequency.
One of these signals, which we will label B, goes directly into a high-speed RF switch (“pulse switch", PS_B in Fig. <ref>), while signal A goes through a power divider (PD), with half of the power going to the detection mixer (see below), and the other half to its own pulse switch (PS_A).
Pulsed experiments generally use both outputs to exploit the coherent phase control between the two signals that our source enables, which we will discuss later.
The pulse switches are controlled by the FPGA, opening and closing as needed to generate either CW or pulsed outputs.
The signal from both switches is combined in a power combiner (PC) and optionally put through a bandpass filter (F_1) to reduce noise.
We found that even with the pulse switches open there is still enough of the carrier signal leaking through to be detected, so the signal then goes through a blocking switch (LB, for Leakage Block) to more completely attenuate that leaking signal.
This combined signal is then sent through a power amplifier (PA) and a digital step attenuator (DSA).
The DSA is controlled by the FPGA, and allows independent control of the power of each pulse, and potentially shaping the power within a single pulse (that capability is not implemented in the default configuration).
Finally, the signal is routed to an antenna coupled to the resonator through a circulator (C) to prevent reflected power from damaging the amplifier.

The spectrometer can work in reflection mode when a single antenna is coupled to the resonator, and both reflection and transmission mode with two antennas. 
In reflection mode (pictured in Fig. <ref>), after the signal has interacted with the resonator it travels back up the same coaxial cable to the circulator, which routes it to the detection electronics. 
In transmission mode, the third output from the circulator is capped with a 50 Ω terminator, and a coaxial cable attached to the second antenna brings any transmitted signal to the detection electronics.



  §.§.§ Monochromatic Detection

The returning signal first passes through a power limiter (L) to protect the detection circuitry from the initial high-power pulses.
This protection could also be achieved using another switch, and indeed both configurations have been tested with similar results, but the limiter, being a passive component, is simpler.
We have found that under normal operating conditions there will often not be enough power reaching the detection side to require the limiter, but we leave it in for convenience and safety, as it causes minimal loss (<0.6 dB at these frequencies).

The remaining signal optionally passes through another bandpass filter (F_2) and then enters a low-noise amplifier (LNA).
At this point, the signal is an RF voltage that has been modified in some way by interactions with the resonator and sample. 
Note that while we output only one frequency, the return signal may have several, due to those interactions. 
It is possible to directly detect signals at these other frequencies by exciting at one frequency and detecting at another, but for this work we will focus on homodyne detection of the signal oscillating at ν_c.
The amplified return signal goes into the RF port of an IQ mixer (M), and with the carrier frequency coming into its LO port the mixer downconverts the signal into low-frequency in-phase (I) and quadrature (Q) signals, which can then be read out on an oscilloscope (Scope).



  §.§.§ Two-tone Electronics

We create two-tone pulses with two different output frequencies, ν_A and ν_B, by simply changing the frequency of one of the RF source outputs.
With the exact same electronics this only provides a two-tone source, as with the above configuration detection is still only possible at ν_A.
To also do two-tone detection, fairly minor changes need to be made to the electronics, adding a splitter and a mixer to output B to match those on output A, and then splitting the returning signal after the LNA to send it to the RF ports of both mixers; see Appendix <ref> for a circuit diagram showing these changes.
Note that in this case the spectrometer is doing homodyne detection for both frequencies, though if the frequencies are too close together there will be some crosstalk.

If desired, either or both of the source or detection electronics can be more comprehensively separated to provide more independent two-tone functionality.
For instance, putting in pulses at different frequencies and different powers would require at least separate DSA's, and maximizing the pulse and signal size at each frequency could be done by using two separate antennas both coupled to the resonator, so each frequency gets dedicated power and low-noise amplifiers.
The preliminary two-tone results presented below use the simpler changes described above.




 §.§ Resonator

As discussed above, the resonator must be able to couple the spins to an oscillating magnetic field.
For this spectrometer, samples are placed into the loop of a loop-gap resonator (LGR) <cit.>.
This style of resonator is formed by shaping a conductor into a connected loop and gap, such that current must flow around the loop to get from one side of the gap to the other. 
The resonator can thus be modeled as a lumped-element RLC circuit with resonant frequency ν_r, with the loop forming the inductor, the gap the capacitor, and the copper body the resistor. 
This design concentrates the oscillating magnetic field inside the loop; the loop dimensions (mm-scale) are generally much smaller than the RF wavelength (cm-scale for GHz frequencies), such that there is a high filling factor for a sample placed in the loop. 
With this design, a relatively low input power translates into a sizable magnetic field strength at the sample, and a small oscillating field from the sample can more easily couple back into the resonator.
In this work, the resonator is a 1/8"-thick disc of OFHC copper 0.6" in diameter with a 0.06" diameter loop and a 0.03" wide gap milled out of it, as shown in Fig. <ref>.

To interact with the sample, the resonator must transduce an oscillating voltage into an oscillating loop field. 
The voltage is introduced to the resonator via an antenna formed from the exposed inner conductor at the end of a coaxial cable, which is placed near the surface of the resonator. 
A capacitive antenna is made by bending the exposed inner conductor into an `L' shape, then lowering it over the gap, while an inductive antenna sits over the loop and is made by making a circle with the inner conductor and electrically connecting it to the outer conductor. 
The resonator sits inside a copper RF shield at the bottom of a sample probe that is inserted into a cryostat, and coaxial cables transmit signals between the rest of the spectrometer and the resonator through one or more antennas inside the shield. 
For reflection mode a single antenna serves to both excite the resonator and detect signals coming from the sample, and for transmission mode one antenna does the excitation and a different antenna does the detection. 
Transmission mode thus requires two sets of antennas and coaxial cables in the sample probe, which can be terminated if unused. 
A detailed description of a substantially similar antenna-resonator arrangement can be found in Ref. .

The resonator is characterized by both ν_r and the width of its resonance peak Δν_r.
These values combine to give the quality factor of the resonator,

    Q=ν_r/Δν_r

The resonator used in this work has a room-temperature unloaded Q of 170, and as characterized below a low-temperature Q of ∼200.
This low Q is good for pulsed experiments as the pulse shape is less distorted with lower Q values, and also allows us to do two-tone ESR using a monomodal resonator, by operating on either side of the resonance peak, though the frequency separation achieved this way is limited to 10s of MHz.
A truly bimodal resonator<cit.> would enable ESR at frequencies with essentially arbitrary separations, that could be addressed either with one antenna for both frequencies or with dedicated antennas for each.



 §.§ Controller

Aside from the RF source, most of the components described above are “dumb”, in that they do not require any control input telling them what to do, they just perform their assigned task with whatever signal inputs they get. 
There are three significant “smart” exceptions that take TTL inputs: the switches have control inputs that open and close them; the digital step attenuator has an input that controls the attenuation of the signal; and the oscilloscope has a trigger input. 
All of these control inputs are provided by a low-cost FPGA, running a custom configuration written in the hardware description language Verilog.
For the FPGA's targeted in this work, the control logic can be simulated and verified using open-source tools, enabling fast iteration and development.
As currently configured, the FPGA runs at 200 MHz, setting a minimum timestep of 5 ns for any of the control inputs.
The frequency is configurable, and this frequency was chosen to minimize the timestep while ensuring stable operation of the control program, as running it any faster could create situations where signals propagating within the FPGA would not be able to reach their destinations in time for the next clock tick, introducing random deviations from the programmed behavior.

Using an FPGA in this way provides a great deal of expandability for the spectrometer. 
Adding additional “smart” components, or the ability to create more complex pulse sequences, can be done by writing additional control logic into the Verilog configuration. 
Such expansion would likely require the reduction of the clock speed, but increasing the timestep to 10 ns, for example, would allow for a significant amount of expansion, such as the addition of more pulses to perform Carr-Purcell-Meiboom-Gill sequences <cit.>.

The initial configuration of the FPGA is done by writing out the control logic in Verilog and loading it onto the board. 
All code used to configure and interact with the FPGA is provided in the supplementary materials.
As currently configured, the FPGA supports up to two independent RF output channels, either in CW mode (with one or the other output enabled at any given time), or in pulsed mode with up to three independent pulses on one output channel and up to two independent pulses on the other.
While fully reconfiguring the FPGA takes tens of seconds, a variety of the control parameters are directly adjustable via USB at a fraction of that time.
These parameters will be highlighted when they are introduced in the following section.



§ SPECTROMETER OPERATION

This section will focus on the important operational details of our spectrometer and will not cover the basics of ESR operation; for a more general overview of both CW and pulsed ESR, see Refs.  and .
The general procedure is to find the resonant frequency of the resonator and then perform one or more pulsed experiments to manipulate and probe the spin sample.

Characterization was performed at zero applied magnetic field using the resonator described above with a resonant frequency of ν_r=3.902 GHz, coupled to a capacitive antenna in reflection mode, while immersed in a bath of liquid helium at 3.0 K in a Janis SHI-950T cryostat.
The sample was a solution of the MNM
 Cr_7Mn ([Pr_2NH_2][Cr_7MnF_8(Piv)_15(O_2C-py]),<cit.> diluted to 10% concentration by volume in toluene, and the input power was 21 dBm.
This sample was chosen based on its zero-field clock transition around 3.9 GHz, our familiarity with it from previous work,<cit.> and because future work will explore using dimers of this MNM as spin qubits <cit.>.
All the data presented here was taken on a 4-channel Tektronix MSO24 oscilloscope; monochromatic experiments can be done with a two-channel oscilloscope, but two-tone experiments require either one four-channel scope or two two-channel scopes.




 §.§ Determining ν_r

As previously mentioned, the current version of this spectrometer uses CW ESR only for preliminary resonator characterization, letting us scan the input frequency for dips in the background associated with the resonant frequency ν_r.
Precisely determining ν_r is done by sending in a short (∼50 ns) pulse at ν_c and monitoring the response immediately after it ends.
When ν_c is far from ν_r, there will be effectively no signal. 
However, when ν_c is at or near ν_r, the resonator will absorb energy from the pulse, and when the pulse ends that energy will radiate back into the antenna over a decay time

    t_decay=Q/πν_r


That produces a “ring-down" signal near ν_r; Fig. <ref>(a) shows the in-phase (I) portion of such a signal for the resonator used in this work, and Fig. <ref>(c) shows the amplitude of the same signal, calculated from the in-phase (V_I) and quadrature voltages (V_Q) as

    V_Amp=√(V_I^2+V_Q^2)


Finding ν_r from this data can be done in two ways.
The first takes a fixed-time cut of the amplitude data and fits that voltage versus the carrier frequency to a Lorentzian function, giving the central frequency ν_r and the width of the peak Δν_r.
An example of this is shown in Fig. <ref>(d), for the vertical cut indicated by the yellow dashed line in (c) at 178 ns.
The second method uses the oscillations that are present in either the I or Q signals, as seen in Fig. <ref>(a).
Applying a carrier frequency ν_c that is detuned from the resonator frequency ν_r results in a time-dependent phase shift, oscillating the signal between I and Q at the beat frequency ν_b=|ν_c-ν_r|.
Taking the Fourier transform of the I or Q data, as shown in Fig. <ref>(b), allows the extraction of that beat frequency as a function of ν_c, and fitting that data to find where it goes to zero gives ν_r.
Once ν_r is found by either method, the quality factor Q can be found by simply fitting the ring-down signal at ν_r to a decaying exponential and plugging the resulting time constant and frequency into Eq. <ref>.
For the resonator used in this work, the Lorentzian and Fourier methods give resonant frequencies of ν_r=3901.6±0.3 MHz and ν_r=3901.8±0.5 MHz, respectively, and the quality factor is Q=198±1.



 §.§ Pulse Generation

Once ν_r is determined, a variety of spin echo experiments can be done by generating one or more sets of pulses, depending on whether the spectrometer is operating at a single frequency on one or two outputs (monochromatic), or at two different frequencies for the two outputs (two-tone).



  §.§.§ Monochromatic Pulses

As noted above, the spectrometer can create up to three pulses at a single frequency, using either one or both outputs; the third pulse is currently only enabled on output B.
The default configuration has two of the pulses forming a Hahn echo pulse sequence, the first having a width of t_π/2 and the second a width of t_π, with a delay τ between them.
The third pulse is optionally used as a nutation pulse, occurring an amount of time τ_nut before the first Hahn pulse and having a width of t_nut.
As currently configured, all pulses are programmatically-adjustable in increments of the timestep, 5 ns, with a minimum value of 20 ns and a maximum value of 327.68 μs, with the exception of t_nut which has a maximum value of 1.28 µs; the 20 ns minimum pulse time is set by the on/off time of our switches.
The delays between any of the pulses are adjustable from 5 ns to 327.68 μs, and the sequence repetition time T is adjustable from 5 ns to 21.47 s.
The first Hahn pulse is also attenuated by 3 dB relative to the other pulses, giving it half the power so that the width of the two Hahn pulses can be the same.
The scope trigger rising edge is coincident with the rising edge of the first Hahn pulse; the included acquisition code is configured to trigger the scope on the rising edge and automatically offset to put the expected echo on the scope screen. 



  §.§.§ Two-tone Pulses

The two RF outputs can also produce two different frequencies, either for two-tone ESR or for heterodyne detection.
As discussed in the two-tone electronics section (<ref>), minor changes to the electronics are required for simultaneous detection at both frequencies.
The spectrometer can create one or two pulses for each output, and as with the monochromatic setup output B can also have a third pulse.
The details are mostly same as with the monochromatic pulses, and the extra attenuation is timed to coincide with the first Hahn pulse on output A.
It is trivial to swap the frequencies of each output.




 §.§ Background Subtraction

Once ν_r is determined, a Hahn echo sequence is performed.
If there is a strong enough echo signal over the desired experimental parameters, experiments can be done directly by integrating the signal over an appropriate range. 
Unfortunately, for many situations the signal to noise ratio is too low for this to be an effective approach, due to a variety of factors including small samples and large amounts of background noise.
To address these issues we have two main tools: averaging over multiple signals, and background subtraction.
The oscilloscope has built-in averaging, but it is often necessary to take more averages than the maximum provided by the scope.
Our acquisition software in the supplementary materials enables additional averaging as needed, with no upper limit (the functional upper limit is the amount of time required).

If white noise is the only factor making it difficult to see the signal, averaging can suffice to bring out the signal.
However, there can also be systematic offsets, coming from the other electronics for a variety of reasons, and averaging will not remove these.
Instead, we combine averaging with background subtraction, where we take two different measurements where ideally the only thing that changes is the spin echo signal, and then subtract one from the other to remove a variety of non-sample-dependent background effects, such as ground fluctuations when the RF switches open and close, or cavity ring-down.
We do this either by changing the Hahn echo delay τ, or by changing the relative phase between outputs A and B.
Not all backgrounds can be removed with these methods, but they have proved quite effective at removing a range of electronic artifacts that would otherwise obscure the spin echo signal; because the resonator ring-down depends on the phase of the input signal, it can only be subtracted using the delay-based method.
For our background subtraction comparison, shown in Fig. <ref>, we chose τ such that the background includes some resonator ring-down.
Note that delay-based subtraction can be done with any configuration of the spectrometer, but phase-cycling requires a monochromatic experiment with output A providing the reference frequency (going into the LO port of the mixer) and output B providing the excitation pulses.
Phase-cycling generally results in a higher signal to noise ratio, when the background is not phase-dependent.

For our delay-based background subtraction, we set up our Hahn echo sequence and take a signal measurement V_sig, as shown by the blue squares in Fig. <ref>(a).
We then change τ (increasing it by 5 µs if τ<10 µs, otherwise decreasing it by 5 µs) but keep the scope window fixed to shift the expected echo signal out of the scope window while leaving the background mostly unchanged, and take a background measurement V_back as shown by the orange triangles in Fig. <ref>(a)).
Subtracting V_back from V_sig gives us our data, V_data=V_sig-V_back, as shown by the green circles in Fig. <ref>(a), where the resonator ring-down background has been mostly removed.

For our phase-cycled background subtraction, we exploit the fact that if we shift the phase of an input signal by Δϕ=180^∘, the portion of the output signal that was downconverted from the carrier frequency (and thus was coming from the resonator and sample) will be inverted.
With our Hahn echo sequence set up and with a relative phase of Δϕ=0, we can take an in-phase measurement V_0, as shown by the blue squares in Fig. <ref>(b).
If we then change the relative phase to Δϕ=180^∘, we can take an inverted measurement V_180, as shown by the orange triangles in Fig. <ref>(b).
The low-frequency background in each measurement will be largely identical, because much of it does not depend on the phase, but the echo signal will now be inverted; as noted above, one background that does depend on the phase is resonator ring-down, so if that significantly contributes to the background in an experiment the delay-based background subtraction should be used. 
If we then take V_data=V_180-V_0 we will be subtracting the background but doubling the echo signal, as shown by the green circles in Fig. <ref>(b); this data is usually halved to maintain a constant signal size.
The resonator ring-down background, being phase-dependent, is still present.



 §.§ Standard Experiments

While many different experiments can be made with the spectrometer as currently configured, here we present several that are included as turn-key options in the acquisition code.
Many other experiments using 1-5 pulses can be set up with minor changes to the acquisition code, and experiments using more pulses can be implemented by changing the FPGA configuration.
For each experiment, phase-cycling was used for background subtraction, with delay times τ chosen to exclude any resonator ring-down from the background.




  §.§.§ Monochromatic Experiments

Three different single-frequency experiments are shown in Fig. <ref>.
In addition to these experiments, the acquisition software is also configured to easily do pulse sweep measurements (sweeping t_π/2 and t_π), phase sweep measurements (sweeping Δϕ), and frequency sweep measurements (sweeping ν_c).

In (a), we present a Rabi oscillation experiment.
A Hahn echo pulse sequence is set up with t_π/2=t_π=110 ns (based on a rough preliminary sweep of the pulse widths) and τ=250 ns.
A nutation pulse is sent 5 µs before the π/2 pulse, and the width of the nutation pulse t_nut is swept from 0 to 400 ns.
Integrating the area under the echo results in the data shown in <ref>(d), and fitting that data to a decaying sinusoid (the green dashed line) returns a π/2 pulse time of 106±2 ns.


In (b), we present an inversion recovery experiment.
t_π/2 and t_π are both set to the 110 ns pulse width determined in the Rabi experiment, with the same τ as in (a), and a nutation pulse of width t_nut=110 ns; because the attenuator only adds attenuation to the π/2 pulse, this nutation pulse performs a π rotation.
The delay between the nutation pulse and the π/2 pulse, T, is swept from 5-100 µs to determine the longitudinal relaxation rate T_1.
Fitting the integrated echo to an exponential (as shown in Fig. <ref>(e)) yields T_1=23±1 µs.

Finally, in (c) we present a Hahn echo delay sweep.
The same Hahn echo pulse times from (b) are used, but now τ is swept from 250-1500 ns to determine the transverse relaxation rate T_2.
Fig. <ref>(f) shows the integrated echo data and an exponential fit that yields T_2=390±10 ns (this value is limited by our temperature and level of dilution).



  §.§.§ Two-tone Experiments

With two separate output frequencies, all of the monochromatic experiments can be done on each frequency individually (with the exception of the phase sweep), but it is also possible to do inherently multi-frequency measurements.
As noted above, only delay-based background subtraction is possible with two-tone experiments.
Our resonator has a low enough Q that we are able to detune more than 40 MHz from ν_r, and we have tested both double-electron-electron-resonance (DEER) pulses and two-tone hole-burning (HB) experiments.
Our sample, however, is made up of monomers of Cr_7Mn, so even though we can excite spins at two frequencies, there is no coupling between those spins and thus there is no DEER signal.
However, we can demonstrate that pulses at both frequencies are working properly with a HB experiment <cit.>.
As shown in Fig. <ref>(a), a nutation pulse with width t_nut (so labeled because for this spectrometer it is the same nutation pulse as that used in the Rabi experiment) is sent at one frequency such that it performs a π rotation to “burn a hole” in the spin spectrum at that frequency, followed after time T by a Hahn echo sequence at the other frequency.
The initial nutation pulse is longer and lower-power than the Hahn pulses, to have a narrower bandwidth.
Integrating the echo signal both with (solid lines in Fig. <ref>(a)) and without (dashed lines) the nutation pulse, and then subtracting one from the other, allows the hole to be characterized, with variation of either frequency, the nutation width and power, or delay, yielding information about the pulse bandwidth and/or spin diffusion in the sample.

For our HB experiment we kept ν_B constant at 3925 MHz and varied ν_A from 3911-3940 MHz, and the results are shown in Fig. <ref>(b-d); note that our echo signal is ∼23 MHz above ν_r due to the interaction between the resonant width of the resonator and the inhomogeneous broadening of our Cr_7Mn sample, whose central spin transition frequency is ∼3.95 GHz.
Our nutation pulse had a width of t_nut=275 ns and a power of 15 dBm, with T=5 μs, t_π/2=t_π=80 ns, and τ=300 ns.
Echo signals, using delay-based background subtraction, are shown both without (b) and with (c) the nutation pulse.
The signal from (b) is subtracted from (c) and the resulting data is integrated to show the hole in Fig. <ref>(d).
A Lorentzian fit to the hole gives a width of 5.6±0.7 MHz, much narrower than the overall echo linewidth of ∼75 MHz.
This result demonstrates the ability of the spectrometer to operate at two frequencies simultaneously.



§ CONCLUSIONS

The spectrometer described herein is effective, flexible, and inexpensive. 
In its current form it can perform many common ESR experiments, and can be relatively easily extended to incorporate additional measurements in the future. 
As of the time of writing, the components of the spectrometer itself (not including the cryostat, oscilloscope, or DC voltage source) cost about , with the majority of that coming from the RF source, the mixer, and the amplifiers.
With the appropriate choice of resonator, it enables experiments both within and outside the standard ESR frequency bands. 
Additionally, the multiple-output nature of the spectrometer can enable two-tone ESR, where two different frequencies are sent to the resonator simultaneously. 
As demonstrated, the spectrometer itself currently supports such experiments, and our lab is working on bimodal resonator designs that will allow simultaneous ESR for spin samples with frequencies separated by more than 1 GHz.



§ SUPPLEMENTARY MATERIAL

We provide the code used to program the FPGA, as well as the data acquisition code. Instrument drivers are provided for all equipment used, including the temperature controller/thermometer (Lakeshore 335) and programmable power supply (GPD-3303S).


We thank Jonathan Friedman, Brendan Sheehan, Kai-Isaak Ellers, and Andrew Mounce for useful conversations and advice, and Richard Winpenny for samples and useful conversations. Part of this work was enabled by the use of pyscan (github.com/sandialabs/pyscan), scientific measurement software made available by the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy. Support for this work was provided by the Hamilton College Dean of Faculty, the Muhlenberg College Provost, and the Muhlenberg College Dean of Students.




§ AUTHOR DECLARATIONS

CC: Conceptualization (lead); formal analysis (lead); investigation (lead); methodology (lead); resources (lead); software (lead); supervision (lead); writing – original draft (lead); writing – review and editing (lead). SD: Investigation (supporting); software (supporting); writing – review and editing (supporting). AA: Resources (supporting); writing – review and editing (supporting). GT: Resources (supporting); writing – review and editing (supporting).

The authors have no conflicts to disclose. The data that supports the findings of this study are available from the corresponding author upon reasonable request.





§ SPECTROMETER COMPONENTS

[c]| p.1 | p.15 | p.07 | p.07 |
	Component     Part No.     Cost ($)     Freq. Range (GHz)

	RF Source     WindFreakTech SynthHD     1999     0.054-15

	RF Switch     MiniCircuits (MC) ZFSWA2R-63DR+     88.48     0.5-6

	RF Splitter     MC ZX10R-14-S+     88.48     0-10

	Power Amplifier     MC ZVA-183-S+     1682.17     0.7-18

	Digital Step Attenuator     MC ZX76-31R75PP-S+     166.91     0.009-6

	Circulator     Fairview Microwave (FM) SFC4080B / SFC2040A     325.99 / 407.99     4-8 / 2-4

	RF Limiter     MC VLM-63-2W-S+     65.71     0.03-6

	Low-Noise Amplifier     FM SLNA-060-40-09-SMA     1440     2-6

	IQ Mixer     Marki Microwave MMIQ-0218LXPC     1925.10     2-18

	FPGA     Lattice ECP5 Evaluation Board     154.69     N/A

	





§ DIFFERENT FREQUENCY RANGES

Most of the original components work in the 2-6 GHz range, the main exception being the circulator; two circulators are provided above, for the 2-4 GHz range and the 4-8 GHz range. Accessing a higher frequency range (6-15 GHz) can be done by substituting the following components (listed with component name, price, and frequency range):


  * RF switch: Analog Devices HMC547ALP3 (evaluation board EV1HMC547ALP3), $244.87, DC-20 GHz

  * RF Splitter: This component can be removed, using output A as the mixer LO and output B as the pulsed output.

  * Digital Step Attenuator: This component can be removed, and different tip angles achieved by varying the pulse width.

  * RF Limiter: This component can be removed or replaced with an RF switch.

  * Low-Noise Amplifier: Fairview Microwave FMAM1071, $2240, 6-18 GHz

  * Circulator: Fairview Microwave SFC0818, $431.99, 8-18 GHz




§ TWO-TONE ELECTRONICS DIAGRAM








 Paper Title: Local analysis of a single impurity on a graphene Josephson Junction.







Dipartimento di Fisica e Astronomia "Ettore Majorana",  Universitá di Catania, Via S. Sofia, 64, I-95123 Catania, Italy
INFN, Sez. Catania, I-95123 Catania, Italy.


francesco.pellegrino@dfa.unict.it
Dipartimento di Fisica e Astronomia "Ettore Majorana",  Universitá di Catania, Via S. Sofia, 64, I-95123 Catania, Italy
INFN, Sez. Catania, I-95123 Catania, Italy.
CNR-IMM, Catania (University) Unit, via Santa Sofia 64, I-95123 Catania, Italy


Dipartimento di Fisica e Astronomia "Ettore Majorana",  Universitá di Catania, Via S. Sofia, 64, I-95123 Catania, Italy
INFN, Sez. Catania, I-95123 Catania, Italy.
CNR-IMM, Catania (University) Unit, via Santa Sofia 64, I-95123 Catania, Italy


Dipartimento di Fisica e Astronomia "Ettore Majorana",  Universitá di Catania, Via S. Sofia, 64, I-95123 Catania, Italy
INFN, Sez. Catania, I-95123 Catania, Italy.
CNR-IMM, Catania (University) Unit, via Santa Sofia 64, I-95123 Catania, Italy



§ ABSTRACT 
In this work, we investigate the local effects of a single short-range impurity on the electron system of a short ballistic graphene Josephson Junction. Within the Dirac-Bogoliubov-De Gennes approach, we systematically analyze the local density states, whose subgap energy dependence enables us to distinguish between elastic and inelastic scattering processes and identify the magnetic nature of the impurity. Furthermore, 
we observe that the spatial dependence of the local density of states is a sensitive probe of the microscopic processes resulting in subgap impurity-induced bound states. The Fourier analysis evidences the wavevectors related to the momenta of the high-transmissive channels in ballistic graphene.





Local analysis of a single impurity on a graphene Josephson Junction
    Elisabetta Paladino
    August 5, 2024
====================================================================





§ INTRODUCTION



The potential of quantum technologies is being explored by developing hybrid systems that combine different physical components with complementary functionalities <cit.>.
Within this context, gate-tunable hybrid superconducting qubits, known as gatemons, have been implemented using semiconducting nanowires <cit.>, InAs Josephson Junctions (JJs) <cit.>, two-dimensional materials <cit.>, van der Waals heterostructures <cit.> and graphene <cit.>. These devices offer reduced dissipative losses, crosstalk, and compatibility with high magnetic fields <cit.>, paving the way for the realization of fault-tolerant topological qubits based on Majorana zero modes <cit.>. High-quality graphene superconductor heterostructures with clean interfaces, obtained by encapsulating graphene in hexagonal boron nitride (hBN) <cit.>, have enabled ballistic transport of Cooper pairs over micrometer-scale lengths, gate-tunable supercurrents that persist at large parallel magnetic fields  <cit.>, and different features of two-dimensional Andreev physics <cit.>. Moreover, thanks to the low specific heat of graphene embedded in hBN, highly sensitive graphene Josephson Junction (GJJ) based microwave bolometers have been realized for circuit quantum electrodynamics applications <cit.>. Single near-infrared photon detection has also been achieved by coupling photons to localized surface plasmons of a GJJ, which can be integrated into JJ-based computing architectures as high-speed, low-power optical interconnects <cit.>. GJJs are also an excellent platform for generating exotic quantum states, such as long-lived Floquet-Andreev states generated by applying continuous microwave light without significant heating <cit.>.


The Josephson effect in heterostructures is described microscopically by the proximity effect and constructive interference between Andreev processes at the two normal-superconducting (N-S) interfaces, which lead to coherent electron-hole superpositions known as Andreev bound states (ABSs). In the short junction regime, the current-phase relation (CPR) resulting from the phase dependence of the ABSs spectrum differs from the sinusoidal CPR of the tunnel JJs <cit.>. 

