Inverse Design of Winding Tuple for Non-Hermitian Topological Edge Modes
Zihe Yang Kunling Zhou Bowen Zeng Yong Hu







These authors contribute equally to this work.
School of Physics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China
Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, P. R. China


These authors contribute equally to this work.
School of Physics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China
Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, P. R. China


[]zengbowen@csust.edu.cn
Hunan Provincial Key Laboratory of Flexible Electronic Materials Genome Engineering,
School of Physics and Electronic Sciences, Changsha University of Science and Technology, Changsha 410114, P. R. China

[]huyong@hust.edu.cn
School of Physics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China
Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, P. R. China







§ ABSTRACT 

The interplay between topological localization and non-Hermiticity localization in non-Hermitian crystal systems results in a diversity of shapes of topological edge modes (EMs), offering opportunities to manipulate these modes for potential topological applications. The conventional strategy for characterizing the domain of EMs is to calculate the topological invariants, but which does not provide the wavefunction forms of EMs. This leads to the the bulk-boundary correspondence typically being verified only through numerical methods. In this work, by recognizing EMs as specified solutions of eigenequation, we derive their wavefunctions in an extended non-Hermitian Su-Schrieffer-Heeger model. We then inversely construct a winding tuple { w_  GBZ,w_  BZ} that characterizes the existence of EMs and their spatial distribution. Moreover, we define a new spectral winding number equivalent to w_  BZ, which is determined by the product of energies of different bands. The inverse design of topological invariants allows us to categorize the localized nature of EMs even in systems lacking sublattice symmetry, which can facilitate the manipulation and utilization of EMs in the development of novel quantum materials and devices.





Inverse Design of Winding Tuple for Non-Hermitian Topological Edge Modes
    Yong Hu
    September 9, 2024
========================================================================




Introduction—The topological edge modes (EMs) localized at edge corresponds to nontrivial bulk topological invariants performed on the Brillouin zone (BZ), a phenomenon known as the bulk-boundary correspondence (BBC), which is a central concept in the filed of celebrated topological band theory <cit.>. In non-Hermitian open systems, where non-Hermiticity arises from interactions with the environment, the conventional BBC may break down due to the systems' sensitivity to the boundary <cit.>. This sensitivity leads to the accumulation of macroscopic bulk states at the boundary under open boundary condition (OBC) called the non-Hermitian skin effect m(NHSE) <cit.> and equivalently reshapes the range of allowed wavevector, the collection of which is referred to as generalized Brillouin zone (GBZ) <cit.>. Note that the NHSE respects spectral topology, rather than band topology, which refers to