Recently, there has been interest in the spatial control properties of ballistic GJJ. In fact, an experimental methodology has been developed to measure and control the real-space current density in two parallel ballistic GJJs, based on Fourier and Hilbert transformations of the magnetic field-induced modulation of the critical current <cit.>.
Tunneling spectroscopy measurements in GJJs revealed the possible presence of weakly coupled microscopic quantum dots to proximitized graphene, which act as energy filters in the tunneling process <cit.>. 
A theoretical analysis has suggested that a homogeneous dilute distribution of impurities in a GJJ can alter the critical current and the skewness of the CPR <cit.>. 

Moreover, defects in the nearby substrate can act as carrier traps and induce fluctuations in the carrier density of the graphene channel <cit.> and lead to critical current fluctuations with a 1/f spectrum <cit.>. 


Motivated by findings related to point systems weakly interacting with proximitized graphene and the latest interest in controlling the spatial properties of ballistic GJJs, we investigate the local effect of a short-range impurity on the low-energy properties of a short planar GJJ, employing an analytical approach based on the Dirac-Bogoliubov-de Gennes model <cit.>. Two paradigmatic descriptions of a single impurity are considered: the Anderson model <cit.> and the Lifshitz model <cit.>. According to the latter, an impurity interacts with the density of conduction electrons through a scalar potential <cit.>, this model has been used to describe a vacancy in a graphene crystal <cit.>. The Anderson model, which has been used to study the effect of adatoms on the graphene electron system <cit.>,  it includes the possibility of electron transfer from the host to some energy level that belongs to the adsorbed atom <cit.>.
Here, we take into account an extra magnetic term for both models to describe the exchange interaction between the local spin on the impurity site and the electrons that flow through the GJJ. 

We focus on a single impurity located in the middle of the normal phase region and analyze how it affects the LDOS within the superconductive gap |E|<Δ.
This quantity is experimentally detectable by scanning tunneling spectroscopy <cit.>. 
Here, the LDOS reveals that ABSs are sensitive to the type of scattering processes caused by the impurity.
In proximity to the impurity, the energy dependence of the LDOS is sensitive to the magnetic properties of the impurity, and it is different for the Anderson or Lifshitz model. 
We analyze the conditions for the occurrence of an impurity-induced subgap bound state. Finally, at the energy corresponding to this impurity-induced bound state, we analyze the spatial dependence in the normal phase region along the transverse direction and apply a Fourier analysis.

This paper is structured as follows. Section <ref> provides technical details on the Dirac-Bogoliubov-De Gennes model used to describe ABSs. Sections <ref> and <ref> present the Anderson and Lifshitz models, respectively, and we analyze the energy dependence of the LDOS near the impurity placed in the middle of the normal phase region. Section <ref> examines the spatial dependence of the LDOS along the normal phase region, and we analyze it in Fourier space. Finally, the conclusions are reported in Section <ref>.







§ MODEL



We analyze a GJJ in the ballistic regime, as depicted in Fig. <ref>, which is composed of two identical metallic superconductive electrodes on the sides and a graphene monolayer in the middle.
The entire device has a finite width W and is considered infinite along the x̂-direction, the normal graphene stripe has length L, and we assume that W≫ L.
The electron system is described by the following Dirac-Bogoliubov-de Gennes (D-BdG) Hamiltonian <cit.>


    ℋ̂_D-BdG=   ∑_ζ= ±∫ d^2rΨ̂_ζ^†(r) H_D-BdGΨ̂_ζ(r),
    
    H_D-BdG=   τ_z[U(r) 1_σ+ħ v_D/i(∂_xσ_x+∂_yσ_y)]
       +τ_x1_σReΔ(r)-τ_y1_σImΔ(r) ,


where ζ=± denotes valley indices.
Here, the four components spinors Ψ̂_±(r) are defined as

    Ψ̂_+(r)=[ψ̂_A, K, ↑^†(r), ψ̂_B, K, ↑^†(r), ψ̂_A, K^', ↓(r), ψ̂_B, K^', ↓(r)]^† ,
       Ψ̂_-(r)= [-ψ̂_B, K^', ↑^†(r), ψ̂_A, K^', ↑^†(r),-ψ̂_B, K, ↓(r), ψ̂_A, K, ↓(r)]^† ,


where v_ D∼ c/300 is the Fermi velocity in monolayer graphene, the set composed by the two-dimensional matrix identity _σ (_τ) and the Pauli matrices {σ_x,σ_y, σ_z } ({τ_x,τ_y, τ_z }) act on the two-dimensional sublattice A-B (electron-hole) subspace.
The superconducting order parameter Δ(r), and the scalar potential U(r) are defined across the junction as follows


    Δ(r)   =Θ(|x|-L / 2) Δ e^i ϕ_0(x), 
    ϕ_0(x)   =Θ(x) ϕ_R+Θ(-x) ϕ_L,


    U(r)=-μ_0Θ(L / 2-|x|)-U_0Θ(|x|-L / 2),


where Θ(x) is the Heaviside step function, and we assume that U_0 ≫ |μ_0| <cit.>.



The multiple coherent Andreev reflections at the interfaces between the normal phase region and the superconductive sides at x=± L/2 generate a set of ABSs <cit.>. These eigenstates of the D-BdG Hamiltonian have subgap eigenenergies, i.e. |E| <Δ. In the short junction limit L≪ξ, where ξ = ħ/(πΔ) is the coherence length, the eigenstates with energies above the gap, i.e. |E| >Δ, form the Andreev continuum. These states have eigenenergies that are independent of the phase difference ϕ=ϕ_ R-ϕ_ L, so ABSs are the only ones responsible for carrying the Josephson equilibrium supercurrent <cit.>.
In this study, we disregard the continuum and focus on the low-energy characteristics of the GJJ.


To obtain ABSs, we seek the eigenstates of the D-BdG Hamiltonian by exploiting the uniformity of the system along the y-direction


    φ_k, E,ζ(r)=e^i k y/√(W)φ̃_k, E(x),


where  k determines the wavenumber along the y direction, 
and E is the eigenenergy.
Since the D-BdG Hamiltonian in Eq. (<ref>) is independent of the valley index ζ, we omit it in φ̃_k, E(x), and it leads to double degeneracy in the ABS spectrum.

As shown in Fig. <ref>, there are three sharp partitioned sectors along the x direction for which we look for a solution to the stationary D-BdG problem of the following form


    φ̃_k, E(x) =  N_k,Eφ̃_k, E^(S-L)(x)     x≤ -L/2, 
     N_k,Eφ̃_k, E^(N)(x)     -L/2< x< L/2, 
     N_k,Eφ̃_k, E^(S-R)(x)     x ≥ L/2,


and afterward, we impose the continuity at the interfaces x=± L/2. The coefficient N_k,E denotes the normalization factor.
To search for eigenfunctions, we follow the approach used in Ref. pellegrino_commphys_2022 where the junction length was taken infinitesimal L/ξ→ 0. Here, we generalize to the case of a finite junction length L.

We start from the superconductive sides (x<-L/2 and x>L/2).
We write both φ̃_k, E^(S-L)(x) and φ̃_k, E^(S-R)(x) as a linear combination of terms of the form e^s η_λ xa_k, E, λ,s,  where s=±, λ=±. By imposing the exponential form, it solves the D-BdG equation in the superconductive sides for  


    η_λ= i √((U_0+λ√(E^2-Δ^2)/ħ v_D)^2-k^2) .


We exclude pairs of coefficients λ and s such that (λ,s)=(+,+) and (-,-) (where λ s=+) on the left side, and (λ,s)=(+,-) and (+,-) (where λ s=-) on the right side, since they correspond to 

wavefunctions that cannot be normalized.
Then, we obtain the following expressions


    φ_k, E^(S-L)(r)=   e^i k y[x^L_k,E e^-η_+(x-L/2)a_k, E,+,-. 
    
            +   .y^L_k,E e^η_-(x-L/2)a_k, E,-,+],


    φ_k, E^(S-R)(r)=   e^i k y[x^R_k,E e^η_+ (x-L/2)a_k, E,+,+. 
    
            +   . y^R_k,E e^-η_- (x-L/2)a_k, E,-,-] .


The four-dimensional vectors a_k, E, λ, s can be represented as


    a_k, E, λ, s= W_λ, sb_k, E, λ ,
 

where W_λ, s=_τ (Q_λ, sΛ_λ, s) is defined in terms of the matrices 


    Q_λ, s   =1/√(2)[σ_z+i(e^-s z_λσ_--e^s z_λσ_+)],
    Λ_λ, s   =1/√(2[1+e^s(z_λ+z_λ^*)])(1-σ_z)+
       +1/√(2[1+e^-s(z_λ+z_λ^*)])(1+σ_z),


which act on the sublattice space, where z_λ satisfies the identity e^s z_λ=k v_ D(k+s η_λ)/[U_0+λ√(E^2-Δ)], and 


    b_k, E, λ=1/√(2)[ e^i (ϕ_j + λ/2β(E) ), 0, e^-i λ/2β(E), 0]^T,


with β(E)= arccos(E / Δ) and j=R,L.

Assuming that the superconductive sides are in the large doping regime, i.e. U_0 ≫ |μ_0|, and Δ≪ U_0, then η_λ, defined in Eq. (<ref>), can be approximated <cit.> as


    η_λ≈  i U_0/ħ v_D- λ√(Δ^2-E^2)/ħ v_D ,


and z_λ≈ i π /2, which leads W_λ,s≈1/√(2)(σ_z+s σ_x). 

To find the solution of the stationary equation in the normal phase sector φ̃_k, E^(N)(x), we employ the transfer matrix approach <cit.> 
After algebraic manipulations, the stationary problem can be rewritten as the following ordinary differential equation in terms of the transfer matrix


    φ̃_k, E^(N)(x)=𝒯(k, E ; x) φ̃_k, E^(N)(-L / 2),



    d  T(k,E;x)/d x=[i τ_zσ_xE/ħ v_D+i σ_xμ_0/ħ v_D+σ_z k]  T(k,E;x) ,


with boundary condition 𝒯(k, E ; x=-L / 2)=1_τ1_σ.


The transfer matrix which solves the above problem is explicitly expressed as


    T(k,E;x) = _τ+τ_z/2𝕋(k,E;x) +  _τ-τ_z/2𝕋(k,-E;x) ,


where


    𝕋(k,± E;x)   = sin[q_± (x+L/2)] [iμ_0± E/ħ q_±σ_x+k/q_±σ_z ]
       + cos[q_± (x+L/2)]_σ ,


and q_± =√((μ_0± E/ħ)^2 -k^2).

Using the transfer matrix, we can impose the continuity condition on the wave functions at the N-S interfaces, and  we obtain the compact condition


    φ̃^ (S-R)_k,E(x=L/2)= T(k,E;L/2)φ̃^ (S-L)_k,E(x=-L/2) .


After a few algebraic manipulations, it can be rewritten as


    x^ R_k,Ea_k,E,+,+ +y^ R_k,Ea_k,E,-,-= 
       = T(k,E;L/2)[x^ L_k,Ea_k,E,+,-+y^ L_k,Ea_k,E,-,+] .


The expression above is a homogeneous system of four equations for the variables { x^ L_k,E,y^ L_k,E,x^ R_k,E,y^ R_k,E}. The solution of this system is not trivial only if the associated matrix is singular, i.e. its determinant D is zero. 
For each pair of values of the y component of the wave vector k and the phase difference ϕ, the energies E(j,k,ϕ) that nullify the determinant D are the eigenenergies of the ABSs, with j being the subband index.
The determinant D is an even function of energy, which leads to an even number of subbands E(j,k,ϕ), so the subband index runs over the range {± 1, ±2,…, ± n }, where 2n is the number of subbands, and the following property E(j,k,ϕ)=-E(-j,k,ϕ) is valid.
For each ABS, after finding the eigenenergy E(j,k,ϕ), we choose the nontrivial solution that guarantees the condition |x^ L_k,E(j,k,ϕ)|^2+|y^ L_k,E(j,k,ϕ)|^2=1.





Under the approximation in Eq. (<ref>), the normalization factor N_k,E(j,k,ϕ) is determinated for a given subband index j, y-component of the wavevector k, and phase difference ϕ, by enforcing the condition below


    ∫^W/2_-W/2 dy ∫^∞_-∞d x φ^†_k, E,ζ(r)  φ_k, E,ζ(r)|_E=E(j,k,ϕ) =1 ,


which gives


    N_k,E(j,k,ϕ)   =  1/√(1+B(j,k, ϕ))√(√(Δ^2-E(j,k,ϕ)^2)/ħ v_D) ,


where 


    B(j,k, ϕ)=   L/ξ√(1-(E(j,k,ϕ) / Δ)^2)
    ×   [x_La_k, E,+,-+y_La_k, E,-,+]^†𝒜(k, E(j,k,ϕ)) 
    ×   [x_La_k, E,+,-+y_La_k, E,-,+] ,


    𝒜(k, E)   =∫_-L / 2^L / 2d x/L𝒯^†(k, E, x) 𝒯(k, E, x) 
       = _τ+τ_z/2𝔸(k,E) +  _τ-τ_z/2𝔸(k,-E) ,



    𝔸(k,± E)   =[(μ_0± E)^2/ħ^2 ^2 q^2_±-k^2sin(2q_± L) /2q_± ^3 L]_σ
       
    -[k(μ_0± E)/ħ q_±-k(μ_0± E)sin(2q_± L)/2ħ q^3_± L]σ_y 
       +
    k sin^2(q_±L)/q^2_± Lσ_z ,


and x^ L_k,E and y^ L_k,E are two components of the nontrivial vector which solves the system in Eq. (<ref>) with E→ E(j,k,ϕ).


From now on, we focus on the short junction regime L/ξ≪ 1, where the subband index j can assume two values, namely {-1,1}, and we have E(j,k,ϕ)=jϵ(k,ϕ).
In particular, we will consider a finite value of L/ξ≪ 1, and note that in the limiting case L/ξ→ 0, we have B(j,k, ϕ)→ 0.

To describe the low-energy electronic properties, we project the full D-BdG Hamiltonian in Eq. (<ref>) onto the ABS subspace with the projector 𝒫̂_A <cit.>. 

The Andreev Hamiltonian ℋ_A=𝒫̂_A ℋ̂_D-BdG𝒫̂_A is expressed as


    ℋ̂_A=∑_ζ= ±∑_kϵ(k, ϕ) ( γ̂_ζ,+, k^†γ̂_ζ,+, k-γ̂_ζ, -,k^†γ̂_ζ,-,  k),


where  γ̂_j, ζ, k is the fermionic ABS operator. 

Each set of values of ζ,k determines a two-level system with energy splitting 2ϵ(k,ϕ).

Fig. <ref> shows the ABS spectrum as a function of the component k, with the phase difference set at ϕ=3π/4 and the chemical potential in the normal phase at μ_0 = 5 ħ /L (a) and μ_0 = 8 ħ /L (b). The solid black line is the spectrum obtained numerically for a finite short junction L/ξ=π/20, while the red dashed line is the universal analytical expression ± E(k,ϕ)=±Δ√(1-τ(k) sin^2(ϕ/2)), obtained in the limit L/ξ→ 0, where the k dependence is totally included in the transmission probability τ(k) which depends on the nature of the normal phase stripe <cit.>.
For the graphene electron gas <cit.>, the transmission probability of the normal state is τ(k)=(^2-k^2)/[^2-k^2 cos ^2 (L √(^2-k^2) )], with =μ_0/(ħ). 
From the comparison of the dispersion relations in Fig. <ref>, we see that the extrema are located at the same k, and in the limiting case L/ξ→ 0 they correspond to the extrema of the transmission probability <cit.>.
In Fig. <ref> one can see that there is an energy window, which we call mini-gap, in which excitations are prohibited <cit.>. The minigap δ(ϕ)=min_k[ϵ(k,ϕ)] depends on the phase difference ϕ.
In particular, in the limit L/ξ→ 0, we have δ(ϕ)= Δ |cos(ϕ/2)|.


We conclude this Section by introducing the particle density and the charge density operators, which are the quantities of interest.
Within the D-BdG formalism, the particle density and the charge density operators are expressed, in terms of the four component spinors defined in Eq. (<ref>), as


    ρ̂(r)=   ∑_ζΨ̂_ζ^†(r) _τ_σΨ̂_ζ(r),
    ρ̂_ C(r)=   ∑_ζΨ̂_ζ^†(r) τ_z _σΨ̂_ζ(r),


where in ρ̂_ C electrons (holes) have charge +1 (-1).


Applying the projector P̂_A on the density operators in Eq. (<ref>), we define the Andreev particle and charge densities operators as


    ρ̂_ A(r)=   P̂_Aρ̂(r)P̂_A=
            ∑_ζ∑_j,j'∑_k,k'φ^†_k, E(j,k,ϕ),ζ(r)
       ×_τ_σφ_k', E(j',k',ϕ),ζ(r)
            γ̂_ζ,j, k^†γ̂_ζ,j', k'
            ,
    ρ̂_ C,A(r)=   P̂_Aρ̂_ C(r)P̂_A=
            ∑_ζ∑_j,j'∑_k,k'φ^†_k, E(j,k,ϕ),ζ(r)
       ×τ_z _σφ_k', E(j',k',ϕ),ζ(r)
            γ̂_ζ,j, k^†γ̂_ζ,j', k'
            ,


whose support is the ABS subspace.

In the following sections, we study modifications of the spin-resolved local particle density of states (LDOS).
The spin-up electron and the spin-down hole LDOS are defined as


    ρ_ e↑(r, Ω)   = Tr[
         ρ̂_ A, e ↑(r)δ(Ω-Ĥ_ tot)] ,
    ρ̂_ A, e ↑(r)   =1/2[ρ̂_ A(r)+ρ̂_ C,A(r)] ,




    ρ_ h↓(r, Ω)   = Tr[
          ρ̂_ A, h ↓(r)δ(Ω-Ĥ_ tot)] ,
    ρ̂_ A, h ↓(r)   =1/2[ρ̂_ A(r)-ρ̂_ C,A(r)] ,


where we have Tr[·]=∑_ζ,j,k⟨∅|γ̂_ζ,j,k·γ̂^†_ζ,j,k| ∅⟩, where the reference state | ∅⟩ is the vacuum of all field operators γ̂_ζ,j,k.
The spin-down electron LDOS is given by ρ_ h↓(r, Ω)=ρ_ e↓(r, -Ω). 

So far we considered the bare GJJ in the short-junction regime. 
In the following, we include the presence of a localized impurity described in terms of the Anderson or Lifshitz model.
The total Hamiltonian Ĥ_ tot is composed of the Andreev Hamiltonian Ĥ_ A and a term Ĥ_ imp that takes into account the interaction of the ABSs with a single impurity.









§ SINGLE ANDERSON IMPURITY
 


In this Section, we discuss the Anderson model <cit.> which describes an impurity that causes inelastic scattering processes in a GJJ electronic system. Within this model, each impurity has two localized electronic states, distinguished by the spin projection along the z direction. For the spin-up and spin-down cases, the energies are E_d, ↑=μ_0+ϵ_d + ϵ_ Z and E_d,↓=μ_0+ϵ_d - ϵ_ Z, respectively. The
energy ϵ_d is the impurity energy with respect to the Fermi level μ_0, and the Zeeman energy term   breaks the time-reversal symmetry, and here any Coulomb repulsive term <cit.> is neglected.
Within the Bogoliubov-de Gennes approach, the Hamiltonian of the impurity can be expressed in a spinorial form as


    ℋ̂_ D   = Φ̂_d^† (ϵ_dτ_z+ϵ_Z_τ)Φ̂_d


where Φ̂_d^†=[c_d,↑^†, c_d,↓], and the fermionic operators c_d,↑^† (c_d,↓) creates (annihilates) an electron with spin-up (spin-down) bounded to the impurity. 


Here, we describe the interaction between a single impurity and the electron system in a graphene monolayer in terms of a tunneling Hamiltonian <cit.> expressed as


    V̂_ D   = V̂_d+V̂^†_d ,
    V̂_d   =∑_ζ= ±∫ d^2rΦ̂_d^† V_d, ζ(r) Ψ̂_ζ(r),


where the matrices V_d,ζ are defined as


    V_d,+(r)     =[[    v_A, d(r)    v_B, d(r)            0            0;            0            0 -v_A, d^*(r) -v_B, d^*(r) ]], 
    
            V_d,-(r)     =[[ -v_B, d^*(r)  v_A, d^*(r)            0            0;            0            0    v_B, d(r)   -v_A, d(r) ]].


We assume that the impurity is located at r_d in a carbon site and that the tunneling term acts on the electron system in graphene at the atomic scale.
Following Ref. pellegrino_commphys_2022, using a microscopic description based on a tight-binding model for the electron system in graphene, the matrix elements of the potential V̂_ D are expressed as


    v_α, d(r)=     t_0 √(A_c)[m_d  δ_α, A  + (1-m_d) δ_α, B] δ(r-r_d),



where A_c=3√(3)a^2/2 is the area of the unit cell <cit.>, with a=1.42 Å, t_0 is the tunneling amplitude which, without loss of generality, is taken real and positive. If the impurity is located at a carbon site belonging to the sublattice A (B) then one has m_d=1 (m_d=0). 

Here, we consider that the carbon site where the impurity is located belongs to sublattice A, and we have v_α, d(r)= t_0 √(A_ c)δ_α, Aδ(r-r_d).
The results remain unchanged regardless of the sublattice chosen, as a result of the sublattice symmetry.


The local interaction potential projected in the ABS subspace is explicitly written as


    𝒱̂_dP̂_ A = t_0∑_j ∑_ζ∑_k Φ̂_d^† w_d, ζ,j,kγ̂_ζ,j,k


where


    w_d,±,j,k   =g_d,±φ_k, E(j,k,ϕ),±  (r_d)
       =g_d,±e^i k y_d/√(W)φ̃_k, E(j,k,ϕ)(x_d),



    g_d, +   =  √(A_c)[  1  0  0  0;  0  0 -1  0 ] ,
    
     g_d, -   =  √(A_c)[  0  1  0  0;  0  0  0 -1 ] .


To study the local effect induced by an Anderson impurity on the electron system in a GJJ, we use the total Hamiltonian ℋ̂_ tot=ℋ̂_ A+ℋ̂_ imp, where

ℋ̂_ imp=Ĥ_ D+ 𝒱̂_d 𝒫̂_A+ 𝒫̂_A𝒱̂^†_d, which consists of the impurity Hamiltonian Ĥ_ D, and the tunneling terms which connect the ABSs and the Anderson impurity. 

To calculate LDOS, we use the following expressions


    ρ_ e ↑(r, Ω)   =-1/π Im Tr[  ρ̂_ A,e ↑(r) 𝒢̂_ tot(Ω+iη)],
    ρ_ h ↓(r, Ω)   =
    -1/π Im Tr[  ρ̂_ A,h ↓(r) 𝒢̂_ tot(Ω+iη)],

which involves the total Green function 


    𝒢̂_ tot(Ω)=(Ω-ℋ̂_ tot)^-1 ,


with η=0^+.
After algebraic manipulations, we write the projection of the total Green function onto the ABS subspace as


    𝒢̂(Ω)=   𝒫̂_A𝒢̂_ tot(Ω) 𝒫̂_A=(Ω-ℋ̂_eff)^-1,
    ℋ̂_eff=   ℋ̂_A+𝒫̂_A𝒱̂_d^†(Ω-ℋ̂_d)^-1𝒱̂_d𝒫̂_A.



Before calculating the LDOS in the presence of a single Anderson impurity, we consider the clean GJJ.
For the unperturbed GJJ system, the ABS Green function takes the simple form 


    𝒢̂_0(Ω)   =(Ω-ℋ̂_A)^-1
       =∑_j=±∑_ζ= ±∑_k1/Ω-j ϵ( k, ϕ)γ̂_j, ζ, k^†γ̂_j, ζ, k,


and by replacing it in Eq. (<ref>), we find the LDOS


    ρ_0,ℓ(r, Ω)=-1/π Im Tr[ρ̂_ A,ℓ(r) 𝒢̂_0(Ω+iη )].




It is useful to introduce the following 4×4 matrix Green function


    G̅_0(r,r',Ω)=∑_j ∑_k
    φ_k, E(j,k,ϕ),ζ (r) φ^†_k, E(j,k,ϕ),ζ (r')/Ω -jϵ(k,ϕ)
       =  ∑_k  e^ik(y-y')/W[ ∑_j φ̃_k, E(j,k,ϕ)(x) φ̃^†_k, E(j,k,ϕ)(x')/Ω -jϵ(k,ϕ)],


which acts on the direct product of the electron-hole subspace and sub-lattice subspace on which the spinors of the D-BdG formalism are built. It allows one to write a compact and intuitive form of the LDOSs. 
Although the valley index ζ is indicated in the initial line, the spinorial wavefunction does not depend on it, so it is omitted in G̅_0(r,r',Ω).
Furthermore, looking at the second line of Eq. (<ref>), each entry of G̅_0(r,r',Ω) has the form of a Fourier transform, and we calculate them using the Fast Fourier transform algorithm <cit.>, see details in Appendix <ref>. 
Starting from the definition in Eq. (<ref>), applying the cyclic property of the trace, and using Eq. (<ref>), the LDOS in the clear limit takes the following compact form


    ρ_0,ℓ(r, Ω)=-1/π Im{ tr[M_ℓG̅_0(r,r,Ω+i η)] },


where ℓ={ e ↑, h ↓}, M_ e ↑=1/2(_τ + τ_z) _σ, and M_ h ↓=1/2(_τ - τ_z) _σ, and tr[·] denotes the trace over the direct product of the electron-hole subspace and sub-lattice subspace.

To calculate LDOS in the presence of a single impurity, one has to find the ABS Green function 𝒢̂(Ω) expressed in Eq. (<ref>). 
For this aim, we rewrite Eq. (<ref>) in the form of a Dyson equation as


    𝒢̂(Ω)=𝒢̂_0(Ω)+𝒢̂_0(Ω)  Σ̂(Ω) 𝒢̂(Ω) ,


where  the self-energy is


    Σ̂(Ω)   =  𝒫̂_A𝒱̂_d^†(Ω-ℋ̂_ D)^-1𝒱̂_d𝒫̂_A.



The exact solution to the Dyson equation, as detailed in Appendix <ref>, is expressed in the following closed-form 
 

    𝒢̂(Ω)  =𝒢̂_0(Ω)+∑_ζ,ζ'∑_j, j^'∑_k, k^' F_ζ,j,k,ζ',j',k' (Ω)γ̂_ζ,j,  k^†γ̂_ζ^', j^', k^',


where the analytical expression of F_ζ,j,k,ζ',j',k'(Ω) is reported in Appendix <ref>, see Eq.(<ref>).

Based on the results in Eq.(<ref>), for both electrons and holes, the LDOS is written as the sum of two terms ρ_ℓ(r, Ω)=ρ_ℓ,0(r, Ω)+δρ_ℓ(r, Ω), where



    δρ_ℓ(r, Ω)=-1/π Im Tr{ρ̂_ A,ℓ(r)[𝒢̂(Ω+iη )- 𝒢̂_0(Ω+iη )]}.


Using Eq. (<ref>), we have the following


    δρ_ℓ(r, Ω)=-1/π Im∑_ζ∑_j,j'∑_k,k'φ^†_k, E(j,k,ϕ),ζ(r) M_ℓ
       ×φ_k', E(j',k',ϕ),ζ(r) F_ζ,j',k',ζ,j,k(Ω+i η) ,


where η=0^+.
Applying the cyclic property of the trace and using Eqs. (<ref>) and (<ref>), the expression above can be rewritten as



    δρ_ℓ(r, Ω)=-1/π Im tr{M_ℓG̅_0(r,r_d,Ω+iη)
    ∑_ζ g_d,ζ^† X(Ω + i η) [_τ - Y(Ω+iη)]^-1 g_d,ζG̅_0(r_d,r,Ω+iη)  } ,



where


    Y(Ω)= ∑_ζ g_d,ζG̅_0( r_d, r_d,Ω) g^†_d,ζX(Ω) ,


and the nature of the impurity is included in the 2×2 matrix X(Ω). For an Anderson impurity, one has 


    X(Ω) → t_0^2 G_d(Ω) ,


with


    G_d(Ω)    =  [(Ω-ϵ_Z)_τ -ϵ_dτ_z]^-1
       =
    [ 1/Ω-(ϵ_d+ϵ_ Z)              0;              0 1/Ω+(ϵ_d-ϵ_ Z) ] .


Fig. <ref> shows the LDOS for electrons with spin-up (solid black line) and holes with spin-down (red dashed line) as a function of energy Ω. The junction length, Fermi level and phase difference are set to L/ξ=π/20, μ_0=5 ħ/L, and ϕ=3π/4, respectively. To obtain the LDOS numerically, a finite value of η=10^-3Δ is used, which produces a fictitious finite linewidth of the energy levels within the mini-gap and a smoothing of the square root divergences that are expected at the extrema of the dispersion relations E(j,k).

Figs. <ref> (a) and (b) illustrate the case of clean GJJ, evaluated in the normal phase region at r_0=(0,0) and at the N-S interface r_0'=(-L/2,0), respectively. In both panels, the LDOS ρ_0, e ↑ and ρ_0, h ↓ are zero for energies within |Ω| < δ(ϕ). 


Figs. <ref> (c) and (d) illustrate LDOS in r_0=(0,0) and r'_0=(-L/2,0), respectively, when a non-magnetic Anderson impurity with parameters (ϵ_d,ϵ_ Z)=(0.2,0) Δ is present in r_d=r_0. Figs. <ref> (e) and (f) show the same LDOS when the impurity is magnetic, with parameters (ϵ_d,ϵ_ Z)=(0.2,0.1) Δ.
For both Anderson impurities, we fix the tunneling amplitude at t_0=√(Δħ W/(10 A_ c)  ).
In the absence of any spin-splitting term, such as the Zeeman term, the total Hamiltonian Ĥ_ tot is particle-hole symmetric. This symmetry leads to the identity ρ_ e,↑(r,Ω)=ρ_ h,↓(r,-Ω). The relation between the spin-down electron and the spin-down hole LDOS, which is independent of the Hamiltonian, is ρ_ e,↑(r,Ω)=ρ_ e,↓(r,Ω). Therefore, with a particle-hole symmetric Hamiltonian, the electron LDOS is independent of the spin z-projection, i.e. ρ_ e,↑(r,Ω)=ρ_ e,↓(r,Ω).
In the presence of a non-magnetic Anderson impurity where the particle-hole symmetry is present (see Fig. <ref> (c) and (d)), two bound states within the mini-gap, due to the hybridization of ABS and impurity, arise at opposite energies, with Ω_0=-0.185 Δ≲ -ϵ_d and Ω_0'=-Ω_0.
In Fig. <ref> (e) and (f), the effect of a magnetic Anderson impurity is taken into account, which breaks the particle-hole symmetry. This is evidenced by the emergence of two sharp peaks within the minigap at energies Ω_0=-0.098 Δ≳ -ϵ_d + ϵ_ Z and Ω_0'=0.267 Δ≲ϵ_d + ϵ_ Z, which are no longer opposite in energy, Ω_0'≠ -Ω_0.



















§ SINGLE LIFSHITZ IMPURITY



In this Section, we consider an impurity that acts as a source of elastic scattering processes for the electron system in a GJJ, and it is modeled by the Lifshitz model <cit.>. 
Similarly to Sect. <ref>, we consider a localized impurity that is placed at position r_d and at a carbon site that belongs to sublattice A, which generates the potential below


    Û_ D   = 
    ∑_κ'=K,K'∑_κ=K,K'∑_α=A,B∑_α'=A,B∑_s=↑,↓∫ d^2 r u_s
       ×δ_s,s'δ_α,Aδ_α',A A_ cδ(r-r_d) 
     ψ̂_α,κ, s^†(r)  ψ̂_α', κ', s'(r) ,


here u_s is the spin-resolved interaction energy, but spin-flip processes are excluded.
Due to the short-range potential of the impurity, there is a mixing between the valleys K and K'.
We focus on the low-energy electronic system and project the total Hamiltonian onto the ABS subspace. The ABS Green function can be expressed in the form of a Dyson equation, see Eq. (<ref>), where the self-energy is given by


    Σ̂(Ω)   =   P_ AÛ_ D P_ A
       =∑_ζ,ζ'∑_j,j'∑_k,k'w^†_d,ζ,j,k U_d w_d,ζ',j',k'γ̂_ζ,j,  k^†γ̂_ζ^', j^', k^' ,


with


    U_d=u_d τ_z + u_ Z_τ ,


u_d=(u_↑+u_↓)/2 and u_ Z=(u_↑-u_↓)/2.

Following the procedure used for a single Anderson impurity (details in Appendix <ref>), we find that the ABS Green function Ĝ(Ω) has the form shown in Eq. (<ref>), with



    X(Ω) → U_d.


Moreover, for both electrons and holes, the LDOS can be expressed as ρ_ℓ(r, Ω)=ρ_ℓ,0(r, Ω)+δρ_ℓ(r, Ω), where ρ_ℓ,0(r, Ω) is equal to Eq.(<ref>), while the modification induced by a single Lifshitz impurity is given by Eq. (<ref>) with the substitution in Eq. (<ref>).

Fig. <ref> shows the LDOS for electrons with spin-up (solid black line) and holes with spin-down (red dashed line) as a function of energy Ω, by selecting the junction length, Fermi level and phase difference used in Fig.  <ref>.


Fig. <ref> (a) [(c)] and (b) [(d)] show LDOS at r_0=(0,0) and r'_0=(-L/2,0), respectively, in the presence of a non-magnetic [magnetic] Lifshitz impurity located at r_d=r_0 and characterized by (u_d,u_Z)=(-1/8,0)  ħ W/A_ c [(u_d,u_Z)=(-1/8,0.1225) ħ W/A_ c]. 

A non-magnetic Lifshitz impurity acts as a source of elastic scattering that does not disrupt Cooper pairs <cit.>. Therefore, for any value of u_d and u_ z=0, the mini-gap is not affected by impurity scattering.
In fact, in Fig. <ref> (a) and (b), the LDOS ρ_0, e ↑ and ρ_0, h ↓ are exactly zero within the energy range Ω < δ(ϕ), despite the presence of a strongly attractive non-magnetic Lifshitz impurity. 
In Appendix <ref>, we verify semi-analytically that the emergence of a bound state within the mini-gap energy generated by the non-magnetic Lifshitz impurity is impossible.

The spin-independent strong attractive interaction considered produces a pair of symmetric peaks, Ω_0=-0.623 Δ and Ω_0'=-Ω_0, which appear outside the mini-gap energy window, δ(ϕ)<|Ω_0|<Δ, and preserves the particle-hole symmetry property ρ_ e ↑(r,Ω)=ρ_ h ↓(r,-Ω).

In contrast, a magnetic Lifshitz impurity,  which breaks the particle-hole symmetry, is a source of elastic scattering processes that can disrupt Cooper pairs <cit.>. In Fig. <ref> (c) and (d), we consider a spin-resolved potential, with u_↑=u_d+u_ Z=0.25 × 10^-2ħ W/A_ c and u_↓=u_d-u_ Z=-0.2475 ħ W/A_ c. This term u_ Z reduces the interaction with the spin-up channel and increases the attractive interaction with the spin-down channel. Two sharp peaks appear at Ω_0=-0.272 Δ and Ω_0'=0.786 Δ, and only the first one is within the mini-gap energy window.





§ ANALYSIS IN REAL SPACE



In this Section, we investigate the structure of the density fluctuations that are generated near the impurity, which can result in a sharp peak in the LDOS at an energy within the mini-gap energy δ(ϕ).

In the presence of a single impurity at r_d=(x_d,y_d)=(0,0), Figs. <ref> (a)-(b) and Figs. <ref> (a)-(b) [Figs. <ref> (a)-(b) and Figs. <ref> (a)-(b)] show the LDOS for spin-up electrons and spin-down holes, respectively, along the N-S interface r=(-L/2,y) and in the middle of the normal region along the transverse direction r=(x_d,y), with the Fermi level set at μ_0=5 ħ/L [μ_0=8 ħ/L].

In all panels, Ω_0 is the energy at which a sharp peak of the LDOS appears in the range of -δ(ϕ)<Ω_0<0. We have considered an Anderson non-magnetic impurity (solid black line), an Anderson magnetic impurity (green dashed line), and a Lifshitz magnetic impurity (blue dashed-dotted line).
For all the cases considered, the finite signals of LDOSs shown here are consequences of the hybridization between the impurity and ABSs. In fact, the LDOSs for a clean GJJ both ρ_ e ↑,0(r,Ω_0) and  ρ_ h ↓,0(r,Ω_0)  are exactly zero.
Moreover, regardless of the nature of the impurity, the LDOS exhibits a similar pattern of oscillations. This oscillatory behavior reminds us of Friedel oscillations that arise in an electron gas from localized perturbations <cit.>, where their characteristic wavevector 2 is a direct consequence of an important property of the perturbated system, namely the existence of a sharp Fermi surface in an electron gas <cit.>.
We now explore the Fourier analysis of the spatial pattern of the LDOS to determine what information about the ABSs in a GJJ can be obtained.
For this aim, we define the partial Fourier transform as


    ρ̃_ℓ(k;x,Ω)=
      
       ∫_-∞^∞ d y e^-i k yρ_ℓ(r,Ω) ,


where ℓ={ e ↑, h ↓}, and they have been calculated by using the Fast Fourier transform algorithm <cit.> (details in Appendix <ref>). 

Fig. <ref> (c)-(d) and Fig. <ref> (c)-(d)  [Fig. <ref> (c)-(d) and Fig. <ref> (c)  -(d)]  show respectively the partial Fourier transform ρ̃_ℓ(k;-L/2,Ω_0) and ρ̃_ℓ(k;x_d,Ω_0) (ℓ={ e ↑, h ↓}) of LDOS illustrated in  Fig. <ref> (a)-(b) and Fig. <ref> (a)-(b)   [Fig. <ref> (a)-(b) and Fig. <ref> (a)  -(b)], and the Fermi level is μ_0=5 ħ/L [μ_0=8 ħ/L].

Independently of the type of impurity and doping level, ρ̃_ℓ(k;-L/2,Ω_0) and ρ̃_ℓ(k;x_d,Ω_0) have a global maximum at k=0, indicating that the most effective scattering processes that lead to the bound state at Ω_0 have zero transferred momentum.

In addition to the peak at k=0, for both considered doping levels, ρ̃_ℓ(k;-L/2,Ω_0) and ρ̃_ℓ(k;x_d,Ω_0) show a pronunciated profile around finite wavenumbers. 

To understand the origin of these special transferred momenta, we recall that Fig. <ref> shows how the ABS dispersion relation ϵ(k,ϕ) and the analytical expression E(k,ϕ)= Δ√(1-τ(k) sin^2(ϕ/2)) have extreme points located at the same momenta <cit.>. 
In particular, for a given phase difference ϕ,  the minima of E(k,ϕ) occur when the transmission probability τ(k) reaches its maximum value τ(k)=1 (total transmission).
For the normal state of the graphene electron gas <cit.>, the total transmission occurs for k_0=0 (Klein tunneling <cit.>)  and for ± k_n=±√(μ_0^2/(ħ)^2-(n π)^2/L^2) (stationary wave condition), where n=1,…,⌊μ_0L /(πħ)⌋ (⌊⌋ is the integer part). 
Klein tunneling is independent of the doping level, whereas the number of momenta that satisfy the stationary wave condition is a function of the doping level.
For example, at a doping level μ_0=5ħ/L, only ± k_1 satisfy the stationary wave condition. When the doping level increases to μ_0=8ħ/L, ± k_1 and ± k_2 satisfy the stationary wave condition.

At special momenta k_0 and ± k_n, which correspond to highly transmissive channels of the normal graphene stripe, the ABS dispersion relation ϵ(k,ϕ) shows quasi-degenerate minima, and at these energy values, the density of states exhibits square root divergences <cit.>.

At the doping level μ_0=5ħ/L, regardless of the nature of the impurity, along the N-S interface x=-L/2, we observe in Fig. <ref> (a)-(b) an oscillating behavior of spin-up electrons and spin-down holes LDOSs with a characteristic wavenumber 2 k_1, as confirmed by the partial Fourier Transform in  Fig. <ref> (c)-(d), where pronounced peaks around ± 2 k_1 appear. 
Along the x=x_d line within the normal phase region, both the spin-up electron and spin-down hole LDOSs show a pattern in Fig. <ref> (a)-(b) that is reminiscent of beats. In fact, in the corresponding partial Fourier Transforms in Fig. <ref> (c)-(d) pronounced peaks appear around two wavenumbers k_1 and 2 k_1.

This result suggests that the scattering processes with exchanged momentum 2 k_1 and k_1 play a crucial role in the generation of bound states, and that the ABSs involved have momenta around the special ones k_0 and ± k_1.

At the doping level of μ_0=8ħ/L, the spin-up electron and spin-down hole LDOSs along the N-S interface at x=-L/2 display an oscillating behavior with a wavenumber of 2 k_2, as seen in Fig. <ref> (a)-(b). The partial Fourier Transforms in Fig. <ref> (c)-(d) provide further confirmation, with two distinct peaks at ± 2 k_2.
Along the x=x_d line in the normal phase region, both the spin-up electron and spin-down hole LDOSs in Fig. <ref> (a)-(b) show a pattern that resembles beats, and the partial Fourier Transforms in Fig. <ref> (c)-(d) are prominent around ±(k_1-k_2) and ±(k_1+k_2).
This result indicates that the momenta of ABSs more involved in the formation of the bound states are close to the specific values k_0, ± k_1, and ± k_2.
In conclusion, the Fourier analysis of LDOS reveals the momenta mainly involved in the scattering processes with ABSs associated with a high transmission probability τ(k), corresponding to the minima of ϵ(k,ϕ). 










§ CONCLUSIONS



In this work, we investigated the LDOS in the normal phase stripe of a ballistic short GJJ. In particular, we studied local effects in the vicinity of a short-range impurity and the generation of bound states resulting from hybridization with ABSs.
We examined two distinct approaches for describing a single short-range impurity, the Anderson model and the Lifshitz model. The Anderson model treats the impurity as a source of inelastic scattering, whereas in the Lifshitz model the impurity only causes elastic scattering.

We have found that in the absence of magnetic terms the Lifshitz impurity does not produce any bound state within the mini-gap. This result can be seen as a consequence of the Anderson theorem, which states that this kind of impurity cannot disrupt Cooper pairs. However, the Anderson impurity is capable of forming two bound states within the minigap, which are symmetric in energy with respect to the Fermi level.
By introducing a magnetic term, we have observed that the Lifshitz impurity disrupts Cooper pairs, and it can induce a bound state within the mini-gap. For the Anderson model, an additional magnetic term can break the energy symmetry of the two bound states within the mini-gap.
In summary, from the energy features of the LDOS within the mini-gap, one can distinguish the nature of an impurity coupled to the electron system of a short ballistic GJJ.
Moreover, we have studied the spatial structure of the density fluctuations in a bound state within the minigap. 
In these cases, we have observed that, apart from the Lifshitz non-magnetic model, given the phase difference and the Fermi level, LDOS exhibits a unique oscillating and decreasing trend as one moves away from the impurity, regardless of the nature of the impurity.
For each spatial profile, the corresponding Fourier analysis shows that the characteristic wavevectors of the density oscillations are connected to the momenta of the high-transmissive channels in ballistic graphene,  providing insight into normal-phase graphene.



The authors thank G. G. N. Angilella, J. Ankerhold, F. Bonasera, P. Hakonen, C. Padurariu, and V. Varrica for illuminating discussions and fruitful comments on various stages of this work. E.P. acknowledges support from the PNRR MUR project PE0000023-NQSTI. F.M.D.P acknowledges support from the project PRIN 2022 - 2022XK5CPX (PE3) SoS-QuBa - "Solid State Quantum Batteries: Characterization and Optimization". G.F. thanks for the financial support Università degli Studi di Catania, Piano di Incentivi per la Ricerca di Ateneo.
I.V. thanks the Institut für Komplexe Quantensysteme and IQST, Universität Ulm (Germany) for the great hospitality and stimulating environment.





figuresection



§ CALCULATION OF THE ABS'S GREEN FUNCTION WITH A SINGLE IMPURITY



In this Appendix, we show the details for the calculation of the ABS's Green function in the presence of a short-range impurity.
The approach described below is used for both the Anderson model (see Sect. <ref>) and the Lifshitz model (see Sect. <ref>).
In both cases, the ABS's Green function is involved in a Dyson series, see Eqs. (<ref>)-(<ref>), of the form


    Ĝ(Ω)=Ĝ_0(Ω)+Ĝ_0(Ω) Σ̂(Ω)Ĝ(Ω) ,


where Ĝ_0(Ω) is defined in Eq. (<ref>) and


    Σ̂(Ω)=∑_ζ,ζ'∑_j,j'∑_k,k'w^†_d,ζ,j,k X(Ω) w_d,ζ',j',k'γ̂_ζ,j,  k^†γ̂_ζ^', j^', k^' .


For the Anderson model, we have the following correspondence


    X(Ω) →[ t_0^2/Ω-(ϵ_d+ϵ_ Z)                  0;                  0 t_0^2/Ω+(ϵ_d-ϵ_ Z) ] ,


which is equal t_0^2 G_d(Ω), defined in Eq. (<ref>). Whereas, for the Lifshitz model, we have

    X(Ω) →[  u_d+u_z        0;        0 -u_d+u_z ] ,


which corresponds with U_d, defined in Eq. (<ref>).
By expanding the ABS's Green function as


    𝒢̂(Ω)=∑_n=0^∞𝒢̂_n (Ω),


where the n-th term


    𝒢̂_n(Ω)=𝒢̂_0(Ω)[Σ̂(Ω) 𝒢̂_0(Ω)]^n ,


for n>0 it can be expressed as



    Ĝ_n(Ω)   =   ∑_j,j^'∑_ζ,ζ^'∑_k,k^'1/Ω-j ϵ(k,ϕ)
      w^†_d, j,ζ,k
    X(Ω)
     
    Y^n-1(Ω)  w_d,j',ζ',k'  G_0(j^',k^',Ω)  γ̂^†_j,ζ,kγ̂_j^',ζ^',k^'



where


    Y(Ω)  =  ∑_ζ,j,k w_d,ζ,j,k    w^†_d,ζ,j,k/Ω-j ϵ(k,ϕ)     X(Ω) .








It is convenient to write the ABS's Green function as


    Ĝ(Ω)   =Ĝ_0(Ω) +∑_j,j^'∑_ζ,ζ^'∑_k,k^'1/Ω-j ϵ(k,ϕ)w^†_d,ζ,j,k
    X(Ω) 
       ×[ ∑^∞_n=0 Y^n(Ω)]  w_d,ζ',j',k'1/Ω-j' ϵ(k',ϕ)γ̂^†_j,ζ,kγ̂_j^',ζ^',k^',


in the square brackets, we have isolated the geometric series of ratio Y(Ω), which can be expressed in a closed form, and 𝒢̂(Ω) can be written in the form


    𝒢̂(Ω)  =𝒢̂_0(Ω)+∑_ζ,ζ'∑_j, j^'∑_k, k^' F_j,ζ,k,j',ζ',k' (Ω)γ̂_j, ζ, k^†γ̂_j^', ζ^', k^',


where


    F_ζ,j,k,ζ',j',k' (Ω) =1/Ω-j ϵ(k,ϕ) w^†_d,ζ,j,kX(Ω)
       ×[_τ-Y(Ω)]^-1 w_d,ζ',j',k'1/Ω-j' ϵ(k',ϕ).



The result expressed in Eq. (<ref>) is formally valid for both types of impurities, the differences from the Anderson model and the Lifshitz model are generated by the form of the corresponding matrix X(Ω), see Eqs. (<ref>) and (<ref>).



§ NUMERICAL FOURIER TRANSFORM



The discrete Fourier transform (DFT) is a mathematical operation that converts a finite sequence of discrete values { z_n } into another sequence {z̃_j }, according to the definition 


    z̃_j = ∑^N-1_n=0 e^-i 2π j n/N z_n,


and the inverse operation (IDFT) is defined as


    z_n = 1/N∑^N-1_j=0 e^i 2π j n/Nz̃_j.


Both of these can be calculated rapidly using the fast Fourier transform (FFT) <cit.>.

We demonstrate how IDFT has been used to calculate the 4×4 matrix G̅_0(r,r',Ω). According to Eq. (<ref>), each entry of G̅_0(r,r',Ω) is an inverse Fourier transform from the k domain to the y domain, i.e.  


    f(y)= ∑_k e^i k yf̃(k) .
  

In the case of a wide GJJ (W ≫ L,ξ), the summation over k can be replaced by an integration 


    f(y)=  W /2π∫^∞_-∞ dk e^i k yf̃(k) .
 

Now, we take f̃(k) in the range [-k_ max,k_max] and select the finite sequence f̃_j=f(k_j-k_ max), with k_j = 2 k_ max j/N and j=0,…,N-1. In terms of the definition of IDFT, we have the following


    f_n=1/N∑^N-1_j=0 e^ iy_n k_jf̃(k_j-k_ max) .


where y_n=π n /k_ max, where the resolution in the y-domain is Δ y = π /k_ max and the spatial range is [-π N/(2 k_ max),π N/(2 k_ max)].
When k_ max is fixed, the discrete values obtained by the IDFT approach the following expression in the limit of large N≫1:


    f_n= 1/2 k_ max∫^k_ max_-k_ max d k e^i k y_nf̃(k).


Comparing with Eq. (<ref>), we can see that


    f(y_n)= lim_k_ max→∞lim_N →∞ k_ max W/π f_n .


To calculate G̅_0(r,r',Ω), we have approximated the expression above with the finite values of N=23062 and k_ max=100/L.
According to this approximation, the function f(y) is calculated in the spatial range [-y_ max, y_ max] where y_ max=362.2 L. 

To calculate numerically the partial Fourier transform ρ̃_ℓ(k;x,Ω), defined in Eq. (<ref>), we have used an analogous approach.
For a given pair of x and Ω,  ρ̃_ℓ(k;x,Ω) has a form of the type below


    g̃(k)=
    
     ∫_-∞^∞ d y e^-i k y g(y) .


We consider the function g(y) in the interval [-y_ max,y_ max] and select a finite sequence g_n=g(y_n-y_ max), where y_n=2y_ max (n/N) and n=0,…,N-1.
Keeping fixed y_ max, in the limit N→∞ the discrete values obtained by the DFT of Eq. (<ref>) tend to be 


    g̃_j = N/2y_ max∫^y_ max_-y_ max dy e^-i k_j y g(y),


with k_j= π j/ y_ max, comparing it with Eq. (<ref>), we find g̃(k_j)= lim_ y_ max→∞lim_N →∞2 y_ max/Ng̃_j.


To calculate ρ̃_ℓ(k;x,Ω), we have used an approximation of the limit above, taking the finite values of N=23062 and y_ max=362.2 L. This enabled us to compute the partial Fourier transform in the range of [-100/L,100/L].






§ IMPOSSIBILITY OF A BOUND STATE GENERATED BY A NON-MAGNETIC LIFSHITZ IMPURITY








In this Appendix, we show that a single non-magnetic Lifshitz impurity cannot create a bound state in the subgap energy range |E| <Δ.
For a given phase difference ϕ, within the subgap energy window, the only region in which a generic impurity can create a bound state (with infinite lifetime) is within the minigap, δ(ϕ)=min_k[ϵ(k,ϕ)].
In particular, the occurrence of a pole of the function F_ζ,j,k,ζ',j',k' (Ω), see Eq. (<ref>), indicates the emergence of a bound state due to the hybridization of ABSs and the single impurity <cit.>. This is possible if there is an energy Ω that solves the following condition


    Det[_τ-Y(Ω)]=0 ,


where form of Y(Ω) depends on the specific nature of the impurity (see App. <ref>).
Here, we focus on the case of a single non-magnetic Lifshitz. Using Eqs. (<ref>) and (<ref>), after simple algebraic manipulations, below we report an equivalent equation to Eq. (<ref>) in terms of the 4×4 matrix G̅_0(r,r',Ω), defined in Eq. (<ref>), i.e.



    Det[  1/u_d A_ c- G̅_0(r_d,r_d,Ω)_11-G̅_0(r_d,r_d,Ω)_22            G̅_0(r_d,r_d,Ω)_13+G̅_0(r_d,r_d,Ω_0)_24;              G̅_0(r_d,r_d,Ω)_31+G̅_0(r_d,r_d,Ω)_42 -1/u_d A_ c- G̅_0(r_d,r_d,Ω)_33-G̅_0(r_d,r_d,Ω)_44 ]=0.
 

Conversely, we see that for any energy Ω within the mini-gap |Ω|<δ(ϕ), Eq. (<ref>) is an algebraic quadratic equation where the unknown variable is 1/(u_d A_ c). This algebraic equation has at least a real solution only if its associated discriminant D(Ω) is non negative, D(Ω)≥0, where


    D(Ω)=   [G̅_0(r_d,r_d,Ω)_11+G̅_0(r_d,r_d,Ω)_22+G̅_0(r_d,r_d,Ω)_33+G̅_0(r_d,r_d,Ω)_44]^2
       -4[G̅_0(r_d,r_d,Ω)_13+G̅_0(r_d,r_d,Ω)_24][G̅_0(r_d,r_d,Ω)_31+G̅_0(r_d,r_d,Ω)_42] .




Fig. <ref> illustrates the discriminant D(Ω) as a function of the energy Ω within the mini-gap δ(ϕ), for three different values of the phase difference: ϕ=3π/4 (solid lines), ϕ=0.1π (dashed lines) and ϕ=0.9π (dashed-dotted lines), and two values of the Fermi level: μ_0=5 ħ/L (black lines) and μ_0=8 ħ/L (red lines).
In all the cases presented, we have a finite short junction length L/ξ=π/20, and the impurity is placed in the middle of the normal phase stripe r_d=(0,0). Here, the discriminant D(Ω) is always negative, which confirms that a non-magnetic Lifshitz single impurity cannot generate a bound state within the mini-gap.







 Paper Title: Topological transition as a percolation of the Berry curvature.


	
	
	
	
	
	
	sangjin.sin@gmail.com
	Department of Physics, Hanyang University, Seoul 04763, Republic of Korea
	
	
	
	§ ABSTRACT 

		We first study the importance of the sign of the Berry curvature in the Euler characteristic of the two dimensional topological material with two band. Then we report an observation of a character of the topological transition as a percolation of the sign of the Berry curvature. 
		The Berry curvature F has peaks at the Dirac points which enables us to divide the Brillouin zone into two regions depending on the sign of the F : one with the same sign with  a peak and the other   with the opposite sign. We observed that when the  Chern number is non-zero, the oppositely signed regions are localized, while in the case of a trivial topology, the oppositely signed regions are delocalized dominantly.   Therefore, under the topological phase transition from non-trivial to trivial, the oppositely signed region will percolates.   We checked this for several models including Haldane model, the extended Haldane model and the QWZ model. 	Our observation may serve as a novel feature of the topological phase transition.
	
	
	
	Topological transition as a percolation of the Berry curvature
    Sang-Jin Sin
    August 5, 2024
==============================================================



§ INTRODUCTION 

	
	Much progress has been made in the study of topological materials  in  both  theory and experiment, profoundly impacting our understanding of electronic properties and phase transitions. Initiated by the  Quantum Hall Effect<cit.>, which demonstrated  capacity of topological concepts  to elucidate complex quantum behaviors, research has diversified into areas such as topological insulators <cit.> and topological superconductors <cit.>. These studies have grounded theoretical frameworks on the topological invariant such as the Chern number, the integral of the Berry curvature over the Brillouin Zone (BZ). 
	
	 
		More recently, quantum geometry has become a pivotal point within the study of topology <cit.>. Quantum geometry fundamentally describes how geometric and topological properties of electron wave functions influence material behavior at the quantum level <cit.> . This field explores a variety of phenomena, from the influence of quantum metrics on superconducting states to the broader implications of electron band topology in designing novel materials <cit.> . Specifically, quantum geometry plays crucial role in unraveling the geometric contributions to physical properties such as superconductivity and super fluidity, particularly evident in the study of super fluid weight in flat bands and similar system <cit.>.   
	
	Despite the theoretical and experimental importance of the quantum metric, a comprehensive understanding of how local spatial properties, such as the percolation of Berry curvature, influence the characterization of topological phases remains elusive. As a imaginary part of the quantum metric tensor, the significance of the spatial distribution of the Berry curvature must be underscored <cit.>. Our study aims to bridge this gap by focusing on the spatial dynamics of Berry curvature and its impact on the topological properties of materials. By elucidating the localization and delocalization behaviors of Berry curvature in various topological models, we hope to provide a refined   distinction between trivial and non-trivial topological phases.
	
	This paper explores the phenomenon of Berry curvature percolation and its impact on the topological phases of various models, including the Qi-Wu-Zhang (QWZ) model, the Haldane model and the Extended Haldane model. We specifically focus on the localization and delocalization properties of Berry curvature in relation to the Chern number, which helps to differentiate between trivial and non-trivial topological phases.
	
	

§  QUANTUM METRIC AND TOPOLOGICAL INVARIANT


	
	In this section, we describe about the relation of the quantum metric and the Berry curvature.
	
	
	For  an orthonormal eigenvector |n⟩ of the system with non-degenerate eigenvalues the Quantum Geometry Tensor (QGT) is
	
    Q_ij^(n)= ⟨∂_in| ∂_jn⟩ -⟨∂_in| n⟩⟨n| ∂_jn⟩.

	The real part of the QGT  serves as the Quantum Metric   (QM), while the imaginary part of the QGT is  the Berry curvature :
	
    g_ij^(n)=ReQ_ij^(n),   F_ij^(n)= -2Im Q_ij^(n).

	
	If we consider two band system in two dimension, the Berry curvature can be express as F_xy.
	A general two band Bloch Hamiltonian reads 
	
    H=h_0(k)I_2+h(k)·σ,

	where I_2 is the 2 × 2 identity matrix, k=(k_x,k_y),  
    h=(h_1(k),h_2(k),h_3(k)),   σ=(σ_1,σ_2,σ_3) ,
  and σ_i's  are the Pauli matrix. With ĥ = h/|h|, the Berry curvature can be written as
	
    F_xy
    		=1/2|h|^3[(h·(∂_k_xh×∂_k_yh)]. 
    		=	1/2ĥ·(∂_k_xĥ×∂_k_yĥ).

	In this formula, ∂_k_xĥ×∂_k_yĥ terms describes the orientation of the surface on the Bloch sphere. 
	
	The Berry curvature possesses both positive and negative values, indicating that its orientation is not well defined;   the    orientation reversal occurs along line of  F_xy=0.
	
	
	To see the relation of the quantum metric and the Berry curvature, we calculate the quantum metric.
	quantum metric that can be simplified as the following:
	
    g_ij=1/4∂_k_iĥ·∂_k_jĥ.

	With the Eq.(<ref>), the determinant of the quantum metric can be written as
	
    (g)
    		=1/16[(ĥ·(∂_k_xĥ×∂_k_yĥ)]^2. .

	From this and   Eq.  (<ref>),
	
    (g)=1/4F_xy^2,

	for the two-dimensional two band system. 
	The Chern number  
	
    C =1/2π∮_BZF_xy dk_x dk_y
    		,

	is an integer as one may verify explicitly.   
	
The Euler characteristic, χ, is  a topological invariant  that can be defined by   integral of Euler class <cit.>. Here we follow slightly different path, using  the Kronecker's index<cit.>. 
    Let S be an n dimensional closed Riemann manifold on which a set of n+1 functions    V^i (x)  satisfying V^i (x)V^i (x)=1 is defined. With this set of functions, we can consider a continuous mapping of S upon the unit n sphere. Define determinant D with V class as below:

    D =
     
     	V^1    ...     V^n+1
    ∂_x^1 V^1    ...    ∂_x^1 V^n+1
    ⋮    ...    ⋮
    ∂_x^n V^1    ...    ∂_x^n V^n+1.

Then, we can  show that  

    D^2 = G,      
    G_αβ  =  
    	∂_x^α V ·∂_x^β V ,

and define the Euler characteristic in even  dimension   by 
 
    χ =2/vol(S_n)∫±√(D^2) d^nx ,

 where vol(S_n) is the area of n-dimensional sphere  and  the sign is chosen to allow for overlapping of the covering <cit.>. 
In our study, we consider only two dimensions in momentum space on the Bloch sphere represented by  ĥ. 
In two dimension case, we can express D = 2ℱ_xy and  G_ij=4g_ij. 
 By using Eq. (<ref>) and Eq. (<ref>), specifically we can simplify D^2 as 

    D^2= G= 16  g  = 4 ℱ_xy^2.


From Eq .(<ref>), the Euler characteristic number can be expressed as follows :


    χ= 2/π∫_S±√( g) dk_x dk_y = 1/π∫_Sℱ_xy dk_x dk_y.


Therefore, χ can be simply   represented by: 

    χ =2C.

  The important  point to notice is  that the existence of overlapping regions in the mapping of the Brillouin zone (BZ) to the Bloch sphere causes an orientation reversal. Specifically, there is an induced orientation on the BZ due to the pullback of the mapping from momentum space to the manifold of quantum states, denoted as

    h: BZ→Bloch  sphere ,

which plays the role of the Gauss map in the theory of the surface. 
It turns out that the sign of the Berry curvature determines this orientation <cit.>. Thus, due to the orientation reversal across the BZ along the line where F_xy=0, the orientation sign must be considered carefully—an aspect often neglected in many preceding studies. Therefore, the sign of the Berry curvature serves as a crucial indicator for observing the quantities associated with topology.







	
	
	

§ THE PERCOLATION OF THE BERRY CURVATURE 

	In the last section, we observed that   the sign of the Berry curvature serves as crucial means  in the mechanism of  the  topological observable to work. 
	In this section, we explores the crucial phenomenon of Berry curvature percolation and its substantial role in determining topological phases. A fundamental aspect of this discussion is understanding how localization and delocalization of Berry curvature influence topological properties. Initially, We define the terminology "localization" of the specific regions as implying that the region is discontinuous along any trajectory initiated from a specific point within it, thus preventing a return to the starting point.  Conversely, "delocalization" refers to the continuous property of a region, allowing trajectories to return to their starting point.
	
	

 §.§ Percolation of the Berry curvature

	Berry curvature tends to be peaked  at specific locations where energy bands exhibit near crossings or touching. 
	To see this notice that in the 2-band system, the Berry curvature associated with + band can be expressed as
	
    F_xy(k)
    		=    i1/(ϵ_+-ϵ_-)^2[  ⟨n_+| ∂_k_x H|n_-⟩⟨n_-| ∂_k_y H|n_+⟩
         -⟨n_+| ∂_k_y H|n_-⟩⟨n_-| ∂_k_x H|n_+⟩  ], 
    
    		=    -2 Im(⟨n_+|∂_k_x H|n_-⟩⟨n_-| ∂_k_y H|n_+⟩)/(ϵ_+-ϵ_-)^2,

	where ϵ_± is eigenvalue and  | n_±⟩ is the eigenvector for each   band.
	When there is a degeneracy of eigenvalues at a band touching point, the value of (ϵ_+-ϵ_-)^2 approaches zero, making the peak in the Berry curvature at the near touching point. 	
	
	Now let's concentrate on the sign of the F_xy. The sign flip of  F_xy  obviously changes the distribution of the  Peak Sign Regions (PSR) of the Berry curvature as well as that of the Oppositely Signed Regions(OSR). 
	From Fig. <ref>, the PSR are depicted in red, and the OSR in blue. When the Chern number is non-trivial, the OSR is localized, resembling islands, while the PSR is predominantly delocalized. Conversely, in cases where the Chern number is trivial, the OSR is extensively delocalized, akin to a river. This pattern suggests that the topological phase transition is accompanied by the percolation of the OSR. We consistently verify this intriguing phenomenon across various models, confirming its general applicability in describing topological transitions. Furthermore, as discussed in Section <ref>, the distribution of the Berry curvature can be equated to the distribution of the metric tensor within the framework of the QGT, further elucidating the geometric underpinnings of these topological phenomena.
	
	
	
	
	
	

 §.§ Behavior of the Berry curvature
	
	
	In this section, we elucidates the reasons for these observations, focusing on how variations in the Berry curvature's distribution directly relate to changes in the Chern number.
	
	The primary factor influencing changes in the distribution of the Berry curvature is band inversion.
	The band inversion occurs when the energy levels of orbitals, typically of differing parities, interchange at specific high-symmetry points within the BZ through a process where the band gap initially narrows, closing, and then reopens. After the band gap is reopened, orbitals in the lower valence band and upper band conduction band  around the gap closing point are inverted with respect to their trivial phase <cit.>. 
	
	Near the band touching point, there is a distinct region where eigenvectors from a different band are mixed (See Fig. <ref> ). 
	Such band mixing regions  always  exists around the band touching point. 
	Consequently, in the vicinity of band touching point, the Berry curvature is predominantly constituted by eigenvectors from the other band, contrasting with other regions where the Berry curvature is composed of eigenvectors from the original band.  It  signifies that there is a eigenvector jumping near the peak. 
	
	

	This phenomena  in eigenvector configuration is crucial in understanding underlying nature of the distribution of the Berry curvature  under the topological phase transition: due to the mixing of eigenvectors, the sign of the Berry curvature changes near the band-touching point. 
	In brief, the band inversion modifies localization property of the Berry curvature. 
	
	
	
	Furthermore, according to Eq. <ref>, when a sufficient band gap exists, the Berry curvature values within the OSR are significantly lower than those at the peaks. Therefore, the distribution of the OSR plays a crucial role in quantitatively compensating for the peak's value, directly affecting whether the Chern number is trivial or non trivial.
	Specifically, when the Chern number is non-trivial, the OSR are localized, leading to minimal compensation for the peak's value, thereby ensuring that the Chern number remains non-zero. Conversely, when the Chern number is trivial, the flatness of the OSR necessitates their dominant distribution across the Brillouin Zone to sufficiently compensate for the peaks’ values, consequently resulting in their delocalization.
	Overall, the analysis of the Berry curvature distribution serves as a critical indicator of topological characteristics, determining whether the topological phase is trivial or non-trivial.
	
	

 §.§ Percolation rate

	Percolation theory, a critical concept in statistical mechanics, studies the behavior of connected clusters in a random medium. It provides insights into phase transitions by analyzing the probability and geometry of clusters as they form and spread through a system <cit.>.
	Percolation refers to the phenomenon where localized clusters spread and merge, during a phase transition. 
	
	In this context, the delocalization of the OSR can be quantified by considering the OSR as clusters in the percolation theory. We define S_i as the size of individual i-th OSR clusters, considering periodicity as shown in Fig. <ref>. Then, the percolation rate of the Berry curvature can be written as follows,
	
    P=S_max/N,

	where N is the size of the BZ, and S_max is the size of the largest cluster. 	
	When the Chern number is non trivial, the OSR remains localized, forming several small clusters, resulting in a low percolation rate of under 0.5. Conversely, when the Chern number is trivial, the OSR delocalizes into one connected cluster, leading to a high percolation rate of over 0.5. 

	The intriguing point is that delocalization of the OSR occurs always dominantly in the BZ when the Chern number is trivial. If there are narrow delocalized river of the OSR, and the dominant localization of the PSR, then there sholud exist non zero Chern number. But the important thing is that case never happened : if there is delocalization of the OSR, the OSR occupy the BZ dominantly, thus Chern number of that system is always zero with a delocalization of the OSR.
		
	
	

§ APPLICATION

	We now demonstrate how the percolation of the Berry curvature appears when the topological phase is trivial or non-trivial in three different 2D models. These models' topological invariants are defined by the Chern class. 
		
	
	

 §.§ Qi-Wu-Zhang model 

	The Qi-Wu-Zhang (QWZ) model is a two-dimensional lattice model that describes the quantum anomalous Hall effect, serving as a fundamental framework for understanding topological phases in condensed matter physics<cit.>. The Bloch Hamiltonian of the QWZ model is given by 
	
    h_QWZ= Asin k_x σ_1 +Asin k_y σ_2 + (cos k_x+cos k_y+Δ) σ_3 .

	We can directly compute the Berry phase with the Hamiltonian above, as in Eq (<ref>).
	
	In the QWZ model, the Chern number depends on the ratio of A to Δ as follows.
	
	
    C = 1    for   0<Δ/A<2 
    
    			C = -1    for   -2<Δ/A<0 
    
    			C = 0    for  Δ/A>2  or  Δ/A< -2

	
	
	
	In the Fig. <ref>, dashed lines represent the BZ. The red color denotes the PSR, while blue indicate the OSR. When the Chern number is non-trivial, the PSR are predominantly delocalized while the OSR are localized like a island. Δ=0 is also a topological phase transition point from -1 to 1. However, since the Chern number of both sides are non trivial, the localization of the the OSR remains. Conversely, in the trivial Chern number case, the OSR are delocalized. 
	
	Furthermore, we calculate the percolation rate with Eq (<ref>). In the Fig. <ref>, the red line denotes P=0.5, the blue line is the percolation rate of the OSR in the QWZ model. When the Chern number is zero, the percolation rate is higher than 0.5 which means that the OSR are delocalized dominantly in the BZ. In the other case, when the Chern number is non zero, the percolation rate is lower than 0.5. 
	Consequently, observing the distribution of the Berry curvature's each signed region serves as the topological feature that determines the topological phase is trivial or non trivial.

	
	


	

 §.§ Haldane model 

	We also verify the percolation property of the Berry curvature in the Haldane model. 
	Haldane Model's tight binding Hamiltonian is given by
	
	
	
	
    H_H=t_1∑_⟨ ij ⟩c_i^†c_j+t_2∑_⟨⟨ ij ⟩⟩e^-iν_ijϕc_i^†c_j+λ_v∑_iϵ_ic_i^†c_i,

	where  ⟨ ij ⟩ denotes the summation over nearest-neighbor (NN) hopping and ⟨⟨ ij ⟩⟩ represents the sum over next-nearest-neighbor (NNN) interactions. t_1 is the hopping constant which determines NN hopping. The inclusion of a second term accounts for next-nearest-neighbor (NNN) hopping, influenced by both the internal magnetic flux ϕ and the hopping parameter t_2. Furthermore, the phase factor associated with this term depends on the direction of the NNN hopping:
	
    ν_ij=sgn(d̂_i×d̂_j)_z=±1,    (i,j)∈{1,2}
 where d̂_1,2, is the vectors along the directions of NNN hoppings. The last term is on-site energy, which depends on the sublattices:  ϵ_i=1 and ϵ_i=-1 for each site. 
	Here, we define a primitive lattice vector as
	
    d⃗_1=√(3)/2a(√(3),1),   d⃗_2=√(3)/2a(-√(3),1),

	where a is a lattice spacing. 
	Using the Pauli matrix, we can express the Bloch Hamiltonian by the following way:
	
    h_H(k)= 2t_2cos(ϕ)(2cos(k̃_x)cos(k̃_y)+cos(2k̃_y))I_2
         -t(1+2cos(k̃_x)cos(k̃_y))σ_1 -2tsin(k̃_x)cos(k̃_y)σ_2
    		
       + (λ_v+2t_2sin(ϕ)(2cos(k̃_x)sin(k̃_y)-sin(2k̃_y)))σ_3.

	where k̃_x=3ak_x/2, k̃_y=√(3)ak_y/2. 
 
	

	Utilizing Eq (<ref>), the Chern number of the Haldane model is determined by the ratio of the three independent parameters : t2, ϕ, and λ_v. A detailed representation of these dependencies is illustrated in the phase diagram, presented in Fig. <ref>.
	
		
	


	In the Fig. <ref>, dashed lines represent the BZ. We fix ϕ=π/2 while varying the ratio of the λ_v to t_2 and investigate the delocalization properties of the PSR and the OSR. Similar to the observations in the QWZ model, in the Fig. <ref>, the red areas represent the PSR, whereas the blue areas is the OSR. As demonstrated in Fig. <ref>, in the non-trivial phase where λ_v < 1 , delocalization of the PSR,  become extensive, akin to a river while the OSR are localized. Conversely, in the trivial phase, when the Chern number is zero and λ_v > 1, the OSR become  delocalized with a topological transition which represent the percolation of the OSR. 

	


	Furthermore, the percolation rate is observed as Fig. <ref>. When the -1< λ_v < 1, the OSR are localized in specific regions, thus the percolation rate is lower than 0.5. On the other hand, when the -1> λ_v  or  1< λ_v , the delocalization of the OSR occurs dominantly in the BZ, so the percolation rate is higher than 0.5.
	Consquently, the percolation of the OSR occurs with the topological transition.
	
	

	

 §.§ Extended Haldane Model

	Extended Haldane model has next to next nearest neighbor (N3 neighbor) hopping in addition to the Haldane model <cit.>. 
	The Extended Haldane Model's tight binding Hamiltonian is given by
	
    H_EH=     t_1∑_⟨ ij ⟩c_i^†c_j+t_2∑_⟨⟨ ij ⟩⟩e^-iν_ijϕc_i^†c_j 
       +t_3∑_⟨⟨⟨ ij ⟩⟩⟩c_i^†c_j+λ_v∑_iϵ_ic_i^†c_i.

	In addition to the Haldane model,  ⟨⟨⟨ ij ⟩⟩⟩ means the summation over N3 neighbor hopping and t_3 is the 
	
	Under the Fourier, the tight binding Hamiltonian can be written in Bloch Hamiltonian, 
	
	
    h_0 = 2 t_2 cos(ϕ)[cos(k·a_1) +cos(k·a_2) +cos(k·(a_1-a_2))],

	
    h_1 =     t_1 [cos(k·a_1) +cos(k·a_2)] 
       +t_3[cos(k·(a_1+a_2))+2cos(k·(a_1-a_2))],

	
    h_3 = λ_v + M_H

	
    M_H = 2 t_2 sin(ϕ)[sin(k·a_2)-sin(k·a_1)+sin (k·(a_1-a_2))]

	
    h_0 =      2 t_2 cos(ϕ) [cos(k·d_1) + cos(k·d_2) 
        + cos(k· (d_1 - d_2))], 
    
    		h_1 =      t_1 [cos(k·d_1) + cos(k·d_2)] 
         + t_3 [cos(k· (d_1 + d_2)) + 2 cos(k· (d_1 - d_2))], 
    
    		h_3 =     λ_v + M_H,

	

	where M_H is the Haldane mass M_H = 2 t_2 sin(ϕ) [sin(k·d_2) - sin(k·d_1) + sin(k· (d_1 - d_2))] and d_1=a/2(√(3),3) , d_2=a/2(-√(3),3) like in the Haldane model. From Fig. <ref>, we can see the the Chern number phase diagram for the Extended Haldane model.
	
		
	Similar to the Haldane model, we fixed the ϕ=-π/2 and varied the ratio λ_v/t_2 continuously. When the Chern number is trivial, which occurs when the magnitude of the ratio exceeds approximately 5.2, delocalizaion of the PSR is observed. And the OSR is localized in specific areas. The topological transition occurs from C=-1 to C=2 where λ_v/t_2 = 2.8. Analogous to the QWZ model, the tendency of delocalization within OSR does not change between non-trivial phase. On the contrary, when the Chern number is non-trivial (- 5.2 < λ_v/t_2 < 5.2), the OSR exhibit delocalization. 

	
	Based on the results, we calculate the percolation rate using Eq (<ref>). In Fig. <ref>, the red line represents P=0.5, while the blue line indicates the percolation rate of the OSR in the QWZ model. When C=0, the percolation rate exceeds 0.5, indicating that the OSR are predominantly delocalized in the BZ. Conversely, when C is non zero, the percolation rate is below 0.5. Consequently, the distribution of the Berry curvature in both OSR and PSR acts as the topological feature that determines whether the topological phase is trivial or non-trivial.
	



	
	
	
	

§ DISCUSSIONS
 In this work, we have demonstrated the percolation of the Berry curvature serves as a pivotal factor in determining the topological phases of materials by using the fundamental attributes of the topology.
	By analyzing the Berry curvature in the perspective of the topological transition, we have established a clear criterion for distinguishing whether the Chern number is trivial or non-trivial based on the distribution of Berry curvature and the percolation rate. We have successfully classified it accordingly through analysis of the QWZ model, Haldane model, and Extended Haldane model.
	
	The transition between trivial and non-trivial topological phases is marked by a clear change in the percolation characteristics of PSR and OSR.
	Notably, behavior of the Berry curvature around the topological phase transition points provide a robust method for predicting and understanding the topological properties of materials. 
	
	Experimental efforts to directly measure Berry curvature in systems such as cold atoms and photonic lattices have been conducted<cit.>. Direct measurements of Berry curvature provide essential data for developing advanced materials with tailored electronic properties. These measurements not only enhance our understanding of quantum and topological physics but also pave the way for innovative applications in quantum technology<cit.>. We propose that observing the percolation of Berry curvature could lead to more precise determinations of the Chern numbers. Incorporating observations of Berry curvature behavior into real experimental measurements may allow for more accurate characterization of the Chern number.
	
	This study not only reaffirms the significance of Berry curvature in topological physics but also enhances the theoretical foundation for future experimental and practical applications in the field of condensed matter physics.
	
	
	
	
	
	


 Paper Title: Strained topological insulator spin-orbit torque random access memory (STI-SOTRAM) bit cell for energy-efficient Processing in Memory.


APS/123-QED



mm8by@virginia.edu
Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904, USA=-1

Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, NE 68588, USA=-1

Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904, USA=-1

Department of Electrical and Computer Engineering, Virginia Commonwealth University, Richmond, VA 23284, USA=-1

Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904, USA=-1

Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904, USA=-1
Department of Physics, University of Virginia, Charlottesville, VA, 22904, USA=-1







             
§ ABSTRACT 
We present a novel design of a strained topological insulator spin-orbit torque random access memory (STI-SOTRAM) bit cell comprising a piezoelectric/magnet (gating)/topological insulator (TI)/magnet (storage) heterostructure that leverages the TI's high charge-to-spin conversion efficiency coupled with the piezo-induced strain-based gating mechanism for low-power in-memory computing. The piezo-induced strain effectively modulates the conductivity of the topological surface state (TSS) by altering the gating magnet's magnetization from out-to-in-plane, facilitating the storage magnet's spin-orbit torque (SOT) switching. Through comprehensive coupled stochastic Landau-Lifshitz-Gilbert (LLG) simulations, we explore the device dynamics, anisotropy-stress phase space for switching, and write conditions and demonstrate a significant reduction in energy dissipation compared to conventional heavy metal (HM)-based SOT switching. Additionally, we project the energy consumption for in-memory Boolean operations (AND and OR). Our findings suggest the promise of the STI-SOTRAM for low-power, high-performance edge computing.


Valid PACS appear here

Strained topological insulator spin-orbit torque random access memory (STI-SOTRAM) bit cell for energy-efficient Processing in Memory
    Avik W. Ghosh
    August 5, 2024
=====================================================================================================================================









§ INTRODUCTION

Electronics in the edge computing era poses unique challenges that may need heterogeneous combinations of materials with diverse attributes, and devices and circuits that capitalize on those attributes <cit.>. Along this path, the intersection of magnetism, topology, and strain offers a highly complementary suite of symmetries and tunable properties that could play off of each other strategically. For instance, strain can alter the energy landscape <cit.> of a magnet and rotate it by 90^∘  <cit.>. Rotating a magnet out of plane can modulate spin conductivity of a 3D topological insulator (TI) by breaking time-reversal symmetry and gapping its topological surface states (TSS)   <cit.>. Conversely, a TI can act on the magnet when its spin-momentum locked TSS write spin information onto the free layer of a magnetic tunnel junction (MTJ) with a current driven spin-orbit torque (SOT), its spin Hall angle (SHA) exceeding unity <cit.>. Integrating these properties into a single stack may allow us to naturally integrate multiple functionalities into a compact, vertical monolithic structure. 

In this paper, we will discuss an example stack, a multifunctional device composed out of a four-layer piezo/magnet/TI/magnet structure (Fig. <ref>), which exploits the above attributes to achieve both logic and memory functions in a single bit cell. Here strain actuates a bottom magnet in and out of the TI plane with a voltage-gated piezo, which modulates the TSS and drives a top storage magnet into a write mode, a read mode, or a logic mode using a sense amplifier. 
Since selector and storage magnets are co-located in a vertical geometry, the structure is scalable and naturally suited for a Processing-in-Memory (PiM) architecture.
In contrast to traditional Von Neumann architecture (separate memory and logic cores with data transfer latency) <cit.>, key processing tasks are performed locally within a PiM  <cit.>, allowing the pre-processed data to be transferred to the primary processing unit with reduced latency <cit.>. The individual memory cells in a PiM architecture (bit cells) are often organized in a crossbar layout. Selectors control each row and column of the crossbar grid, enabling the bit cells for read or write operations. Utilizing sense amplifiers, the entire row of the crossbar can be read by comparing the state of the bit cells with a known reference voltage or current to perform logic operations within the memory <cit.>. 

The specific design of the stack depends on the convenience of fabrication (e.g., here we assume the thick piezo layer sits at the bottom, the bottom magnet on it to share the strain, then the TI to feel its proximal magnetization, and finally, the top magnet to get a write operation on it). Other lateral or flipped structures may well serve as better designs, if only for ease of fabrication. The core functionality, however, is employing SOT from a single channel material, in this case, the TI, that delivers both high SHA and simultaneously gate tunability. Let us now discuss these two features separately. 

 Magnetic random access memory (MRAM) is a leading contender for a PiM bit cell due to its non-volatility, high speed, low power consumption, high endurance, scalability, and excellent compatibility with complementary metal-oxide-semiconductor (CMOS) process technology <cit.>. The fundamental building block for MRAM devices is an MTJ, which consists of a thin insulator sandwiched between two magnetic layers—a “pinned layer” whose magnetization is fixed and a “free layer” whose magnetization can be reoriented by a spin current. Conventional MRAM uses current in the overhead to apply a magnetic field to write data on the MTJ free layer. Recently, current-induced SOT mechanism, resulting from either the spin Hall effect <cit.> or the Rashba effect <cit.>, has emerged as a promising approach for energy-efficient switching of the MTJ's free layer. SOT-based switching addresses several limitations of its already commercialized counterpart, spin-transfer torque (STT)-based switching, such as the need for high write voltages, shared read-write paths creating read-disturbs, and degradation of the insulating layer <cit.>. SOT-based MRAM (SOTRAM) is a three-terminal device featuring a decoupled read-write path and in-plane charge current flow in the non-magnetic layer underneath the MTJ free layer with a high charge-to-spin conversion efficiency, yielding a low write current and infinite endurance <cit.>. SOTRAM is attractive as a high density last level-embedded cache <cit.>, with distinct advantages for PiM-like edge-based motion detection and cross-modality object recognition <cit.>. They do suffer from the higher footprint associated with the third terminal, a higher charge current needed to supply the spin current orthogonal to it, and the need for a field-assist when the free layer spin stagnates onto a symmetry direction in the magnet/heavy metal (HM) interfacial plane.  
 Consequently, achieving a significantly high charge-to-spin conversion efficiency (large SHA) is essential to bypass these challenges for energy-efficient applications. 

Traditional SOT devices typically utilize HMs such as Pt <cit.>, Ta <cit.>, and W <cit.> as spin current injectors; however, these HMs exhibit low SHA (Pt: 0.08, Ta: 0.15, and W: 0.4 <cit.>), leading to low charge-to-spin conversion efficiency <cit.>. Recently, TIs have garnered attention as an attractive spin source for SOT switching <cit.>. TIs are characterized by their spin-momentum-locked TSS, insulating bulk states, and SHA greater than unity <cit.>. The conducting TSS arise from band inversion at the surface, mediated by strong spin-orbit coupling, and are topologically protected by time-reversal symmetry <cit.>. More encouragingly, sputtered TIs such as Bi_2Se_3 show very high (> 10) SHAs, primarily because of the reduction in bulk conductivity that would otherwise shunt current away from the TSS, as well as better defect control that suppresses unwanted surface scattering.
These attributes make TIs highly efficient spin current injectors for SOT switching, facilitating energy-efficient information writing <cit.>. 





Conversely, a magnet's ability to modulate the spin conductivity of the TSS of a TI, when in proximity with the TI, based on its magnetization direction, offers a pathway for an intrinsic gating mechanism to control the TI surface current <cit.>. Switching the magnet (gating magnet hereafter) from in-plane to out-of-plane orientations, gaps in the TSS can be opened, thereby modulating the spin conductivity <cit.>. This 90^∘ switching of the gating magnet can be achieved through various mechanisms, such as strain <cit.>, voltage control magnetic anisotropy <cit.>, and changing the anisotropy by an applied voltage <cit.>. Nonetheless, strain-induced 90^∘ switching of magnets is very energy-efficient <cit.>, where a piezoelectric material induces electrical strain in response to an applied gate voltage to facilitate switching from in-plane to out-of-plane and vice versa. 



In this paper, we model the aforementioned potentially compact and energy-efficient four-layer piezoelectric/magnet/TI/magnet (MTJ) stack, which we refer to as a strained topological insulator spin-orbit torque random access memory (STI-SOTRAM). We argue that this structure is naturally suited as a compact bit cell in an in-memory computing architecture. We employ stochastic Landau-Lifshitz-Gilbert (LLG) simulation to analyze the device dynamics. We present the device's functionality, the required phase space for stress generated by the piezo-induced strain and the gating magnet's anisotropy, and the writing condition's phase space (writing voltage vs. time/delay) of the MTJ. We estimate the energy cost for gating and writing mechanisms and find that energy dissipation is significantly reduced compared to HM-based SOT switching, which indicates the benefits of utilizing TI as a spin source and an intrinsic gating mechanism as opposed to an access transistor traditionally used. Furthermore, we project the energy cost for 2-bit AND and OR operations, which shows lower energy costs than traditional SOT switching. Finally, we show the impact of various material parameters on the device metric.








§ STI-SOTRAM BIT CELL DESIGN FOR PIM


Figure <ref>(a) depicts the schematic of the compact four-layer structure of the device comprising a piezoelectric/gating magnet/TI/MTJ stack in the vertical direction. The device contains a gating (selector) magnet at the bottom whose magnetization can be switched in and out of the plane of the TI in nanoseconds with a voltage-gated piezoelectric at a low energy cost (∼ 10 aJ) <cit.>. An out-of-plane selector magnet gaps the bottom surface states of the TI, leading to breaking the current path (Fig. <ref>(b)) and places the upper MTJ free layer in storage mode (Fig. <ref>(a), left). Activating the TI bottom surface with applied voltage restores the current path and drives the top MTJ free layer into one of three modes – (i) altering its magnetization with drain bias for data writing (Fig. <ref>(a), middle and right), (ii) discharging its stored magnetization state for data reading, the output set by its low (parallel)/high (antiparallel) resistance relative to the MTJ pinned layer; (iii) or execute a logic operation (e.g., bitwise AND, OR, etc.) using a sense amplifier (Fig. <ref>(c)). Two selected bit cells can feed their voltage/current to a sense amplifier that processes the local inputs by comparing them with a reference voltage/current and performing Boolean logic operations. For example, for current-based sensing, if the current of the bit cells is I_c, AND and OR operations can be performed by selecting a reference current of 1.5I_c and 0.5I_c, respectively. Similarly, we can choose different reference voltages for a voltage-based sensing scheme (discussed later). Since selector and storage magnets are co-located in a vertical geometry, the structure is scalable and naturally suited for a PiM architecture that pre-processes stored data locally, all with the same vertically integrated, compact bit cell as shown in Fig. <ref>(c). Figure <ref>(d) shows the biasing condition for various operations.

From Mars rovers to mine robots, edge computing avoids cloud access that is either unavailable or unreliable. Energy and latency are its main concerns. Our device gating energy is low because the strain is remarkably energy-efficient, while our write energy is low because TIs offer a very high SHA. Our results (Fig. <ref>) show that at the individual bit cell level, our device requires much lower energy (∼2–6×) than 45-nm Dynamic and Static Random Access Memory (DRAM, SRAM) and HM-based SOTRAM with a lower/comparable speed (2 ns), further improvable with larger bandgap TIs. Moreover, at a PiM architecture level, we significantly outperform DRAM on both non-volatility and energy cost by rolling memory and logic into a single unit, reducing data transit latency from ∼mm-cm data bus lengths between separate memory-logic cores to just ∼ μm-mm array dimensions, as the sense amplifier processes suitably segmented data from locally accessed storage magnets. At data transfer rates of ∼Mbps-Gbps, this enables a 2–3× speed-up over traditional Von Neumann architecture, in addition to the energy gain. Our device exploits the TI as a unique semiconducting channel that delivers high SOT without field assist for low energy write and voltage gating with strain for low power row-column select in PiM architecture.






§ METHODS

To characterize the magnetization dynamics in our device, we solve a coupled stochastic LLG equation in the macrospin limit using the fourth order-Runge Kutta Method. As explained in detail in section <ref>, the device consists of a piezoelectric/gating magnet/TI/MTJ heterostructure. We simultaneously solve the magnetization dynamics of the gating magnet and the MTJ free layer. In the case of the gating magnet, the LLG equation is described as:


    1+α_1^2/γ·m_1t   = - μ_0·(m_1H_eff1) 
        - α_1 μ_0·m_1 (m_1H_eff1),

where m_1 = M_1/M_s1 is the normalized magnetization and M_s1 is the saturation magnetization of the gating magnet. α_1, μ_0, and γ are magnetic damping coefficient, permeability of free space, and gyromagnetic ratio, respectively. We consider H_eff1 = H_k1 + H_stress + H_th, where H_k1 and H_stress are the effective anisotropy field and stress field of the gating magnet, respectively <cit.>. H_k1 = 2K_u1/μ_0 M_s1 - M_s1 (K_u1 is the uniaxial anisotropy of the gating magnet). H_stress = 3 λ_s σ_s/μ_0M_s1 (λ_s is the magnetostriction coefficient of the gating magnet, and σ_s is the stress generated by the electrical strain induced by the piezoelectric). 


For the MTJ free layer, the switching is facilitated by the SOT arising from the TI, and the LLG equation takes the form: 

    1+α_2^2/γ·m_2t   = - μ_0· (m_2H_eff2) 
        - α_2 μ_0·m_2 (m_2H_eff2)
        - ħ/2e·θ_sh^eff J/M_s2t_f2·m_2 (m_2σ_p),

where e is the elementary charge, ħ is the reduced Plank constant, θ_sh^eff is the effective SHA of the TI, t_f2 is the thickness of the MTJ free layer, J is the surface current density of the TI, and σ_p is the unit vector along the spin polarization direction. The other variables and constants have the same meaning as defined in Eq. (<ref>), and a subscript of 2 represents the parameter for the MTJ free layer. θ_sh^eff = θ_sh(1-sech(t_TI/λ)), where θ_sh, t_TI, and λ are the SHA, thickness, and spin diffusion length of the TI. In Eq. (<ref>), H_eff2 = H_d + H_th. H_d is the demagnetization field of the in-plane MTJ free layer and expressed as H_d = - M_s2[N_dx N_dy N_dz], where N_dx, N_dy, N_dz are the demagnetization factors of the free layer along x, y, z axis, respectively.  

In both Eqs. (<ref>) and (<ref>), H_th is a random thermal field with zero mean (μ = 0) and standard deviation

    SD  = √(2 α k_BT/μ_0^2 γ M_s V Δ t),

where α, M_s, and V are the damping coefficient, saturation magnetization, and volume of the respective magnets. Δ t is the simulation time step. μ_0, γ, k_B, and T have their usual meanings. 

Throughout our study, we use TbCo as both the gating magnet and the MTJ free layer magnet, Bi_2Se_3 as the TI, and Pb(Zr,Ti)O_3 (PZT) as the piezoelectric materials unless otherwise specified. The used parameters are listed in Table <ref>.

The modulation of the surface current density J of the TI by the gating magnet is modeled separately with an empirical function, described next. 







§ RESULTS AND DISCUSSION

The first functional block of the entire stack is the piezoelectric/gating magnet heterostructure. 
Applying a voltage to the piezoelectric (PZT), electrical strain is generated and transferred to the adjacent gating magnet (TbCo) via the magnetostriction effect. This strain is then converted into stress via the elastic modulus of the gating magnet, which counteracts the uniaxial anisotropy of the gating magnet. When the stress is sufficient, it can switch the gating magnet from out-of-plane to in-plane with minuscule energy consumption in the order of ∼ 10 aJ <cit.> (energy cost estimation for our device is shown later). The strain-induced switching of the gating magnet facilitates the opening and closing of the band gap in the TSS of the TI (Bi_2Se_3), depending on the out-of-plane and in-plane magnetization orientation of the gating magnet, respectively.






Figure <ref>(a) shows the magnetization dynamics of the gating magnet under the influence of the stress induced by the piezoelectric strain. We started with an out-of-plane magnetization (m_1z = + 1) for the gating magnet, which corresponds to the OFF state of the device as it opens a band gap in the TSS and, hence, no current flow. 
Under the influence of stress, it takes a very short time, ∼ 1 ns for the 90^∘ switching (m_1z = 0) of the gating magnet for a stress σ_s = 100 MPa. The switching delay depends on the strength of σ_s and the material parameters of the gating magnet. The out-of-plane magnetization component (m_1z) is fed to the next block (TI) dynamically, which modulates the band gap in the TSS and, hence, the surface current. We consider a Dirac Hamiltonian form to model the surface state of the TI. In the presence of a magnet in proximity to the TI, the Hamiltonian can be expressed as H = ħ v_F(σk) ·ẑ + M_0M_1·σ <cit.>, where v_F is the Fermi velocity, k is the wavevector,  σ is the Pauli spin matrices, M_0 is the exchange strength between the gating magnet and TI, and M_1 is magnetization of the gating magnet. From simple algebra, for the case of an out-of-plane magnetization orientation of the gating magnet, we can show the energy dispersion takes the form E = ±√(ħ^2v_F^2|k|^2 + (M_0m_1z)^2), where |k|=√(k_x^2 + k_y^2). This gives rise to a band gap E_gap = √((2M_0m_1z)^2) at k=0. The band gap will modulate the surface current of the TI as I_surf = I_0,surfe^-E_gap/k_BT, where I_0,surf represents the surface current needed to switch the MTJ free layer for a specific switching delay. Figure <ref>(b) shows the evolution of the E_gap and the I_surf with time in response to the magnetization dynamics of the gating magnet (Fig. <ref>(a)). Initially, when m_1z = +1, a band gap of 2M_0 opened, and we get vanishing surface current. As the gating magnet switches (m_1z = 0) by the stress, the conductivity of the TSS is restored since the band gap is closed. We use M_0 = 0.1 eV, a typical value for Bi_2Se_3 <cit.>. Note that we cannot open an infinite band gap as it is limited by the bulk band gap of the TI (0.3 eV for Bi_2Se_3 <cit.>). We use I_0,surf = 6I_c,surf while generating Fig. <ref>(b), where I_c,surf is the critical surface current needed to switch the MTJ free layer. 















We utilize an in-plane magnet-based MTJ for our device for primarily two reasons: (i) SOT can provide deterministic switching without an external field assist only for an in-plane magnet because of in-plane spin polarization (orthogonal to the charge current) <cit.>, and (ii) we are limited by the interaction between the TI and the MTJ free layer as such an out-of-plane free layer will open gap in the TSS and hence no current conduction <cit.>. While utilizing an in-plane magnet occupies more area than a perpendicular magnetic anisotropy (PMA) magnet, our design integrates the gating and storage magnets within the same stack. This configuration offers both area and energy efficiency advantages, along with the added benefit of field-free switching. Extensive research is underway for field-free switching of PMA magnets. However, these require a graded structure or an added magnet that provides the symmetry-breaking field and adds structural complexity <cit.>. We use an in-plane magnet of “type y” (easy axis in y-direction). The critical current density for switching J_c for this type of MTJ is expressed as <cit.>:


    J_c = 2eα_2μ_0M_s2t_f2/ħθ_sh^eff(H_in+H_out/2)


H_in is the in-plane shape anisotropy of the free layer, and H_out is the out-of-plane demagnetization component. The other variables and constants have the same meaning defined in section <ref>. Using the parameters listed in Table <ref>, we find J_c = 1.88 × 10^10 A/m^2, which is at least one order of magnitude less than conventional HM-based SOT switching <cit.> and laid the foundation of energy-efficient switching. This amount of critical current density needs to be supplied by the top surface states of TI for the switching of MTJ free layer, from which we can estimate the critical surface current I_c,surf = J_cWt_surf=0.75 μ A (W is the device width, t_surf = 1 nm is the thickness of the TI top surface <cit.>). 

Figure <ref>(c) shows the magnetization dynamics of the MTJ free layer in response to the SOT generated by the TI surface current. We can clearly see the magnetization switching (m_2y = +1 to m_2y = -1) with a switching delay of ∼ 4 ns for a surface current of 6I_c,surf. Note that Figs. <ref>(a)-(c) are generated in the absence of a thermal field. Figures <ref>(d) and (e) show the magnetization dynamics of the gating magnet and the MTJ free layer, respectively, under the influence of the thermal field for 50 simulation runs for the same current as without thermal case, and we can see the stochastic behavior of the magnetization dynamics. Figure <ref>(f) shows the histogram from 1000 simulations runs, and we find a switching time of 10.75 ns, calculated by t_sw = μ + 6SD, corresponding to a write error rate WER = 10^-9, which is the industry standard for memory applications <cit.>. In the subsequent part of the paper, we calculate t_sw similarly unless otherwise specified.

Figure <ref> demonstrates the functionality of the device. We will now delve into the details and constraints of the individual blocks. Beginning from the bottom to up, the piezoelectric-gating magnet stack enables the strain-driven out-plane to in-plane switching. As discussed earlier, the switching is governed by the competition between the uniaxial anisotropy energy and the stress energy induced by the strain. PZT typically can generate an electrical strain of 0.05% - 0.1% (500 -1000 ppm) in response to an applied voltage <cit.>, which sets the achievable stress energy limit to counteract the anisotropy energy. The relation between the applied voltage and strain is V_G/t_piezo = ϵ/d_31 <cit.>, where V_G, t_piezo, ϵ, d_31 are the applied gate voltage, thickness of piezoelectric, strain, and piezoelectric constant, respectively. The typical d_31 constant for PZT is 1.8 × 10^-10 <cit.>. TbCo has Young's modulus of 100 GPa <cit.> and λ_s=400 × 10^-6 <cit.>, which gives a range of stress σ_s from 50 to 100 MPa. Figure <ref>(a) shows the gate voltage required for generating the strain (stress) for a 100 nm thick PZT. The critical question is whether this stress is sufficient to overcome the anisotropy barrier. TbCo has a low uniaxial anisotropy K_u1 of 64 kJ/m^3 <cit.>. Considering the demagnetization effect, the effecting anisotropy becomes 38.9 kJ/m^3 (K_eff = K_u1 - 1/2μ_0M_s1^2). Conversely, the stress energy E_stress = 3/2λ_sσ_s <cit.> ranges from 30 kJ/m^3 to 60 kJ/m^3 depending on the amount of strain (stress) shown in Fig. <ref>(a). Figure <ref>(b) presents the phase space for anisotropy and stress for the switching probability of the MTJ-free layer (colormap). The P_sw is calculated from 10^5 stochastic LLG runs <cit.> for 10 ns with a surface current of 6I_c,surf. The figure illustrates the reciprocal relation between the K_u1 and σ_s, and the range of K_u1 and σ_s for a working device. This phase space is important because K_u1 can be tuned by adjusting the composition of TbCo and alloying <cit.>, while σ_s is adjustable through applied gate voltage.

Next, we consider the voltage requirement for the MTJ write operation (switching the MTJ free layer). For the write operation, three important quantities are the switching time, switching current/voltage, and WER. We calculate the switching voltage vs. switching time for various stress values in Fig. <ref>, showing the typical reciprocal relation for spin-torque switching <cit.>. The switching voltage shown in Fig. <ref>(a) correspond to WER = 10^-9. As σ_s increases, the out-to-in-plane switching time of the gating magnet is reduced, and it requires less time for the band gap to close, which is attributed to the lower overall switching time of the device. We vary the surface current from 3.5 to 20 times the critical surface current and let the magnetization evolve until it reaches 95% of its final value (m_2y = -0.95) <cit.>. We denote the time as the switching time for the individual switching event, and from the switching time distribution of 1000 simulations, we calculate the t_sw corresponding to WER = 10^-9. While calculating the total current requirement for the required surface current, we consider a I_surf/I_tot = 15% (see more discussion later). 











Furthermore, we estimate the energy consumption for the switching process. The energy has two components: the gating energy and the switching energy. The energy associated with the gating mechanism comes from the applied voltage in the piezoelectric and can be estimated as E_piezo = 1/2C_pV_G^2 <cit.>, where C_p is the capacitance of the PZT, and V_G is the applied gate voltage. For the case of 0.1% strain, we get V_G=0.56 V and C_p = ϵ_r ϵ_0 W L/t_piezo = 0.071 × 10^-15 F (L is the length of the device, ϵ_0 and ϵ_r = 1000 <cit.> are the permittivity of free space and the relative permittivity of PZT, respectively). We obtain E_piezo = 11.13 aJ, which is minuscule and consistent with previous studies <cit.>. The other part of the energy is the I^2R loss in the TI. TI is typically modeled as parallel surface and bulk channels (bulk channel accounts for the shunting through the bulk states) <cit.>. The bulk resistance is R_bulk=L/σ_cWt_bulk = 1.46 kΩ, where σ_c is the average conductivity of the TI (σ_c = 5.7 × 10^4 Ω^-1m^-1 for Bi_2Se_3 <cit.>), and t_bulk is the thickness of the bulk state of TI. We use t_bulk = 6 nm since the TI is 8 nm thick and the thickness of the top and the bottom surface is 1 nm each <cit.>. We estimate R_surf based on the current distribution in the top surface channel, which is found to be 30% of the total current as previously reported in both theory <cit.> and experiments <cit.>. This makes R_surf = 1.94 kΩ. Nonetheless, the bottom surface needs to be grounded through the gating magnet for the gating mechanism to work for our vertical structure (Fig. <ref>(b)). Therefore, we divide the bottom surface resistance equally with a ground in the middle and estimate the equivalent resistance R_eq = 633.5 Ω, and eventually, we achieve 15% current through the top surface channel of the TI for the resistance configuration shown in Fig. <ref>(b). Shunting through the bulk in TI is an open question, and it requires materials with a higher band gap to get more current in the surface <cit.>. For high-speed applications, we want the switching time in single-digit nanoseconds <cit.>. We estimate the energy used in this operation regime. 
Figure <ref>(a) shows the voltage requirement for t_sw for 2-10 ns (WER = 10^-9), while Fig. <ref>(b) shows the energy consumption for the writing operation for a range of I_surf/I_tot. I_surf/I_tot can be varied by tuning the conductivity, bulk band gap, and thickness of the TI <cit.>. Our analysis suggests energy consumption in the TI is in the order of <100 fJ, even with a low current ratio of 15% in the top surface, which is energy efficient compared to HM-based SOT switching (Fig. <ref>(c)), consistent with previous studies <cit.>. If we increase the top surface current, the energy consumption will be decreased further. Fig <ref>(c) shows the energy consumption comparison between HM and TI-based switching, which shows an order of magnitude reduction in energy consumption for TI. Although the conductivity of HM is very high, the current dictates the I^2R loss, and TI requires a significantly low current because of its higher SHA <cit.>. 

The total energy consumption for writing operation E_write = E_piezo + E_TI, which is dominated by the E_TI because of the minuscule energy in the gating mechanism. This suggests the advantage of the stain-based intrinsic gating mechanism because it eliminates the need for an access transistor for the write operation, costs negligible energy, and has a reduced footprint compared to a CMOS access transistor. The energy consumption in TI will be much lower if we incorporate other TIs with higher SHAs. For example, for BiSb (SHA = 10 and σ_c = 1.5 × 10^5 Ω^-1m^-1 <cit.>), the energy consumption is ∼ 0.9 fJ and ∼ 0.2 fJ for a current ratio of 15% and 20%, respectively, which is ∼ 100 × lower than Bi_2Se_3 and ∼ 1000 × lower than traditional HM. 



































The above-estimated write energy is at the bit cell level. Now, we project the energy cost for a PiM array from the bit cell energy. In particular, we show the projection for a 2-bit Boolean AND and OR operation using the PiM crossbar array (Fig. <ref>(c)). Before doing the projection, we need to estimate the read energy cost. For reading operation, we use a small sense current I_sense = 1 μ A, significantly less than the write current, eliminating the risk of an accidental switch. We assume a resistance area product for the MTJ R_P A = 2 Ωμ m^2 and a TMR = 100% <cit.>, which makes R_P = 2.5 k Ω and R_AP = 5 k Ω, respectively (R_P and R_AP are the resistance of MTJ in parallel and anti-parallel state, respectively). We consider a 16 nm PTM HP model for the read access transistor with a R_ON = 5 k Ω. We need to keep the gate voltage ON for the read operation so that the TI surface state is conductive, adding the gating energy during the read operation. Considering this, we found a read energy E_read of ∼ 45 aJ and ∼ 55 aJ for parallel (bit `0') and antiparallel (bit `1') states at a read speed of 4 ns <cit.>. For in-memory 2-bit AND and OR operations, we need to select two bit cells simultaneously, and the PiM operation is done by comparing the sense voltage from the bit cells with a reference voltage V_ref through a sense amplifier (see Fig. <ref>(b)), where V_ref is set to a value depending on the operation we want to perform. For two bit cells, we will have {R_AP, R_AP}, {R_AP, R_P}, {R_P, R_AP}, {R_P, R_P} combinations in the resistance states of the MTJ, corresponding to sense voltages {V_AP, V_AP}, {V_AP, V_P}, {V_P, V_AP}, {V_P, V_P}. Setting a V_ref = (V_AP,AP + V_AP,P)/2 ensures the sense amplifier will produce a logic `1' output only when both the bit cells are in the antiparallel state (logic AND). Similarly, for logic OR, V_ref = (V_AP,P + V_P,P)/2 will serve the purpose. For the resistance and sense current used in our study, we get V_AP,AP = 5 mV, V_AP,P/V_P,AP = 4.29 mV, V_P,P = 3.75 mV, which give V_ref, AND=4.65 mV; V_ref, OR=4.02 mV. We can set the reference voltage by tuning the reference resistance of the sense amplifier, and therefore, we can perform re-configurable in-memory computing by tuning the reference voltage. 
The energy components for a full cycle (write-read) in-memory Boolean computing are summarized in Fig. <ref>, which shows that E_write is the dominant contribution. This suggests the energy efficiency of the STI-SOTRAM bit cell for in-memory computing because the write energy of STI-SOTRAM is much smaller than that of a conventional HM-based device, as shown in Fig. <ref> (see the comparison with other emerging technology in Table <ref>). Note that we assume a symmetric write operation <cit.> for both bits `0' and `1', and the energy associated with the sense amplifier is estimated as 1/2C Δ V^2, where C is the capacitance of the sense amplifier (typically 1 pF <cit.>) and Δ V is the difference between the sense voltage and the reference voltage. The estimated total area for the Boolean operation is 6720 nm^2, which includes the area of the bit cells and the read access transistor (W/L = 10:1).  

Finally, we present the effect of various key parameters on device switching time/delay t_sw. We show the effect of the thermal stability factor Δ_1, exchange constant M_0, and damping coefficient α_1 of the gating magnet on t_sw for various σ_s values. In Fig. <ref>, the top panel shows the t_sw while the bottom panel shows the histogram of t_sw from 1000 simulations. For the case of varying Δ_1, as it increases, the switching time of the gating magnet will increase for a specific σ_s, which increases the overall switching time of the device. While for a larger α_1, t_sw reduces as the gating magnet switching reduces with a large damping coefficient because the gating magnet dynamics is governed by precessional and damping torques. Lastly, if we increase M_0, t_sw increases because the gap opening is larger, and it takes longer to provide sufficient surface state current in the TI. Note that even if we get a large M_0, the TSS band gap cannot be infinitely large as we are limited by the bulk bandgap of the TI. In contrast, we need a larger value of M_0 for a high ON/OFF ratio, which suggests the need for a TI with larger bulk bandgap <cit.>.







§ CHALLENGES AND OPPORTUNITIES 
 
The above results show excellent promise for utilizing the STI-SOTRAM for an energy-efficient bit-storage device. However, it is worth discussing the challenges associated with implementing STI-SOTRAM devices. We will separate these discussions into two categories - material issues vs. device issues.


 §.§ Material Challenges

There exist material issues <cit.> since we need three separate interfacial processes with input-output isolation – (a) strain rotating the bottom selector magnet, (b) modulating the TSS with such rotation, and (c) TSS writing information onto the top storage magnet. Let us discuss these three processes one by one.

(a) Can strain switch a magnet by 90^∘?
The magnetocrystalline anisotropy density K_u is often larger than stress energy. We will need to compositionally tune sputtered magnets like amorphous TbCo  <cit.>, or tailor film thickness of epitaxially grown CoFe to approach a small K_u near the out-of-plane to in-plane crossover  <cit.>. The resulting low K_u can be overcome with stress, while being large enough to avoid spontaneous thermal fluctuations. The condition for switching is (3/2)λ_s σ_s > K_u, where σ_s is the uniaxial stress. The magnets referenced above have magnetostriction coefficient λ_s ∼ 200-400 × 10^-6 <cit.> with elastic moduli ∼ 100 GPa <cit.> and strain ∼ 0.1%, which gives (3/2)λ_sσ_s ∼ 30-60 kJ/m^3 and suggests successful switching of the magnetization if we can keep K_u lower.

(b) Can a rotated magnet kill the TSS?
Theory <cit.> and experiments <cit.> show that an interfacing out-of-plane magnet opens a TSS bandgap, leaving only chiral quantum anomalous edge states with significantly diminished spin surface current. A 3D tight binding-based study suggests the same, even accounting for bands along the patterned TI side walls <cit.>. Critically, the magnet must adjoin the TI, providing at least 50 mV exchange coupling. The g-factor for the Zeeman gap is highly material dependent (∼ 18 for Bi_2Se_3, ∼ 6 for Sb_2Te_2Se <cit.>), while that depends on bulk bandgap due to the magnetic proximity effect (∼ 20 meV for EuS/Bi_2Se_3 <cit.>, ∼ 100 meV for MnBi_2Se_4/Bi_2Se_3 <cit.>).

(c) Can an activated TI write efficiently on the top magnet, accounting for small TI bandgap and current shunting?
SHA, even in sputtered TI, can be large (∼ 10-20, about 100 × greater than conventional HMs, e.g., Pt), switching magnets very efficiently <cit.>. TIs have small bulk bandgaps ∼ 100-300 meV that shunt current into the bulk and sidewalls. However, previous studies suggest that even for a modest bandgap TI, an ON-OFF ratio of ∼ 10 might be able to flip spins of a magnet with a low  WER  <cit.>.  This self-correction happens through internal anisotropy fields <cit.>, an increased bandgap with thin film quantization <cit.>, and reduced charge current with gap opening.  To avoid current shunting into the storage magnet, a thin layer of insulating NiO <cit.> or MgO <cit.> can be grown that transmits magnon torque between the magnet/TI, or else use an insulating BaFe_12O_19 as the storage magnet’s free layer <cit.>.

(d) Can we grow and pattern the heterogeneous stack?

To grow the stack with compatible processing temperatures, we will need to sputter the Bi_2Se_3 directly onto a magnet on a piezo such as Pb(Zr,Ti)O_3(PZT)/Pb(Mg_1/3Nb_2/3)O_3-PbTiO_3 (PMNT). 
Previous experiments show that amorphous TI retains bandgap, topological protection, and high SHA <cit.>, even after patterning with wet etching <cit.>. Besides mechanical decoupling, the stacks are also electrically decoupled, as the strains on a common piezo substrate stay localized around the contacts <cit.>.



 §.§ Device/circuit/architecture issues


Table <ref> shows projected performance metrics for our device vs. competing memory technology, especially for PiM/embedded applications. 

(a) Key application space. Our device is similar to a DRAM crossbar, complementary to non-volatile memory in the low power/embedded space, and competitive in NAND space with similar trade benefits as MRAM/SOTRAM. It capitalizes on SOT technology, which is attractive as last level embedded cache for digital AI and stand-alone off chip cache for analog AI. The device uses a superior TI channel that is both spin-selective with high SOT efficiency and gate tunable with low power strain gating. For in-memory computing in embedded applications (image processing, combinatorial optimization) for edge intelligence, error tolerance and retention needs are low while energy and latency are premium. Our energy budget is low with write voltage <100 mV for thin film (∼ 1 μm on conducting subtrate)/2-D monolayer piezos. Our device projects excellent metrics with ∼ 2-3X improvement over DRAM in energy-delay products. 

(b) Critical current/voltage. Our projected current density ∼ 10^5-10^6A/cm^2 is typical of STTRAMs and can be further reduced with higher SHA. Since the TSS are metallic in ON-state (conductivity of BiSe ∼ 8,000-60,000 Ω^-1m^-1), our room-temperature write voltage is projected to be < 100 mV. 

(c) Back-end-of-the-line (BEOL) integration. TIs can be sputtered and wet etched while retaining high SHA, so usual BEOL metallization steps should work. Both TI and magnets should survive 85^∘-100^∘C processing temperatures. Thermal degradation of TI does not usually occur below 300^∘C. 

(d) Endurance, stability and device-to-device variations. While HM-based SOTRAM endurance is  ∼ 10^15, piezo fatigue and TI endurance limit our STI-SOTRAM device to ∼ 5 × 10^6 cycles <cit.>. While this is significantly lower endurance, we believe this would suffice for hyperdense, nonvolatile persistent cells in PiM applications, where the probability of data overwrite per cell is very low (∼ 10^-3, easily error corrected <cit.>).

(e) Data retention, read disturb tolerance and scalability. The lower selector magnet needs small out-of-plane anisotropy for strain gating, while the upper storage magnet needs higher barrier consistent with SOTRAM technology with 10 year retention. Like SOT, we avoid read disturbs with separate read-write paths (unlike SOT, we do not need field-assist to circumvent stagnation, as our TI spins and storage magnet are in-plane). The top MTJ requires ion-milling and mesa etching, which the TSS are known to survive.  

(f) Density. Our device footprint is similar to three terminal SOT, slightly bigger than STTRAM, DRAM, and flash. We compete on density (and latency) by rolling storage and processing into one bit cell, along with non-volatile magnetic storage and ultra-low energy field-free write using strain gating (∼ 10 aJ), and high SHA (∼ 10), key metrics for embedded applications and edge intelligence. 




§ CONCLUSION
 
In summary, we demonstrated a novel four-layer piezoelectric/magnet/TI/MTJ bit cell design with reduced energy cost and footprint for in-memory computing. Eliminating an access transistor with a built-in strain-based gating mechanism has proved energy-efficient, yielding an overall reduced energy cost. High-speed operation is achieved with significantly less energy than the traditional SOT metals at the device and array level. Our results suggest this heterogeneous stack may provide a compact and energy-efficient design for low-power, high-speed in-situ applications.


The challenges at this time are the yet-to-be-demonstrated ability to fabricate the entire stack, the lower endurance associated with TIs (which is a lesser issue with PiM as the data overwrite per cell is low), and the established challenges with SOT, such as higher footprint (again, PiM saves at the circuit level by rolling memory and logic into one unit), and higher current (but lower voltage since the HM has lower resistance). The in-plane spins help with field assist but hurt with scaling. Out-of-plane field-free switching has been demonstrated with SOTs and with Weyl semi-metals <cit.>. 







§ ACKNOWLEDGMENTS

We thank S. Joseph Poon at the University of Virginia, Supriyo Bandyopadhyay at Virginia Commonwealth University, Joseph A. Hagmann at Mitre, Patrick Taylor, George de Coster, and Mahesh R Neupane at Army Research Laboratories, Andrew D. Kent at New York University, Daniel B. Gopman at the National Institute of Standards at Technology, and Steve Kramer at Micron for insightful discussions. This work was supported in part by the NSF I/UCRC on Multi-functional Integrated System Technology (MIST) Center; IIP-1439644, IIP-1439680, IIP-1738752, IIP-1939009, IIP-1939050, and IIP-1939012. The calculations were performed using the computational resources from High-Performance Computing systems at the University of Virginia (Rivanna).


apsrev4-1


 Paper Title: Information Scrambling at Quantum Hall Interfaces and Their Analog to Black Hole Event Horizon.









Ken K. W. Ma Hometown: Orlando, Florida 
 
kwokwai_ma@alumni.brown.edu
Kun Yang () Department of Physics, Florida State University and the National High Magnetic Field Laboratory, 

 Address: 1800 E Paul Dirac Dr, Tallahassee, Florida 32310 

kunyang@magnet.fsu.edu





Information Scrambling at Quantum Hall Interfaces and Their Analog to Black Hole Event Horizon
    Ken K. W. Ma and Kun Yang
    August 5, 2024
==============================================================================================



The black hole information paradox has been hotly debated for the last few decades without a full resolution. This makes it desirable to find analogues of this paradox in simple and experimentally accessible systems, whose resolutions may shed light on this longstanding and fundamental problem. Here, we review and resolve the apparent “information paradox" in two different interfaces separating Abelian and non-Abelian quantum Hall states. In both cases, the information carried by the pseudospin degree of freedom of the Abelian anyons get scrambled when they cross the interface and enter the non-Abelian quantum Hall liquid. Nevertheless, it is found that the scrambling mechanism depends on the nature of the interface. The corresponding analogues of different concepts in black hole physics such as event horizon, black hole interior, Hawking radiation, and Page curve will also be discussed.





§ INTRODUCTION


The existence of black holes has received strong support from recent observations <cit.>. Instead of being a region which nothing can escape from, Hawking predicted that a black hole emits radiation and evaporates slowly <cit.>. He also concluded that the radiation carries no information except mass, angular momentum, and charge of the black hole <cit.>. This result points to possible loss of information in black holes. On one hand, it is consistent with the no-hair theorem <cit.>. On the other hand, quantum mechanics forbids information loss due to the unitarity of time evolution. This apparent contradiction leads to the black hole information paradox <cit.>. It is believed that the resolution of this paradox may provide important clues on how to combine quantum mechanics and general relativity.

Various approaches have been proposed to resolve this black hole information paradox. Among them, the holographic principle <cit.> supports the preservation of unitarity and information. In particular, information can be encoded holographically on surfaces, such as the event horizon. This belief is substantiated by the discovery of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence <cit.>. Recently, the firewall scenario <cit.> was proposed to resolve the conflict between black hole complementarity <cit.> and monogamy of entanglement <cit.>. If this conjecture is correct, the firewall at the event horizon (or black hole's boundary) may also break the entanglement between the outgoing and the in-falling particles. Thus, the boundary can be as important as, or even more important than, the interior of a black hole.

Suppose black hole evaporation is a unitary process. Then how is information hidden in the black hole released from Hawking radiation? Page argued that the release of information starts slowly at the beginning, but becomes faster in the later stage of the evaporation <cit.>. If the system was initially in a pure state, entropy of the radiation (coming from its entanglement with the remainder of black hole) would first increase from zero but eventually decrease back to zero when the black hole evaporates completely, thus recovering the pure state nature of the system and all the (quantum) information it carries. This feature is now known as the Page curve. Based on quantum information theory, the thought experiment by Hayden and Preskill (Hayden-Preskill protocol) has provided further insight on retrieving information from Hawking radiation <cit.>. Suppose the black hole has already passed its Page time and become maximally entangled with its previously emitted Hawking radiation. If the internal dynamics of black hole can be described by an instantaneous random unitary transformation, then any additional information entering the black hole can be recovered from Hawking radiation almost immediately (a very short time compared to the lifetime of the black hole) <cit.>. The protocol has postulated the existence of information scrambling <cit.>, which describes the dispersal of local information into entanglement and correlation throughout the entire system. Thus, the original information is stored nonlocally and cannot be recovered via local measurement. This concept has attracted considerable attention in the context of many-body dynamics <cit.> and quantum neural network <cit.>, which has been verified in recent experiments <cit.>. In addition, recent studies have recovered the Page curve for AdS black holes <cit.>. However, a full resolution of the paradox remains an open problem <cit.>. It is thus desirable to mimic the information paradox in simple and experimentally accessible systems, that allows for a complete understanding of this process.

Somewhat similar to the holographic principle, the bulk-edge correspondence relates the topologically protected edge modes and bulk topological orders in fractional quantum Hall (FQH) systems <cit.>. This allows us to learn about the bulk by probing the edge of the system <cit.>. Comparatively speaking, interfaces between a pair of FQH states are explored much less <cit.>. In fact, the physics and dynamics of interfaces are much richer than simple edges <cit.>. For example, as we demonstrate below, certain interfaces allow quasiparticle tunneling between two different FQH states, even if the quasiparticles are of very different nature. If they have different internal degrees of freedom, (local) information carried by them needs to be transmuted (or scrambled in a specific way) to prevent information loss. This motivates us to explore analogues of black hole information paradox in quantum Hall interfaces.

In this chapter, we review and resolve a condensed matter analogue of information paradox in two different quantum Hall (QH) interfaces <cit.>. We first consider the interface between Halperin-331 <cit.> and Pfaffian (Moore-Read) <cit.> QH states. The latter is a famous non-Abelian state which hosts non-Abelian anyons that may be useful in topological quantum computation <cit.>. Meanwhile, both 331 and Pfaffian states may be realized in bilayer systems or wide quantum wells at filling factor ν=1/2 <cit.>. Due to the competition between interlayer tunneling and intralayer Coulomb interaction, a phase transition between the 331 and Pfaffian states was predicted <cit.>. This suggests the possibility of creating a 331-Pfaffian interface by controlling the tunneling strengths in different regions of the bilayer system<cit.>. Interestingly, the Abelian 331 state has a pseudospin (layer) degree of freedom, which is absent in the Pfaffian state. If the original information carried by the pseudospin degree of freedom becomes irrecoverable after quasiparticles cross the interface and enter the Pfaffian liquid, it leads to an “information paradox". We demonstrate that the information is scrambled and stored nonlocally in the Pfaffian liquid and the interface. We also mimic black hole evaporation in the same system, and find it satisfies the Page curve naturally. In other words, the original pseudospin information is recovered and the “information paradox" in our model is resolved. Following the similar logic, we consider the “information paradox" in another QH interface formed by Halperin 330 state and Read Rezayi state at level four (abbreviated as RR_4 state) <cit.>. On one hand, the setting closely resembles the 331-Pfaffian interface by having a pseudospin degree of freedom in the 330 state that is absent in the RR_4 state. Both states may be realized in bilayer systems at ν=2/3 <cit.>. On the other hand, we will show that the nature of 330-RR_4 interface is different from the 331-Pfaffian interface. This leads to different mechanisms for scrambling the original pseudospin degree of freedom carried by the Abelian anyons in the two QH interfaces. Here, we must emphasize that we are not aiming at a resolution of the original information paradox in astrophysical black holes. This is clearly unachievable by proposing a simple analogy. Instead, we want to simulate some important concepts in resolving the original paradox in a simple and accessible manner.



§ BASICS OF QUANTUM HALL EFFECT


To set the stage for later discussion, we first review briefly some basic concepts in quantum Hall (QH) physics <cit.>. Electrons moving in two dimensions (x-y plane) and a perpendicular magnetic field (in the z direction) have their energy levels being quantized in Landau levels. A very important parameter characterizing the system is the Landau level filling factor ν, which is the ratio between the number of electrons and the number of magnetic flux quanta enclosed by the system. Of particular interest is the case ν<1, where only the lowest Landau level is partially filled by electrons at low temperature, known as the fractional QH (FQH) regime. Since the kinetic energy of these electrons is quenched due to Landau quantization, the interaction between them dominates the properties of the system. Various FQH states, which possess numerous fascinating properties, are realized in this strongly correlated electronic system. The exotic properties of FQH states are associated with the topological order they possess <cit.>. Most prominent among them is the existence of low-energy excitations (quasiparticles) that have fractional charges and obey fractional statistics (between bosonic and fermionic statistics) <cit.>. A famous example is the Laughlin state at ν=1/3 <cit.>, in which quasiparticle with a fractional charge e/3 <cit.> and a fractional statistics 2π/3 can exist <cit.>. Note that both fractional charge <cit.> and fractional statistics were observed experimentally <cit.>. Such exotic quasiparticles are called anyons, and the possible types of anyons are associated with the specific topological order.



 §.§ Bulk-edge correspondence and conformal field theory


The bulk-edge correspondence, another consequence of the topological order, relates the edge structure and the bulk topological order in FQH systems. In particular, it predicts the existence of gapless edge modes described by conformal field theories (CFTs), and there is a one-to-one correspondences between the bulk topological order and edge CFT <cit.>. In our previous example, the edge of the Laughlin state at ν=1/3 has a single chiral bosonic edge mode ϕ, which can be described by the Lagrangian density,

    ℒ_1/3
    =-3/4π∂_xϕ(∂_t-v∂_x)ϕ.

Here, v is the speed of the edge mode, and ϕ is a (chiral) bosonic field. In general, the edge of a FQH liquid can have more than one edge mode. For Abelian FQH states, the corresponding edge theory is described by <cit.>

    ℒ_ edge
    =-1/4π∑_i,jK_ij∂_tϕ_i∂_xϕ_j
    -1/4π∑_i,jV_ij∂_xϕ_i∂_xϕ_j.

Importantly, the K matrix encodes all information of the topological order. For a FQH state in a bilayer system, it may (but not always) be described by a two-component topological order which has a 2× 2 K matrix. In this situation, two different edge modes exist. Furthermore, the possible type of edge modes is not limited to bosonic mode. Other types of modes such as Majorana fermion modes exist when the topological orders are non-Abelian <cit.>.

With the knowledge of the edge structure in hand, different low-energy excitations in the FQH system can be described or created by suitable CFT operators <cit.>. For example, a charge-e/3 quasiparticle and an electron in the ν=1/3 Laughlin state are created by the operators :exp(iϕ): and :exp(3iϕ):, respectively. Here, :𝒱: denotes the normal ordering of the vertex operator 𝒱. When there is no confusion, this normal ordering notation will be dropped in the later discussion. For a FQH state being described by a multicomponent topological order, there are multiple types of anyons (described by different CFT operators) that have the same electric charge. In other words, the anyons have an additional degree of freedom. This point will become clear when we discuss our setup.



§ INFORMATION PARADOX IN 331-PFAFFIAN INTERFACE



Both Halperin-331 and Pfaffian quantum Hall liquids have the same Landau level filling factor ν=1/2, which can be realized in a bilayer system. For the 331 liquid, it is described by a two-component topological order with the K matrix <cit.>,

    K=[ 3 1; 1 3 ].

The two different edge modes are denoted as ϕ_↑ and ϕ_↓. The two most relevant operators creating an electron are exp(3iϕ_↑+iϕ_↓) and exp(iϕ_↑+3iϕ_↓). On the other hand, the Pfaffian liquid is described by a single-component non-Abelian order with K=2. Its edge has a bosonic mode ϕ and a Majorana fermion mode ψ <cit.>. The corresponding electron operator is ψexp(2iϕ). Since the Halperin-331 and Pfaffian edges have opposite chiralities at the interface, the interface is described by the Lagrangian density <cit.>,

    ℒ
    =   -1/4π∑_i,jK_ij∂_tϕ_i∂_xϕ_j
    +2/4π∂_tϕ_l∂_xϕ_l
    -iψ_l∂_tψ_l
    
       -ℋ(ϕ, ψ).

Here, the indices i,j=↑, ↓ denote the layer or the pseudospin.
All ϕ_↑, ϕ_↓, and ϕ_l are charge modes. The first two are right-moving along the edge of the 331 liquid, whereas the last one is left-moving along the edge of the Pfaffian liquid. The Pfaffian liquid also has a left-moving neutral Majorana fermion mode ψ_l along the edge. The edge structures of both quantum Hall states are illustrated in Fig. <ref>(a). As shown by one of us, a relevant random electron tunneling between the Pfaffian and 331 edges can lead to a phase transition at the interface <cit.>.

Now, we follow Ref. <cit.> and briefly summarize how different modes get localized at the interface. This also allows us to introduce useful notations for later discussion. One can define a charge mode ϕ_r=ϕ_↑+ϕ_↓ and a neutral spin mode ϕ_n=ϕ_↑-ϕ_↓ in the 331 liquid. Using this new set of modes, the topological term of the 331 edge becomes

    ℒ_331
    =-2/4π∂_tϕ_r∂_xϕ_r
    -1/4π∂_tϕ_n∂_xϕ_n.

The overall charge density at the interface is given by

    ρ(x)
    =1/2π∂_x(ϕ_↑+ϕ_↓+ϕ_l)
    =1/2π∂_xϕ_c.

The random electron tunneling between the Pfaffian and 331 edges is described by

    H_ T
    =   ∫ξ(x)ψ_l
    (e^3iϕ_↑+iϕ_↓+2iϕ_l
    +e^iϕ_↑+3iϕ_↓+2iϕ_l) dx
    +H.c.
    
    =   ∫ |ξ(x)|ψ_l(x) ψ_r(x)cos[2ϕ_c(x)+φ(x)] dx.

Here, ξ(x) denotes the random tunneling amplitude. In the second line, |ξ(x)| and
φ(x) are the magnitude and the phase of ξ(x), respectively. We have also fermionized exp[iϕ_n(x)]=ψ_r(x)+iψ_R(x). The resulting edge modes are shown in Fig. <ref>(b). If H_ T is relevant in the renormalization group sense, then both charge modes, ψ_l, and ψ_r are localized at the interface. After the localization, only a single right-moving Majorana fermion mode remains gapless and propagates freely at long distance, as shown in Fig. <ref>(c). Notice that this gapless mode is neither an original edge mode of the 331 nor the Pfaffian state.



Since Pfaffian and 331 states are quantum Hall states formed by superconducting pairing between composite fermions <cit.>, both of these two states have quasiparticles with the smallest possible charge of e/4 <cit.>. However, there is a fundamental difference between these quasiparticles. For the 331 state, there are two different types of Abelian e/4 quasiparticles created by the vertex operators, e^iϕ_↑ and e^iϕ_↓. One may view them as quasiparticles with different pseudospins. When we formulate the information paradox in the following discussion, this pseudospin will be regarded as the degree of freedom of the Abelian quasiparticles. For the Pfaffian state, there is only one type of e/4 quasiparticle created by σ e^-iϕ_l/2 <cit.>. Here, σ is the twist field with a scaling dimension 1/16 in the chiral Ising CFT <cit.>. We summarize the three primary fields and their properties in Table <ref>. In particular, σ satisfies the fusion rule σ×σ=ψ+I. Note that we have omitted the subscript l for the Majorana field to make the discussion of Ising CFT general. Its proper meaning should be clear from context. The fusion rule indicates that the quasiparticle is non-Abelian. An interesting question is what happens if a quasiparticle is dragged from the 331 liquid in to the Pfaffian liquid? It seems that the pseudospin information would be lost. In this sense, we can define the interface between the two different QH liquids as the “event horizon with a firewall" in our setup. This definition or analogy makes sense since the interface plays the role of a one-way surface of information in our setup, and the “destruction" of pseudospin information at the interface resembles a firewall conjectured in Ref. <cit.>. In addition, the Pfaffian liquid can be viewed as the interior of a “black hole". Suppose this analogous black hole can evaporate (discussed in Sec. <ref>) and the original pseudospin information cannot be recovered at the end of the evaporation. Then, the lost of information contradicts to the fact that quasiparticle tunneling is a unitary process. Thus, we have identified an apparent “information paradox".





 §.§ 331-Pfaffian interface from anyon condensation



Before resolving the paradox, we reformulate the above discussion in the framework of anyon condensation <cit.>. This technique has been commonly applied to study possible transitions between topologically ordered phases. In the context of quantum Hall physics, it was used to study the interface between Pfaffian and non-Abelian spin-singlet (NASS) quantum Hall states <cit.>. In this section, we first use anyon condensation to deduce the CFT description of the 331-Pfaffian interface. In the next section, we apply the same technique to resolve the paradox. Along the way, we adopt a pedagogical approach and aim at relating the rather abstract technique to the more physical picture in Sec. <ref>. It will allow us to highlight the advantages of applying anyon condensation in studying quantum Hall interfaces.

From Sec. <ref>, we know that the edges of the Halperin-331 and Pfaffian liquids are described by CFTs with central charges 2 and 3/2, respectively. These two edges are counterpropagating at the interface. Hence, we expect the resulting CFT describing the 331-Pfaffian interface has a net central charge of 2-3/2=1/2. To deduce exactly what the CFT is, it is first necessary to separate the charge and neutral sectors for both Halperin-331 and Pfaffian liquids. It is because a charge mode cannot be gapped out by coupling to a neutral mode in a usual situation. Equivalently, we do not consider the possibility of condensing charge bosons, which will break the U(1) gauge symmetry. The separation was already achieved in Sec. <ref>. In particular, the combination of charge modes ϕ_c=ϕ_r+ϕ_l was shown to be gapped out (more precisely, localized) by H_ T <cit.>. Therefore, we can focus our discussion on the neutral sectors.

As stated previously, the neutral sector of the Pfaffian state is described by a chiral Ising CFT. For the Halperin-331 state, its neutral sector is governed by the spin mode ϕ_n, which is described by the U(1)_4 CFT. Different primary fields in this Abelian CFT are summarized in Table <ref>. Note that any two vertex operators in the form
e^iαϕ_n/2 and e^i(α+4ℤ)ϕ_n/2 are identified.



The structure (remaining gapless modes) of the 331-Pfaffian interface is solely determined by anyon condensation in the neutral sectors. This condensation occurs in the
U(1)_4 ×Ising CFT. We emphasize again that the bar denotes conjugation of the Ising CFT due to the opposite chiralities between the 331 and Pfaffian edges at the interface.  Compared to the original CFT, anyons in the conjugate CFT have the same fusion rules, but complex conjugated topological spins and braiding phases. Alternatively, one may interpret the condensation as a coset construction <cit.>. We label a generic anyon as (e^iαϕ_n/2, t̅). Here, the parameter
α=0,1,2,3 determines the corresponding primary fields in the U(1)_4 CFT. Meanwhile, t̅={I̅, ψ̅, σ̅} denotes the primary fields in the conjugate Ising CFT. In the present case, there is only one condensable boson,
B=(e^iϕ_n, ψ̅). The condensation of B leads to confinement of some of the anyons in the condensed phase. An anyon remains unconfined if and only if it has a trivial mutual statistics with B. This condition ensures that an unconfined anyon has a consistent topological spin in the condensed phase. Furthermore, two anyons are identified when they differ from each other by a multiple of B. Using operator product expansion, it is straightforward to deduce the six (or three after identification) deconfined anyons in the condensed phase. They are listed in Table <ref>. Their corresponding topological sectors, namely Ĩ, ψ̃, and σ̃ are defined according to their conformal spins. From the table, we conclude that the 331-Pfaffian interface is described by a chiral Ising CFT. Note that this Ising CFT has an opposite chirality to the one describing the Pfaffian edge at the interface.



Now, one may wonder why going through such abstract and seemingly redundant procedures to find out the CFT describing the interface. Doesn't the net central charge c=1/2 directly indicate that it should be an Ising CFT? There are two reasons for analyzing this simple system by anyon condensation. First of all, it is fortunate that for the 331-Pfaffian interface, the mechanism and consequences of anyon condensation can be visualized in a very transparent and physical manner, but this is a very special case. In Eq. (<ref>), the electron tunneling between counterpropagating edges at the interface couples ψ_l and e^iϕ_n. This leads to a mass term and eventually gaps out the counterpropagating ψ_l and ψ_r. Only ψ_R remains gapless at the interface. This was demonstrated by fermionizing e^iϕ_n=ψ_r+iψ_R. This type of arguments does not always generalize to more complicated interfaces. On the other hand, the condensation of B systematically captures the gaping process and leads to a correct CFT description of the 331-Pfaffian interface. More importantly, anyon condensation relates every primary field in the original and condensed phases. These relations cannot be obtained from the argument in Sec. <ref>.



 §.§ Transmutation of pseudospin information
 


We now discuss the transmutation of pseudospin information when Abelian quasiparticles cross the interface. Let us first comment on the charge sectors. As one will see, they basically play no role in the resolution of the paradox. Since charge-e/4 quasiparticles are allowed in both Halperin-331 and Pfaffian liquids, dragging quasiparticles across the interface does not require any absorption of net charge at the interface. Furthermore, the gapping of ϕ_c indicates that the dragging will not create any low-energy charge excitation at the interface <cit.>.

Thus, we focus on the neutral sectors. An Abelian quasiparticle in the Halperin-331 liquid has its pseudospin degree of freedom carried solely by the neutral mode ϕ_n. This is observed by writing

    e^iϕ_↑   =e^iϕ_r/2e^iϕ_n/2,
    
    
    e^iϕ_↓   =e^iϕ_r/2e^-iϕ_n/2.

Hence, the vertex operators 𝒱_1 and 𝒱_3 encode the spin-up and spin-down states of an Abelian charge-e/4 quasiparticle, respectively. These two operators are not defined in the Pfaffian liquid. To understand the transmutation of quasiparticles when they cross the interface, we need to represent the four primary fields in the U(1)_4 CFT as different products between two Ising CFTs. One of them describes the interface, whereas the other describes the Pfaffian order. Both Ising CFTs now have the same chirality to match the central charges, 1=1/2+1/2. From Table <ref>, one can obtain the inverted expressions:

    𝒱_0
       ≡ I_1
    =I_1/2×Ĩ+ψ×ψ̃,
    
    𝒱_1
       ≡ e^iϕ_n/2
    =σ×σ̃,
    
    𝒱_2
       ≡ e^iϕ_n
    =ψ×Ĩ+I_1/2×ψ̃,
    
    𝒱_3
       ≡ e^-iϕ_n/2
    =σ×σ̃.

The tilded and untilded fields are in the CFTs describing the interface and the Pfaffian liquid, respectively. Also, the subscripts 1 and 1/2 in the identity fields denote the central charges of the corresponding CFTs. When there is no confusion, these subscripts will be skipped.

Eq. (<ref>) suggests that the U(1)_4 CFT is obtained from condensing the boson b=(ψ,ψ̃) <cit.>. This result is consistent with the orbifold construction <cit.>. After the condensation, one of the unconfined particles is (σ, σ̃). We should state clearly that these two twist fields describe excitations (anyons) at different regions of the system, so it is meaningless to consider their fusion. In other words, the present situation is different from the case of a Pfaffian liquid, in which σ in the bulk and σ at the edge created from vacuum must fuse to I for conserving fermion parity. Importantly,

    (σ, σ̃)×(σ, σ̃)
    =(ψ, ψ̃)+(I, Ĩ)+(ψ, Ĩ)+(I, ψ̃).

The first two terms on the right hand side show that two orthogonal copies of vacuum exist, so (σ, σ̃) needs to split into two inequivalent types of anyons in the resulting U(1)_4 CFT <cit.>. We denote them as (σ, σ̃)_1 and (σ, σ̃)_2. Both of them have conformal spins 1/8, which are identified as the vertex operators 𝒱_1 and 𝒱_3 in the U(1)_4 CFT  (see Table <ref>). The fusion rules are consistent by imposing the conditions
(σ, σ̃)_1×(σ, σ̃)_1=
(σ, σ̃)_2×(σ, σ̃)_2
=𝒱_2
and (σ, σ̃)_1×(σ, σ̃)_2=𝒱_0. Following Ref. <cit.>, we interpret the above result as an incoming pseudospin-up quasiparticle transmutes into a neutral anyon σ̃ at the interface, and another anyon σ free to move in the Pfaffian liquid. To be more precise, the last anyon actually carries charge e/4, but we skip displaying its charge sector e^-iϕ_l/2 explicitly. The same conclusion holds for an incoming quasiparticle with pseudospin down. We illustrate the results in Fig. <ref>.





  §.§.§ Matching of Hilbert spaces and analogue of information scrambling



It is obvious that the total quantum dimension of (σ, σ̃)_1 and (σ, σ̃)_2 is two. It matches the two-dimensional Hilbert space spanned by the pseudospin degree of freedom of an Abelian charge-e/4 quasiparticle. This matching is guaranteed mathematically by the commutativity between fusion and restriction in anyon condensation <cit.>. Interestingly, the information of pseudospin is being stored nonlocally at the interface and in the interior of Pfaffian liquid. There is no local measurement that can distinguish between (σ, σ̃)_1 and (σ, σ̃)_2. Hence, it is impossible to recover the original information from any local measurement. This feature resembles a quantum information scrambling, which can be defined as the spreading of local information into many-body entanglement and correlation in the whole system <cit.>.

The situation becomes more interesting when we keep dragging more Abelian quasiparticles across the interface. Suppose N-1 charge-e/4 quasiparticles were already dragged. We assume the corresponding N-1 anyons σ̃ created at the interface are well separated from each other, so that no fusion occurs between them. We also pose the same assumption for the N-1 anyons created in the Pfaffian liquid. Consider dragging an additional charge-e/4 quasiparticle across the interface. This process increases both numbers of σ̃ and σ by one. As a result, there are N neutral anyons
σ̃ at the interface, and N non-Abelian anyons in the interior of the Pfaffian liquid. The dimension of the corresponding topological Hilbert space is then increased by a factor of two, which is consistent with the one bit of information carried by the additional Abelian quasiparticle from the Halperin-331 liquid. We illustrate the example of N=6 in Figs. <ref>(a) and <ref>(b). Now, we relax the confining potential, and allow the anyons to move and braid <cit.>. The braiding can further scramble the original information <cit.>. The N anyons at the interface are indistinguishable, so are those N anyons in the Pfaffian liquid. Meanwhile, the information carried by pseudospins of the original N Abelian quasiparticles is still preserved. Both Hilbert spaces for indistinguishable anyons at the interface and indistinguishable anyons in the Pfaffian liquid have dimensions 2^N/2. It is intriguing that the interface and the Pfaffian liquid store the same amount of information. This does not hold in the Pfaffian-NASS interface <cit.>.

The above discussion also suggests another important feature. In addition to being stored nonlocally, the original pseudospin information is actually “hidden" in the fusion channels of the non-Abelian anyons. Hence, the scrambled information is protected topologically and will not be destroyed by any local perturbation. This property is essential in topological quantum computation (TQC) <cit.>.



 §.§ Upper bound of information storage and holographic principle



Our previous discussion assumed that local anyons in the system can be well separated to prevent fusion. This assumption leads to a natural question. How much information can be stored nonlocally with the topological protection that has been described?

Recall that the minimum separation between two anyons is in the order of the magnetic length ℓ_B, so that they are well defined individually and do not fuse. From this, one may naively think that the maximum amount of information can be stored is N_A∼ A/πℓ_B^2, where A denotes the area of the Pfaffian liquid. This argument is valid if the information is carried solely by anyons in the Pfaffian liquid. However, this is not true in the present case. We have assumed both Pfaffian liquid and 331-Pfaffian interface were initially in the ground state with no excitations. As we discussed, the nonlocal storage of pseudospin information of the Abelian quasiparticles from the Halperin-331 liquid requires both anyons at the interface and in the Pfaffian liquid.

For an interface with a length (perimeter) L, it can only accommodate N_L∼ L/ℓ_B neutral anyons σ̃. Since the radius R of a circular quantum Hall droplet satisfies R≫ℓ_B, one has N_L≪ N_A. When the number of σ̃ gets close to or exceeds N_L, different σ̃ anyons start to fuse. The resulting particles will be either a fermion or a boson that can propagate back to the Halperin-331 liquid. More explicitly, one has

    (I,σ̃)×(I,σ̃)
    =(I,Ĩ)+(I,ψ̃).

For the first fusion outcome, the two σ̃ anyons at the interface can fuse to a neutral boson with its spin part described by 𝒱_0 [see Eq. (<ref>)]. This neutral boson can then split to a pair of quasihole and quasiparticle with opposite charges but the same pseudospin, and propagate in the 331 liquid. For the second fusion outcome, the neutral fermion can split to a pair of quasihole and quasiparticle with opposite pseudospins propagating in the 331 liquid. Consequently, some of the hidden information is released and being accessible by local probes. Therefore, the “black hole” is no longer completely black. Note that the released information is not protected topologically and can suffer from quantum decoherence. The discussion shows that the length of the interface sets an upper bound of storing information nonlocally and topologically via (σ,σ̃) pairs. Furthermore, the magnetic length ℓ_B=√(1/eB) (in the unit of h/2π=c=1) plays the role of Planck length in the present system. Here, B denotes the magnetic field.

The above observation actually resembles the argument from holographic principle in black holes. Based on this principle, the maximum amount of information can be stored in a black hole is not determined by its volume, but bounded by its area <cit.>. This is because the Bekenstein entropy of the black hole is proportional to its area <cit.>, which limits the number of degree of freedoms the black hole can have. In contrast to the quantum Hall interface, the black hole can always store and “hide" more information by increasing its area. Since the length of the quantum Hall interface is assumed to be fixed, the analogous (a weaker version of) holographic principle there implies that Hawking radiation in the form of Abelian quasiparticles and quasiholes will be released when the bound N_L∼ L/ℓ_B is reached. Roughly speaking, any additional incoming information is thus reflected by the interface (event horizon).



§ INFORMATION PARADOX IN 330-RR_4 INTERFACE


In the previous section, we demonstrated that an original Abelian anyon from the Halperin-331 QH liquid would transmute into a pair of anyons, when it crosses the interface and enters the non-Abelian Pfaffian QH liquid. More specifically, one of the resulting anyons is neutral and being created at the interface, and the remaining one is created in the Pfaffian liquid. By proximitizing different pairs of QH liquids, a wide variety of interfaces can be formed. The corresponding quasiparticle transmutation (if any) will depend on the properties of the interface. Hence, different ways of scrambling the original information may be realized in different QH interfaces. Recall that the 331-Pfaffian interface has a chiral central charge of c=2-3/2=1/2. This indicates that there must be a gapless chiral Majorana fermion mode on the interface <cit.>. As a result, some of the scrambled information can be stored at the interface. Meanwhile, an interface with a zero chiral central charge can be created by proximitizing a pair of different QH states with identical central charges for their edges. Interfaces of this kind have been studied extensively, and importantly, can be gapped or gapless. When both QH phases are Abelian, they are described by the K-matrix formalism <cit.>. This enables one to study the gapness of the interface through the concepts of Lagrangian subgroups <cit.> or the null vector criteria <cit.>. For an interface involving non-Abelian QH state(s), its gapness and the corresponding gapped phases can be explored by the anyon condensation approach <cit.>. A rigorous mathematical formalism was developed in <cit.>. We note that gapped interfaces between different non-Abelian topologically ordered states is still an ongoing research topic.

Motivated by the above discussion, we consider in this section the information paradox and the corresponding quasiparticle transmutation in the interface between the Abelian Halperin-330 state <cit.> and the non-Abelian Read-Rezayi state at level four <cit.>. Both states or phases may describe the fractional QH (FQH) state in a bilayer system at the total Landau-level filling factor ν=2/3. Whether these phases are favorable or not in a realistic sample depends on the actual microscopic details of the system. Here, we assume both states and the interface between them can be realized, and study the possible consequences. Previous theoretical work suggested that a continuous phase transition between the Halperin-330 state and the RR_4 state might be triggered by tuning the interlayer tunneling strength in a bilayer FQH system <cit.>. This transition and similar phase transitions in bilayer systems at other filling factors can be studied systematically through the anyon condensation approach <cit.>. In particular, the condensation of a specific type of anyon in the RR_4 state leads to the Halperin-330 state. Based on the “folding trick" <cit.>, it is expected that a gapped boundary between these two FQH states can form. Nevertheless, the precise gapping mechanism and the possibility of having different phases for the gapped interface have not been addressed. Suppose the interface can be gapped, the scrambled pseudospin information may be entirely stored by anyons in the non-Abelian RR_4 FQH liquid with no information stored at the interface. This feature is completely different from the one in the aforementioned 331-Pfaffian interface <cit.>.



 §.§ Review of Halperin-330 and Read-Rezayi states



The Halperin-330 state is a two-component Abelian topological order, which is characterized by the matrix <cit.>,

    K=[ 3 0; 0 3 ],

and the associated t=(1,1)^T. Clearly, one has ν=t^T K^-1t=2/3. In a more explicit form, the corresponding topological term for the Halperin-330 edge is

    L_0
    =-3/4π(∂_tϕ_↑∂_xϕ_↑
    +∂_tϕ_↓∂_xϕ_↓).

Here, the subscripts ↑, ↓ denote the layer or the pseudospin index. Roughly speaking, each layer of the Halperin-330 state is a simple Laughlin state at 
ν=1/3  <cit.>. Since ϕ_↑ and ϕ_↓ have the same chirality, the edge is maximally chiral and has a central charge c=2. 

The two most relevant vertex operators for electrons are exp(3iϕ_↑) and exp(3iϕ_↓). These two different operators create respectively an electron in the upper and the lower layer. We call them as electrons with pseudospin up and pseudospin down. Both electrons operators have scaling dimensions 3/2, indicating that they are indeed fermionic as required. The most fundamental anyon that the Halperin-330 state can host has charge e/3. Depending on its pseudospin, it is created by the operator exp(iϕ_↑) or exp(iϕ_↓). The OPE between any one of them and each of the electron operators is singlevalued. For example, one has the following OPEs:

    lim_z→ w
    e^3iϕ_↑(z)× e^iϕ_↑(w)   ∼ (z-w)e^4iϕ_↑(w),
    
    lim_z→ w
    e^3iϕ_↓(z)× e^iϕ_↑(w)   ∼ e^i[3ϕ_↓(w)+ϕ_↑(w)].

Notice that |α|=1 is the smallest possible nonzero value for the generic operator exp(iαϕ_↑) to have singlevalued OPEs with both electron operators. This fact verifies that the e/3 anyon is the smallest-charge anyon that can be host by the Halperin-330 state.



  §.§.§ Parafermion conformal field theory and Read-Rezayi states


Different from the Halperin-330 state, the Read-Rezayi states are a series of non-Abelian topological orders introduced from the CFT approach <cit.>. Each RR state consists of two different types of CFTs. First, it has a compactified U(1) holomorphic boson ϕ. On the edge of the system, ϕ corresponds to the gapless charge mode described by

    L_ϕ
    =-1/4πν∂_xϕ(∂_t+v∂_x)ϕ.

This mode fixes the charge density ρ(x)=∂_xϕ/2π and the quantized electrical Hall conductance σ_xy=ν e^2/h. The possible values of the filling factor ν depend on the second type of CFT in the RR state, which is the chiral ℤ_k parafermion CFT with the central charge <cit.>,

    c=2(k-1)/k+2.
 

The above parafermion CFT can be obtained from the SU(2)_k/U(1) coset construction <cit.>. Here, k∈ℕ is the level of the corresponding Wess-Zumino-Witten model. Following the notations in Ref. <cit.> (which are more convenient for the general discussion), each Virasoro primary field in the parafermion CFT is labeled as Φ^ℓ_m with ℓ+m≡ 0 (mod  2). The index ℓ=0,1,⋯, k. By imposing the field identification, Φ^ℓ_m=Φ^ℓ_m+2k=Φ^k-ℓ_m-k, the index m can be restricted to the range -ℓ<m≤ℓ, and ℓ>0. Hence, there are in total k(k+1)/2 Virasoro primary fields. Their scaling dimensions are given by

    h^ℓ_m=ℓ(ℓ+2)/4(k+2)-m^2/4k.

Since the CFT is chiral or holomorphic, the scaling dimension of the field is also its conformal spin. The fusion rule between different fields is

    Φ^ℓ_1_m_1×Φ^ℓ_2_m_2
    =∑^min(ℓ_1+ℓ_2, 2k-ℓ_1-ℓ_2)_ℓ=|ℓ_1-ℓ_2|Φ^ℓ_m_1+m_2.

When k≥ 2, the ℤ_k parafermion CFT is non-Abelian. A famous example is the Ising CFT at k=2, which has a non-Abelian anyon σ corresponding to the field Φ^1_1. The Ising CFT is relevant in the description of FQH state at ν=5/2 <cit.>. The parafermion CFTs and their related FQH states have attracted considerable amount of attention as the non-Abelian anyons there are useful in topological quantum computation <cit.>. 

An anyon (including the electron) in the RR state is created by the operator, 
ηexp(iωϕ). Here, η is a Virasoro primary field in the parafermion CFT. It is customary to choose η=Φ^k_k-2 for constructing the electron operator. The conformal spin for Φ^k_k-2e^iϕ/ν is

    h=k-1/k+1/2ν.

By requiring ν>0 and h being an half-integer, the possible filling factor is fixed at ν=k/(Mk+2), where M is a positive odd integer. For k=4 and M=1, the Read-Rezayi state may describe a FQH state of electrons at ν=2/3. This matches the filling factor of the Halperin-330 state. In particular, the Lagrangian density describing the chiral Bose mode ϕ_l on the RR_4 edge is

    L_ϕ_l=-3/8π∂_xϕ_l(-∂_t+v∂_x)ϕ_l.

Since we will eventually study the interface between two FQH liquids, we define the chirality of ϕ_l as opposite to the chiralities of the two Bose modes on the Halperin-330 edge. The subscript l indicates that ϕ_l is a left-moving mode. As a reminder, the complete edge theory for the RR_4 state also involves the neutral parafermionic sector (see the particular discussion in Ref. <cit.>). 

For later reference, the ten different Virasoro primary fields in the ℤ_4 CFT and their fusion rules are listed in Table <ref>. Note that we have switched to another set of notations (somehow more convenient for this particular discussion) for the primary fields. The identification with the notations in the previous discussion:

    I   =Φ^4_4, ψ_1=Φ^4_2, ψ_2=Φ^4_0, ψ_3=Φ^4_-2, 
    σ_+=Φ^3_3,
    
    σ_-   =Φ^1_1, ϵ=Φ^2_0, ρ=Φ^2_2, 
    χ_-=Φ^3_1, χ_+=Φ^3_-1.

After determining the electron operator, all other operators for quasiparticles can be deduced in a “brute-force" manner by requiring the OPE between ηexp(iωϕ) and the electron operator is single-valued. This restricts the possible values of ω in the vertex operator for each separate η as listed in Table <ref>. From the result, the charge of the corresponding quasiparticles 2ω/3 (in unit of e) and the whole spectrum of anyons that the Read-Rezayi state can host is determined. Notice that the quantum dimension of the anyon a is determined by the largest eigenvalue of the fusion matrix N_a with matrix elements (N_a)_bc=N^c_ab. Here, N^c_ab is defined from the fusion between the anyons a and b, and the resulting anyon c:  

    a× b=∑_c N^c_ab c.

The maximum eigenvalue is positive and nondegenerate, as guaranteed by the Perron-Frobenius theorem. For an Abelian anyon, the quantum dimension is one. Otherwise, the anyon is non-Abelian.







 §.§ Mathematical details of anyon condensation



Here, we discuss the interface between the Halperin-330 RR_4 states (abbreviated as 330-RR_4 interface) from the anyon condensation perspective. Readers who are uninterested in the mathematical details may skip this subsection. The corresponding physical picture for the fully gapped interface and mechanism of scrambling the pseudospin degree of freedom in the system are reviewed in Secs. <ref> and <ref>.

Since both edges of the QH states have the same central charge but with opposite chiralities at the interface, the interface has a zero chiral central charge. In principle, this kind of interface can be (but not guaranteed to be) fully gapped. For example, the edge of the ℤ_2 toric code (i.e., the boundary between it and the vacuum) remains gapless if the chosen boundary does not break the translational symmetry <cit.>. However, the edge in a generic situation (with the so-called smooth or rough boundary) is gapped due to the condensation of either one of the self-bosons, e or m on the edge <cit.>. The discussion suggests that there are two different possible gapped boundaries. When the gapped boundary is obtained by condensing e, then m is confined and becomes a boundary excitation, or vice versa. Can the 330-RR_4 interface be fully gapped? If the answer is affirmative, there will be no gapless excitations on the interface. Then, it may be possible to transmute the original information carried by certain types of the Abelian anyons completely into topological information stored by anyons in the non-Abelian RR_4 liquid.

In the following discussion, we first employ the anyon condensation approach to explore the gapness of the 330-RR_4 interface. We use 𝒜 and ℬ to denote the sets of anyons for the Halperin-330 state and RR_4 state, respectively. The possible phases of the interface are determined by the resulting phases from condensing different possible anyons (if it occurs) in 𝒜×ℬ̅. Here, ℬ̅ indicates the conjugation of ℬ. In the present case, we have

    𝒜   ={e^iαϕ_↑e^iβϕ_↓},
    
    ℬ̅   ={η e^iω_ηϕ_l}.

Here, α, β are integers, and η denotes the Virasoro primary field in the ℤ_4 parafermion CFT. For each η, the possible values of ω_η are given by the last row in Table <ref> with m∈ℤ. Since both electrons in the Halperin-330 state and RR_4 state are fermions that have trivial mutual statistics with every anyon in 𝒜×ℬ̅, 𝒜×ℬ̅ is a fermionic topological order. Specifically, we introduce the symbol Ψ_e=e^3iϕ_↑ to denote the electron with pseudospin up for the Halperin-330 state. It is a trivial fermion in 𝒜×ℬ̅.



  §.§.§ Determination of Lagrangian subset


The interface can be fully gapped only if there exists a Lagrangian subset ℒ⊂𝒜×ℬ̅, in which the condensed anyons are the only deconfined anyons in the condensed phase. To identify some possible ℒ, we follow the strategy in Ref. <cit.> by first condensing Abelian anyons. After that, we will analyze the resulting phase and check for the necessity of condensing more anyons in the system to obtain a Lagrangian subset. Finally, we discuss the underlying reason that gives rise to the anyon condensation. It is vital to clarify that one can actually condense non-Abelian anyons, but the strategy below greatly simplifies the analysis. When the Lagrangian subset ℒ only consists of Abelian bosons, then the explicit conditions for having a fully gapped interface are:


  * trivial monodromy for all a∈ℒ: M^aa̅_I=1;

  * trivial mutual statistics for all a, b∈ℒ: M^ab_a× b=1;

  * confinement for all a∉ℒ: there exists at least one b∈ℒ that has a non-trivial braiding phase with a.

The condition (i) ensures that a is a boson. If a is its own antiparticle (i.e., a=a̅ and a×a̅=I), then a should have an integer conformal spin. When a is not its own antiparticle, then it further requires that b=a× a also has an integer conformal spin <cit.>. For condition (ii), it implies that the anyon c=a× b also needs to be a boson, otherwise it needs to be condensed as well <cit.>. Note that one can also condense fermions (by fusing it with the trivial fermion) in a fermionic topological order <cit.>, but this turns out unnecessary in our discussion.



  §.§.§ Separation of charge and neutral sectors


From Table <ref>, it is clear that the four Abelian anyons in the ℤ_4 parafermion CFT are I, ψ_1, ψ_2, and ψ_3. To deduce the set of condensable Abelian bosons, one can in principle analyze all anyons with η={I, ψ_1, ψ_2, ψ_3} and generic values for α, β, and ω_η in the vertex operators. However, this general treatment turns out to be unphysical. As we will discuss in Sec. <ref>, the gapped interface originates from the gapping or localization of counterpropagating edge modes due to electron and quasiparticle tunneling process. Therefore, the charge modes and neutral modes should obtain expectation values independently. This suggests us to separate the charge and neutral sectors in the problem, which is achieved by introducing a new set of modes for the Halperin-330 edge:

    ϕ_r=ϕ_↑+ϕ_↓,
    
    ϕ_n=ϕ_↑-ϕ_↓.

They correspond to the overall charge mode and neutral spin mode for the Halperin-330 edge, respectively. Both modes are right-moving. Using the new set of modes, the topological term in Eq. (<ref>) becomes

    L_0=-3/8π(∂_tϕ_r∂_xϕ_r
    +∂_tϕ_n∂_xϕ_n).
 
The t vector becomes t=(1,0)^T, which verifies that ϕ_n is a neutral mode. Furthermore, we have

    𝒜×ℬ̅
    ={e^iQϕ_r e^iSϕ_n}×{η e^iω_ηϕ_l}.

Here, Q=(α+β)/2 and S=(α-β)/2. Since α∈ℤ and β∈ℤ, both Q and S can only take half-integer or integer values. The corresponding anyon in 𝒜 has charge 2Q/3 (in the unit of e). Furthermore, the conformal spins for the two vertex operators exp(iQϕ_r) and 
exp(iSϕ_n) are

    s_Q=h_Q=Q^2/3 , 
    s_S=h_S=S^2/3.

Each of them will change by an integer value whenever Q or S is changed by 3ℤ. Physically, this corresponds to the fusion between the anyon with a trivial boson in the Halperin-330 state. Thus, we can identify a pair of anyons with 
(Q, S) and (Q+3ℤ, S+3ℤ). More explicitly, one can rescale the fields φ_r=ϕ_r/2 and φ_n=ϕ_n/2 and rewrite Eq. (<ref>) as

    L_0=-6/4π(∂_tφ_r∂_xφ_r
    +∂_tφ_n∂_xφ_n).
 
The possible vertex operators take the form exp(ipφ_r) and 
exp(iqφ_n), where both p and q can only take integer values now. Thus, both φ_r and φ_n (and hence the original fields, ϕ_r and ϕ_n) are compactified bosons in the U(1)_6 CFT which has exp(6iφ) as the trivial boson. For later discussion, the six primary fields involving the neutral spin mode in the U(1)_6 CFT  are summarized in Table <ref>. Moreover, a pair of anyons with (Q,S) and [Q+3(ℤ+1/2), S+3(ℤ+1/2)] differ by the fusion of an odd multiples of electrons in the Halperin-330 state. 





  §.§.§ Condensation of Abelian bosons


Following the above discussion, we look for possible values of Q, ω_η, and S such that both anyons with operators exp(iQϕ_r)exp(iω_ηϕ_l) and ηexp(iSϕ_n) are bosonic. The conformal spin for exp(iQϕ_r)exp(iω_ηϕ_l) is

    s(Q,ω_η)=Q^2-ω_η^2/3.

Therefore, a possible solution for s(Q,ω_η)=0 (or alternatively, the null vector for the K matrix describing the charge sector) is (Q,ω_η)=(1,1). Hence, we first condense the following four bosons:

    b_0
       =e^im_0(ϕ_r+ϕ_l),
    
    
    b_1
       =ψ_1 e^3iϕ_n/2e^i(m_1+1/2)(ϕ_r+ϕ_l),
    
    
    b_2
       =ψ_2 e^im_2(ϕ_r+ϕ_l),
    
    
    b_3
       =ψ_3 e^3iϕ_n/2e^i(m_3+1/2)(ϕ_r+ϕ_l).

Here, all m_i∈ℤ. It is easy to check that all of the above bosons satisfy the conditions (i) and (ii) in the previous discussion. This verifies that all of them can be condensed simultaneously <cit.>. It is worthwhile for clarifying that being a boson does not mean that it must be condensable. For example, the condensation of non-Abelian bosons can be obstructed by a no-go theorem <cit.>. Also, a self-dual boson having Frobenius-Schur indicator ϰ=-1 is not condensable <cit.>. Nevertheless, none of these limitations applies here.

Since a deconfined anyon must have trivial mutual statistics with all the condensed bosons, we first put m_2=0 to eliminate a large set of possibilities. The corresponding monodromy is given by

    M^ηψ_2_η×ψ_2
    =exp[-(2π i)(h_η×ψ_2-h_η-h_ψ_2)].

Hence, all anyons with η={σ_+, σ_-, χ_+, χ_-} are confined. This result was used by Barkeshli and Wen in showing that the phase transition from the RR_4 state to the Halperin-330 state could be described by the anyon condensation of ψ_2 <cit.>. By setting m_2≠ 0, the monodromy between Y=ηexp(iSϕ_n)exp(iQϕ_r)exp(iω_ηϕ_l) and b_2 is given by

    M^Yb_2_Y× b_2
    =exp[4m_2 π i/3(Q-ω_η)] , 
    m_2∈ℤ.

Here, η={I, ψ_1, ψ_2, ψ_3, ϵ, ρ}. Thus, the set of possible deconfined anyons is further reduced to
e^iω_η(ϕ_l+ϕ_r)
e^3iγϕ_r/2
e^iSϕ_n×{I, ψ_1, ψ_2, ψ_3, ϵ, ρ}, with γ∈ℤ. This set of anyons have trivial mutual statistics with b_0 also. By requiring them to have trivial mutual statistics with b_1 and b_3, it further reduces the possible set of deconfined anyons to four apparently different classes. For the first class, we have

    𝒯_1
       =e^im(ϕ_l+ϕ_r)
    e^3i(p+1/2)ϕ_r
    e^i(q+1/2)ϕ_n×{I, ψ_2, ϵ}
       ∼ e^im(ϕ_l+ϕ_r)[e^3i(ϕ_r+ϕ_n)/2]
    {𝒱_0, 𝒱_2, 𝒱_4}×{I, ψ_2, ϵ}
       ∼ e^im(ϕ_l+ϕ_r)Ψ_e
    {𝒱_0, 𝒱_2, 𝒱_4}×{I, ψ_2, ϵ}
       ∼Ψ_e
    {𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}.

All m, p, and q in the above calculation are integers. It is recalled that 
Ψ_e=exp(3iϕ_↑)=exp[3i(ϕ_r+ϕ_n)/2] is the electron in the Halperin-330 state. In the second line, we used the fact that ϕ_n is a compactified boson in the U(1)_6 CFT, and labeled the corresponding vertex operators by the symbols in Table <ref>. Moreover, we used the fact that exp(3ipϕ_r) is a trivial boson to make the identification denoted by ∼. In the last line, the identification is made by fusing the set of anyons in the third line with the condensed boson b_2 with m_2=-m. 

Using similar procedures, we determine the second class of deconfined anyons as

    𝒯_2
       =e^im(ϕ_l+ϕ_r)
    e^3ipϕ_r
    e^iqϕ_n×{I, ψ_2, ϵ}
       ∼{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}.

By setting η={ψ_1, ψ_3, ρ}, we have the remaining two classes of deconfined anyons,

    𝒯_3
       =e^i(m+1/2)(ϕ_l+ϕ_r)
    e^3ipϕ_r
    e^i(q+1/2)ϕ_n×{ψ_1, ψ_3, ρ}
       ∼ e ^i(ϕ_l+ϕ_r)/2{𝒱_1, 𝒱_3, 𝒱_5}×{ψ_1, ψ_3, ρ},

and

    𝒯_4
       =e^i(m+1/2)(ϕ_l+ϕ_r)
    e^3i(p+1/2)ϕ_r
    e^iqϕ_n×{ψ_1, ψ_3, ρ}
       ∼Ψ_e  e ^i(ϕ_l+ϕ_r)/2{𝒱_1, 𝒱_3, 𝒱_5}×{ψ_1, ψ_3, ρ}.

Meanwhile, they are actually equivalent to 𝒯_2 and 𝒯_1, respectively. This is observed by fusing 𝒯_3 and 𝒯_4 with either b_1 or b_3 to obtain

    𝒯_3    ∼{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}∼𝒯_2,
    
    𝒯_4    ∼Ψ_e{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}∼𝒯_1.

Therefore, the above four classes of deconfined anyons can be combined into

    𝒯   ={1, Ψ_e}×{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}
       ={1, Ψ_e}×𝒯_B.

Since 𝒱_2 and 𝒱_4 cannot be obtained from any fusion between the bosons in the set {b_0, b_1, b_2, b_3}, this set is not a Lagrangian subset. In the second line of Eq. (<ref>), 
𝒯_B is a bosonic order with 𝒱_0 I being the trivial boson. The separable form in Eq. (<ref>) is guaranteed for any Abelian fermionic order <cit.>. The Abelianity of 𝒯 becomes transparent after splitting the non-Abelian anyons in 𝒯_B. Notice that (Ψ_e)^2=exp(3iϕ_r)exp(3iϕ_n)∼ 1. 



  §.§.§ Splitting of non-Abelian anyons and the ℤ_3 toric code


Since some of the anyons in the original phase have been identified as the trivial vacuum after the anyon condensation, the unconfined non-Abelian anyons in 𝒯 may need to split. Consider the fusion:

    ϵ×ϵ
    =I+ ψ_2+ϵ∼ 2I + ϵ.

Due to the identification ψ_2∼ I, the vacuum appears twice. Consequently,
ϵ needs to split into two Abelian anyons, ϵ=ϵ_1+ϵ_2. This matches the quantum dimension as 2=1+1. Furthermore, the splitting and the fusion rules are consistent only if

    ϵ_1×ϵ_1=ϵ_2,
    
       ϵ_2×ϵ_2=ϵ_1,
    
       ϵ_1×ϵ_2
    =ϵ_2×ϵ_1=I.

Then, the fusion rule ρ×ρ=I+ψ_2+ϵ implies that the non-Abelian anyon ρ also needs to split, ρ=ρ_1+ρ_2. After the splitting, a possible (but not unique) consistent set of fusion rules are

    ρ_1×ρ_1=ϵ_1 , ρ_2×ρ_2=ϵ_2 , ρ_1×ρ_2=I,
    
       ϵ_1×ψ_1=ρ_2
     , ϵ_2×ψ_1=ρ_1,
    
       ρ_1×ψ_1=ϵ_2
     , ρ_2×ψ_1=ϵ_1,
    
       ρ_1×ϵ_1=ψ_1,
     , ρ_1×ϵ_2=ρ_2,
    
       ρ_2×ϵ_1=ρ_1,
     , ρ_2×ϵ_2=ψ_3.

Note that ψ_3∼ψ_1 in the condensed phase since they differ by a fusion with ψ_2 (i.e, b_2 with m_2=0). Therefore, the fusion rules involving ψ_3 are the same as those involving ψ_1. By making the change(s), ϵ_1↔ϵ_2, or/and ρ_1↔ρ_2, one can still obtain a consistent set of fusion rules. This is reasonable as it is impossible to uniquely define or fix the anyons resulting from the splitting. On the other hand, they must be inequivalent. Without loss of generality, we will stick with the set of fusion rules in Eqs. (<ref>) – (<ref>) in the following discussion.

The above discussion shows that 𝒯_B actually consists of nine different Abelian anyons, so 𝒯 is an Abelian fermionic topological order. Then, what is the topological order 𝒯? By labeling the nine anyons in 𝒯_B as shown in Table <ref>, it is observed that these anyons take the form of e^p m^q, where p, q∈ℤ. Moreover, one has e^3=m^3=𝕀. Hence,𝒯_B is the ℤ_3 toric code. This topological order is a generalization of the more famous ℤ_2 toric code that only has four anyons, {1, e, m, f≡ em} <cit.>. Since it is the topological order built from (mathematically, the modular tensor category constructed over) the quantum double model with the finite group ℤ_3 <cit.>, the ℤ_3 toric code is also denoted as 𝔇(ℤ_3) <cit.>. It was suggested that this special topological order can be realized by proximitizing a bilayer system of electrons and holes in separate Laughlin states with respective filling factors ± 1/3 to a superconductor <cit.>. It was also pointed out that the more general ℤ_p toric code appears as the symmetry-enriched neutral sector of non-diagonal quantum Hall states <cit.>. Here, we have shown that the interface between the Halperin-330 and RR_4 states with suitable anyon condensation also leads to the ℤ_3 toric code. The physical origin that triggers such an anyon condensation will be discussed in Sec. <ref>. On the application side, it was suggested that the ℤ_3 toric code could be used in implementing universal topological quantum computation <cit.>. To summarize, the condensation of b_0, b_1, b_2, and b_3 leads to the condensed phase with deconfined anyons,

    𝒯={1, Ψ_e}×𝔇(ℤ_3).

Note that 𝒯 does not describe a fully gapped interface. This is because 
𝔇(ℤ_3) is not a topologically trivial order. Alternatively, the edge of 𝔇(ℤ_3) is gappable but the edge remains gapless unless a further anyon condensation occurs, which we are going to discuss below.






  §.§.§ Two different phases of gapped interfaces


As we have shown, the set {b_0, b_1, b_2, b_3} is not a Lagrangian subset, and the condensation of this set of bosons does not lead to a fully gapped interface. On the other hand, a fully gapped interface can be achieved by further condensing some bosonic particles in 𝔇(ℤ_3). By recycling the results in Refs. <cit.>, one can immediately conclude that there are two different types of fully gapped interfaces between the Halperin-330 FQH state and the RR_4 FQH state.

The first possible kind of fully gapped interface is obtained by a further condensation of e and e^2 in 𝔇(ℤ_3). It is straightforward to verify that both of them satisfy conditions (i) and (ii) in the previous discussion. This implies that they can be condensed simultaneously. Moreover, their condensation leads to the confinement of the remaining anyons (except the identity 𝕀) in 
𝔇(ℤ_3). Therefore, we determine the first Lagrangian subgroup <cit.> for 𝒜×ℬ̅ as

    ℒ_e
    ={b_0, b_1, b_2, b_3}×{𝕀, e, e^2}.

The corresponding fully gapped interface is usually known as the e-boundary. Notice that the symbol × actually means picking an element from each set on the right hand side of Eq (<ref>) and then fuse them (see Eq. (<ref>) for example). The product structure of ℒ_e ensures that it is a maximal set of condensable bosons.

Another possible type of fully gapped interface is obtained by condensing m and m^2 in 𝔇(ℤ_3) instead. This leads to the second Lagrangian subgroup,

    ℒ_m
    ={b_0, b_1, b_2, b_3}×{𝕀, m, m^2}.

The corresponding fully gapped interface is known as the m-boundary. For both e- and m-boundaries, the remaining deconfined anyons outside the Lagrangian subgroups are ℱ_0={1, Ψ_e}. This is the trivial fermionic topological order, which indicates that the condensation of anyons in ℒ_e or ℒ_m leads to a fully gapped interface <cit.>.



 §.§ Physical picture of the gapped interface
  


While anyon condensation has provided a systematic and mathematical approach in studying the gapness of the interface, it will be also desirable to understand the gapping of the interface in a more physical picture. We claim that the anyon condensation may originate from the electron and quasiparticle tunneling processes at the interface in two different steps.

At the beginning, the Halperin-330 and Read Rezayi states are two topologically distinct phases. Therefore, only electrons can tunnel across the two different FQH liquids. In general, the electron tunneling process is described by the following Lagrangian density,

    L_el, tun
    =    ξ_1,a(x)(ψ_1 e^3iϕ_l/2e^3iϕ_↑)^a
    
       +ξ_2,a(x)(ψ_1 e^3iϕ_l/2e^3iϕ_↓)^a
    +H.c.
    
    =    ξ_1,a(x)[ψ_1 e^3iϕ_n/2 e^3i(ϕ_l+ϕ_r)/2]^a
    
       +ξ_2,a(x)[ψ_1 e^-3iϕ_n/2 e^3i(ϕ_l+ϕ_r)/2]^a
    +H.c.

Note that H.c. stands for the Hermitian conjugation. We include the exponent a>1 to describe multi-electron tunneling processes. As the electron operators ψ_1 e^3iϕ_l/2 and e^3iϕ_σ (where σ=↑, ↓) enter the same number of times, L_el, tun conserves the total electric charge. Furthermore, the tunneling amplitudes ξ_1,a(x) and ξ_2,a(x) are random functions in x. This is because the electron tunneling generally does not conserve momentum, and disorder needs to be involved. Due to the random nature of the tunneling, some of the processes are actually irrelevant in the renormalization group sense <cit.>. In this situation, we will need to assume the tunneling strength W_a, defined as ξ_a(x)ξ_a(x')=W_aδ(x-x'), is sufficiently large so that the charge modes 3a[ϕ_l(x)+ϕ_r(x)]/2 can still be pinned. Then, this combination of counterpropagating charge modes obtains a nonzero expectation value, and indicates the localization (analogous to mode gapping in nonrandom tunneling) of charge modes. For the neutral sector, it is expceted that the combination ψ_1 e^± 3iϕ_n/2 will also be gapped by ℒ_el, tun in the strong coupling regime. By defining (ψ_1)^2∼ψ_2, (ψ_1)^3∼ψ_3, and (ψ_1)^4∼ I, the tunneling processes described by ℒ_el, tun lead to the condensation of b_0, b_1, b_2, and b_3 in Eqs. (<ref>) – (<ref>). 

After the above condensation or charge-mode localization, the interface remains gapless. Meanwhile, quasiparticles with charges e/3 and 2e/3 can tunnel across this “new" gapless interface. Let us specifically consider the tunneling of charge e/3 anyon with the topological sector ρ in the Read-Rezayi liquid. This process can be described by the following Lagrangian density,

    L_e/3, tun
    
    =    ζ_1(x)(ρ e^iϕ_l/2e^iϕ_↑)
    +ζ_2(x)(ρ e^iϕ_l/2e^iϕ_↓)
    +H.c.
    
    =    ζ_1(x)[ρ𝒱_1 e^i(ϕ_l+ϕ_r)/2]
    +ζ_2(x)[ρ𝒱_5 e^i(ϕ_l+ϕ_r)/2]
    +H.c.

Since the charge modes have been localized, we focus on the neutral sector. Now, both combinations ρ𝒱_1 and ρ𝒱_5 have zero conformal spins (see Tables <ref> and <ref> for reference), which indicates that they are bosonic. In principle, they may be condensed or gapped as well. Consider the set of anyons generated from any fusion between ρ𝒱_1 and 
ρ𝒱_5, we have

    {ρ𝒱_1, ρ𝒱_5}×{ρ𝒱_1, ρ𝒱_5}
    ={I, ψ_2, ϵ}×{𝒱_0, 𝒱_2, 𝒱_4}

A further fusion with {ρ𝒱_1, ρ𝒱_5} gives

    {ψ_1, ψ_3, ρ}×{𝒱_1, 𝒱_3, 𝒱_5}.

Hence, the fusion between ρ𝒱_1 and ρ𝒱_5 generates the list of anyons (more precisely, their neutral sectors) in ℒ_e and ℒ_m. In this sense, the condensation described by the Lagrangian subgroup 
ℒ_e or ℒ_m may be understood as originating from the gapping of modes due to e/3 quasiparticle tunneling at the interface. However, this is only possible if we can treat the whole system as a single topological phase. Otherwise, only electrons can tunnel across two topologically distinct phases (for example, between a FQH liquid and a normal metal) <cit.>.



 §.§ Transmutation of pseudospin information



Previously, we found that it is possible (at least in principle) to form a gapped interface between the Halperin-330 state and the Read-Rezayi state at level four. Although both states may describe the FQH state in a bilayer system at total filling factor 2/3, they host different sets of anyons. Specifically, the Halperin-330 state is a two-component state, in which the anyons possess the layer or pseudospin degree of freedom that is absent in the RR_4 state. Meanwhile, the RR_4 state supports non-Abelian anyons that have more complicated fusion rules than the usual Abelian quasiparticles. This observation leads to a natural question: What happens when an Abelian quasiparticle from the Halperin-330 liquid crosses the gapped interface and enters the non-Abelian RR_4 FQH liquid? In the opposite direction, what happens when we drag a non-Abelian quasiparticle from the RR_4 liquid to the Halperin-330 liquid? The answers to these questions lead to the ideas of topological quantum information scrambling and Andreev-like reflection of non-Abelian anyons. 

An anyon a originally in the topological phase 𝒜 can pass through the interface (described by the condensed phase of 𝒜×ℬ̅) and transmutes into an anyon b in ℬ if and only if ab is a deconfined anyon in the condensed phase. This idea was introduced to study the transmutation between anyons in the interface between the Pfaffian and non-Abelian spin-singlet FQH states <cit.>. Now, we employ the same kind of argument to study the transmutation of Abelian anyons in the Halperin-330 FQH liquid when they cross the interface and enter the RR_4 liquid. 



  §.§.§ Transmutation of Abelian charge e/3 anyon


First, an Abelian charge e/3 anyon in the Halperin-330 state can have pseudospin up or pseudospin down. We denote them as (e/3, ↑) and (e/3, ↓), respectively. Their associated CFT operators are

    (e/3, ↑)≡exp(iϕ_↑)
    =𝒱_1exp(iϕ_r/2),
    
    
    (e/3, ↓)≡exp(iϕ_↓)
    =𝒱_5exp(iϕ_r/2).

When they cross the fully gapped 330-RR_4 interface, their transmutation depends on the type of the gapped interface (i.e., whether it is an e- or m-boundary). 

Suppose the fully gapped interface is described by the e-boundary. Then, the deconfined anyons are those in ℒ_e in Eq. (<ref>). In particular, we have the following pair of deconfined anyons:

    e^2× b_1
       = [ρ_1 exp(iϕ_l/2)] [𝒱_1exp(iϕ_r/2)],
    
    
    e× b_1
       = [ρ_2 exp(iϕ_l/2)] [𝒱_5exp(iϕ_r/2)].

These are the only two deconfined anyons that involve (e/3,↑) and (e/3,↓). Since e^2× b_1 and e× b_1 are condensed bosons in 
ℒ_e, they are identified as the vacuum sector in the condensed phase. Based on the interpretation in Ref. <cit.>, (e/3, ↑) and (e/3, ↓) will transmute into ρ_1 exp(iϕ_l/2) and ρ_2 exp(iϕ_l/2) in the RR_4 FQH state, respectively. It is important to clarify that these transmuted anyons are now in ℬ, but not ℬ̅. Since ρ_1≠ρ_2, the resulting Abelian anyons are different.



Consider the case when the fully gapped interface is described by the m-boundary that has the Lagrangian subset ℒ_m in Eq. (<ref>). Now, we have the following pair of deconfined anyons:

    m× b_1
       = [ρ_2 exp(iϕ_l/2)] [𝒱_1exp(iϕ_r/2)],
    
    
    m^2× b_1
       = [ρ_1 exp(iϕ_l/2)] [𝒱_5exp(iϕ_r/2)].

Thus, (e/3, ↑) and (e/3, ↓) will transmute respectively into 
ρ_2 exp(iϕ_l/2) and ρ_1 exp(iϕ_l/2). Notice that the result is different from the one in the e-boundary. We summarize the above results in Fig. <ref>.





  §.§.§ Transmutation of Abelian charge 2e/3 anyon


Similarly, there will be a transmutation of the pseudospin information carried by a charge 2e/3 anyon when it crosses the interface and enters the RR_4 FQH liquid. There are three different pseudospin states for the charge 2e/3 anyon in the Halperin-330 FQH liquid. They are spin up, spin zero, and spin down. Physically, they can be viewed as the combination of two spin-up, one spin-up and one spin-down, and two spin-down e/3 anyons. The corresponding CFT operators are

    (2e/3, ↑)
       ≡exp(2iϕ_↑)
    =𝒱_2exp(iϕ_r),
    
    
    (2e/3, 0)
       ≡exp[i(ϕ_↑+ϕ_↓)]
    =𝒱_0exp(iϕ_r).
    
    
    (2e/3, ↓)
       ≡exp(2iϕ_↓)
    =𝒱_4exp(iϕ_r).


Following the previous discussion, we first discuss their transmutation when the fully gapped interface is described by the e-boundary. In this case, one has the following three deconfined anyons,

    e× b_0
       = [ϵ_1 exp(iϕ_l)] [𝒱_2exp(iϕ_r)],
    
    𝕀× b_0
       = [I exp(iϕ_l)] [𝒱_0exp(iϕ_r)],
    
    
    e^2× b_0
       = [ϵ_2 exp(iϕ_l)] [𝒱_4exp(iϕ_r)],

Thus, (2e/3, ↑) transmutes into ϵ_1 exp(iϕ_l); (2e/3, 0) becomes I exp(iϕ_l); (2e/3, ↓) becomes ϵ_2 exp(iϕ_l). For the case of having the m-boundary, one has

    m^2× b_0
       = [ϵ_2 exp(iϕ_l)] [𝒱_2exp(iϕ_r)],
    
    
    m× b_0
       = [ϵ_1 exp(iϕ_l)] [𝒱_4exp(iϕ_r)].

In this scenario, (2e/3, ↑) transmutes into ϵ_2 exp(iϕ_l), whereas (2e/3, ↓) becomes ϵ_1 exp(iϕ_l). The results are illustrated in Fig. <ref>.



The original pseudospin information of a charge e/3 or a charge 2e/3 anyon from the Halperin-330 state is entirely encoded in the neutral spin mode ϕ_n. Clearly, this is a local information which can be accessed via local measurement. More specifically, we know in which layer the Abelian anyon was created. Depending on both the original pseudospin state and the boundary type of the fully gapped interface, the pseudospin information will be transmuted into ρ_1 or ρ_2 (for the charge e/3 anyon), or ϵ_1, ϵ_2, or I (for the charge 2e/3 anyon). Although one knows that the split anyons, say ρ_1 and ρ_2, are different, it is impossible to distinguish between them via any local measurement. In fact, their inequivalence is only manifested topologically in their braiding and fusion rules with other anyons (including themselves) as summarized in Eqs. (<ref>) – (<ref>). In this sense, we claim that the original local pseudospin information is completely scrambled into a form of nonlocal information being stored and protected topologically by the anyons in the RR_4 FQH liquid. No information is stored on the interface. It is reasonable as the interface is fully gapped, and does not support any low-energy gapless excitations there. This feature is different from the situation of having a gapless QH interface. As a concrete example, our recent work <cit.> showed that the pseudospin information for an Abelian charge e/4 anyon from the Halperin-331 state will be scrambled and stored nonlocally by a pair of vortices (one on the interface, and another one in the Pfaffian FQH liquid) when the Abelian anyon crosses the gapless 331-Pfaffian interface. The comparison here demonstrates the dependence of the scrambling mechanism on the gapness of the QH interface.



 §.§ Some remarks


One may realize that the transmuted particle from the charge 2e/3 anyon can be obtained directly by fusing a pair of transmuted particles from the charge e/3 anyons. This is not a coincidence. It is guaranteed from the commutativity between restriction and fusion in anyon condensation <cit.>. This important property also ensures that the total quantum dimension of the split anyons is identical to the quantum dimension of the original non-Abelian anyon. Moreover, the total conformal spin for the transmuted anyon(s) matches the one for the original anyon. In the present case, we have a fully gapepd interface, and the vertex operators describing the charge sector of the deconfined anyons always have zero conformal spins. Thus, the conformal spin for the parafermionic field η from the ℤ_4 CFT must match with the conformal spin for the vertex operator describing the neutral spin mode ϕ_n. This holds in all our results, and further highlights the advantage of separating the charge and neutral sectors as we did in our analysis.

In principle, one can drag more quasiparticles from the Halperin-330 liquid into the RR_4 liquid. Suppose the resulting anyons are well separated from each other (i.e., with a separation much larger than the magnetic length). Then, the braiding between the transmuted anyons will lead to a further scrambling of the original pseudospin information <cit.>. In order to recover the original pseudospin information, one basically needs to “pull" all the anyons back to the Halperin-330 liquid. However, the recovered information will be in a highly entangled form, which again resembles the definition of quantum information scrambling.



§ SIMULATION OF BLACK HOLE EVAPORATION
 

In order to resolve the information paradox in our model, it is necessary to mimic a black hole evaporation in the quantum Hall interface. To simplify our discussion, we only consider the 331-Pfaffian interface here. As we will show, the process recovers the original information carried by the pseudospin degree of freedom naturally. To simplify the discussion, we assume only charge-e/4 Abelian quasiparticles were dragged across the interface before the “evaporation". In general, one can also drag quasiholes and quasiparticles with other charges. Under the above assumption, we argued in Sec. <ref> that neutral anyons σ̃ and non-Abelian charge-e/4 quasiparticles carrying σ are created. This is illustrated in Figs. <ref>(a) and <ref> (b).



The “black hole evaporation" is simulated by shrinking the Pfaffian region. Experimentally, it may be achieved by reducing the interlayer tunneling in the bilayer system <cit.>. When a non-Abelian quasiparticle reaches the shrinking interface, it is released back to the Halperin-331 liquid. This process plays the role of Hawking radiation in the present setup. There are two different mechanisms for the conversion from a non-Abelian quasiparticle into an Abelian quasiparticle. First, the former may encounter an existing σ̃ at the interface. In this case, they recombine and transmute back to an Abelian quasiparticle [see Eqs. (<ref>) and (<ref>)]. This special scenario is shown in Fig. <ref>(c). In order for this recombination to occur, it requires a highly delicate control of the shrinking process. Thus, it is unlikely to recombine all the existing σ̃ and σ (with charge sector skipped) in this way. On the other hand, it is likely that a non-Abelian anyon reaches the interface at a position with no neutral anyon σ̃. Being already outside the Pfaffian liquid, the non-Abelian anyon still needs to transmute into an Abelian quasiparticle. In this case, a pair of σ̃ needs to be created at the interface. One of them combines with the non-Abelian anyon to covert into an Abelian quasiparticle in the Halperin-331 liquid. The remaining one is left at the interface. This mechanism is shown in Fig. <ref>(d). Since the additional pair of σ̃ are created from the vacuum, they must have their fusion channel in the trivial topological sector. Hence, they do not carry additional information. All above processes are unitary, so information should be preserved.



 §.§ Recovery of Page curve


Now, we show that the above “black hole evaporation" recovers all original information and satisfies the Page curve. Here, the first subsystem consists of anyons remaining in the Pfaffian liquid and the interface. Another subsystem consists of Abelian quasiparticles in the Halperin-331 liquid. For simplicity, we call these two subsystems as (I) and (II), respectively. Since we treat the Pfaffian liquid as the black hole and the interface as an event horizon, reduced density matrices at different stages are obtained by partial tracing out (I). At the beginning of the evaporation, (II) is in a vacuum state with no quasiparticles, so its entropy is zero. When the Pfaffian liquid starts to shrink, the entropy of Abelian quasiparticles originating from their entanglement with (I) increases. However, the increase in entanglement entropy will not continue forever. By keep shrinking the Pfaffian region, the number of non-Abelian anyons and the dimension of the corresponding Hilbert space decrease. Hence, the dimensions of Hilbert spaces of (I) and (II) will become comparable and eventually equal to each other. The entanglement entropy reaches its maximum at this moment <cit.>, which is known as the Page time. The Page time depends on the actual shrinking process. After passing the Page time, the entanglement entropy starts to decrease. 

In the most idealistic (yet most unlikely) situation which one can recombine all N non-Abelian anyons with the originally existing σ̃ (exist before the evaporation) at the interface, the Page time occurs when N/2 quasiparticles are released to the 331 region. This feature does not hold in a generic situation. One can actually deduce the average entanglement entropy of (II) in the most idealistic case. We assume the initial state of the total system (before shrinking the Pfaffian region) is a random pure state |Ψ⟩ in the 2^N-dimensional Hilbert space. The entanglement entropy is averaged with respect to the unitary invariant Haar measure on the space of unitary vectors |Ψ⟩ in the 2^N-dimensional Hilbert space <cit.>. Suppose j non-Abelian anyons have been dragged out from the Pfaffian region and transmuted back to Abelian quasiparticles in the Halperin-331 liquid. The corresponding Hilbert space dimensions of (I) and (II) are given by n=2^N-j and m=2^j, respectively. When m≤ n, the conjecture by Page (later proved by Sen <cit.>) suggests that the average entanglement entropy of (II) takes the form <cit.>,

    ⟨ S_ (II)⟩≡ S_m,n
    =(∑_k=n+1^mn1/k)-m-1/2n.

For m>n, one obtains ⟨ S_ (II)⟩ by interchanging m and n in Eq. (<ref>). By plotting ⟨ S_ (II)⟩ versus lnm, one concludes that it is identical to the one in Fig. 1 of Ref. <cit.>.

From our previous discussion, it is very likely that the number of σ̃ anyons increases during the “black hole evaporation". This leads to two consequences. First, it is possible that all non-Abelian anyons in the Pfaffian liquid have been released to the 331 liquid, but some σ̃ anyons still remain at the interface. These anyons are entangled with the Abelian anyons in the Halperin-331 region, so the entanglement entropy is still nonzero. Second, the bound N_L∼ L/ℓ_B can be satisfied easily during the shrinking process. The discussion in Sec. <ref> showed that pairs of Abelian quasiparticles and quasiholes with opposite charges will be released to the 331 region. After eliminating the Pfaffian region completely, the total number of anyons in the 331 region needs not be equal to the number in the original configuration (before sending the anyons in the Pfaffian liquid). Only in the most idealistic situation that we mentioned previously, these two numbers are equal as shown in Fig. <ref>(a) and (e). In general, the final state of the system will have a superposition of different total numbers of anyons in the 331 liquid. It is reasonable since the total charge in the system is still conserved. This idea is illustrated in Fig. <ref>(f). In particular, Fig. <ref>(f) denotes a superposition state of 6+M charge-e/4 quasiparticles and M charge--e/4 quasiholes, where M is a non-negative integer. This kind of superposition state is actually a closer analogue of the actual Hawking radiation emitted from a black hole, which consists of different types of particles or excitations.

Independent of the actual shrinking process, the system must return to a pure state when the Pfaffian region is eliminated completely. Then, the entanglement entropy goes back to zero and resembles the Page curve. The original pseudospin information is recovered but in a highly entangled form. Thus, the paradox in our model is resolved. Our above discussion suggests that the Page curve in the present system should be more complicated than the one in Fig. 1 of Ref. <cit.>.



§ SUMMARY AND DISCUSSION


To conclude, we have identified and resolved an “information paradox" in the 331-Pfaffian and the 330-RR_4 quantum Hall interface respectively <cit.>. In both cases, the paradox originates from an apparent inability to recover the original pseudospin information carried by Abelian quasiparticles after they cross the interface and enter the non-Abelian single-component QH liquid. 

Employing the technique of anyon condensation, we found that each incoming Abelian anyon needs to be transmuted into new types of anyons when the former crosses the interface. Moreover, the mechanism of transmutation and its corresponding scrambling of pseudospin information depend on the nature of quantum Hall interfaces. For the 331-Pfaffian interface, an incoming Abelian anyon is transmuted into a pair of non-Abelian anyons. One of them is created in the Pfaffian liquid, whereas the other is created on the interface. This is expected because the interface remains gapless and being described by a conformal field theory with central charge c=1/2. On the other hand, the 330-RR_4 interface can be fully gapped. In this case, the incoming Abelian anyon is completely transmuted into a non-Abelian anyon in the Read-Rezayi quantum Hall liquid. Abelian anyons with opposite pseudospins would transmute into non-Abelian anyons satisfying different fusion channels.

Regardless of the transmutation mechanism, the original pseudospin information is stored nonlocally in the system. As a result it cannot be recovered by any local measurement. We believe this is a fair analogy to an object falling into a real black hole, in the sense that while the information it carries is not lost, it does become inaccessible to an (outside) observer. This resembles the idea of quantum information scrambling, which is consistent with the modern viewpoint that black holes are fast (perhaps the fastest) information scramblers <cit.>. The matching between the dimensions of Hilbert spaces for the Abelian quasiparticle and non-Abelian anyons further verifies the preservation of information.

In addition, we considered the case when more quasiparticles are dragged across the interface. Focusing on the 331-Pfaffian interface, we argued that the maximum amount of information the system can store in a topologically protected way is bounded by the length of the interface. This feature is reminiscent of a similar bound in black hole set by its area due to the holographic principle and the Bekenstein entropy. Finally, we discussed the simulation of black hole evaporation by shrinking the Pfaffian region which releases quasiparticles back to the 331 liquid. We demonstrate explicitly that the corresponding entanglement entropy would follow the Page curve. Hence, the original pseudospin information is recovered and the “information paradox" in our model is resolved. Note that remnants may be left at the end of evaporation in actual astrophysical black holes <cit.>. This may provide an alternative resolution of the information paradox, which is not addressed in the present work.

It is quite surprising that the seemingly simple quantum Hall interface has a rich analogy with black hole physics. Nevertheless, we need to point out some potential differences between our model and real astrophysical black holes. For a (semi)classical black hole, the horizon is not expected to have an effect on an infalling object (the so-called “no drama scenario”), including the information carried by it. Whether this remains to be the case or not for a fully quantum-mechanical black hole is unclear. A firewall at the horizon is a possible scenario that is currently under investigation and debate <cit.>. In our model, the Abelian quasiparticles must be transmuted when they cross the interface. This is inevitable as the Abelian and non-Abelian quantum Hall liquids support different degrees of freedom. Thus, the interface in our model behaves like a firewall. In our opinion, this interface may be a very simple and accessible “black hole firewall", which deserves more attention. In future work, it will be tempting to examine possible analogy of black hole thermodynamics in quantum Hall interfaces. It is also interesting to examine whether quantum Hall interfaces can provide an easy simulation of (a topological version of) the Hayden-Preskill protocol. 

The black-hole information paradox is arguably one of the most fundamental problems in physics, which involves gravitation, quantum field theory, and in particular, quantum information science. This long-standing problem is currently being actively studied by physicists in many different areas, and from very different perspectives (but so far only theoretically). Its resolution may well pave the way for the quantum theory of gravity, the holy grail of theoretical physics. While there is a lack of complete parallel between our model and certain “believed” processes in actual black holes (especially in the description of black hole evaporation which should be spontaneous), the analogy presented here provides a simple and accessible platform to simulate (i) apparent information loss, (ii) information scrambling, and (iii) information recovery. We believe these are arguably the most important and central concepts in understanding and resolving the original information paradox. Furthermore, our work may open a new research direction of studying how local information can be transmuted and stored nonlocally in an actual black hole. Since the concept of firewall and many other aspects in the paradox are still under intense debate, it is worthwhile to have simple analogies that capture some of the relevant concepts (but not necessarily all details precisely) in the original problem. In addition, our results have established a connection between quantum information, black hole physics and quantum Hall physics, and may bring experimentalists into this exciting research area.  

Last but not least, it is worthwhile to mention that a deep connection between quantum Hall effect and gravitational physics has been revealed in previous work <cit.>, leading to the recent experimental discovery of chiral graviton<cit.>. In particular, Refs. <cit.> have suggested a possible simulation of Hawking-Unruh effect by scattering quasiparticles in quantum Hall systems. We are optimistic that more connections between black hole physics and quantum Hall physics may be discovered in the future.



§ ACKNOWLEDGMENT


We thank N. E. Bonesteel for a useful discussion in our original work. This work was supported by the National Science Foundation Grants No. DMR-1932796 and DMR-2315954, and performed at the National High Magnetic Field Laboratory which is supported by National Science Foundation Cooperative Agreement No. DMR-2128556, and the State of Florida.

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footnote-gap
The gapping of modes is realized when the tunneling amplitude in H_T is a constant instead of a random variable in Eq. (<ref>). Strictly speaking, this kind of nonrandom tunneling is unphysical as there is a momentum mismatch between different modes. However, gapped modes and localized modes discussed in Sec. <ref> have the same physical consequences in the present discussion.

footnote0
In our opinion, this is a big advantage over the Pfaffian-NASS interface <cit.>. In that case, the NASS quantum Hall liquid at ν=4/7 has quasiparticle with the smallest charge of e/7. Hence, a net charge of e/4-e/7=3e/28 is absorbed, which leads to an accumulation of charges at the interface. Furthermore, the difference between filling factors of the two quantum Hall liquids may complicate the gapping of counterpropagating charge modes at the interface. This difference also introduces additional challenges in realizing the interface experimentally.

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footnote
Although Fig. <ref> may imply the interface is a one-dimensional object, it is not. The interface actually penetrates in both quantum Hall liquids and has a finite width (but much shorter than its length), similar to the interface between Pfaffian and anti-Pfaffian liquids <cit.>. Therefore, braiding of anyons can occur at the interface.

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footnote2
In the Pfaffian-NASS interface, one may consider a similar “information paradox" by dragging a charge-e/7 non-Abelian quasiparticle from the NASS liquid to the Pfaffian liquid. Depending on the species, the neutral sector is given by σ_↑ or σ_↓ in the c=6/5 CFT. Both transmute into (σ̅, σ), where σ̅ is a primary field in the ℳ(5,4) minimal model describing the interface <cit.>. Since σ̅ has a quantum dimension d=(1+√(5))/√(2)>√(2), the Hilbert space of anyons at the interface has a larger dimension.

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foot-TQC
It is reminded that one cannot perform universal TQC (or reproduce any unitary operation to arbitrary accuracy by braiding of anyons) by using anyons in the Ising CFT <cit.>. More complicated anyons such as Fibonacci anyons are required for achieving the universality <cit.>.

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footnote-group
In the general case, the maximal set of condensable bosons may not be closed under fusion due to the splitting of non-Abelian anyons. In this case, a more appropriate terminology should be Lagrangian subset <cit.>. Meanwhile, ℒ_e and ℒ_m are indeed closed under fusion in our present case.

footnote-random
Since the tunneling process is random, the renormalization group (RG) equation reads 
dW/d(ln s)=(3-2Δ)W. Here, s is the scaling factor of time and space, and 
ξ(x)ξ(x')=Wδ(x-x'). For the single electron tunneling process (i.e., a=1), the scaling dimension for the operator involving the neutral modes ψ_1exp(iϕ_n) is already h_n=3/4+3/4=3/2. The scaling dimension of the operator involving charge modes exp[i(ϕ_l+ϕ_r)/2] is non-universal but must be positive. Therefore, the process is irrelevant in the RG sense. On the other hand, the corresponding RG equation for nonrandom tunneling is dW'/d(ln s)=(2-Δ)W', where W' stands for the tunneling strength. Depending on the interaction between 
ϕ_r(x) and ϕ_l(x), the single electron tunneling process can be relevant in the RG sense. 

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In principle, the spin vector s is another required topological quantum number in characterizing the topological order <cit.>. We will ignore s and focus on the K matrix and t vector, which is sufficient for the present discussion.

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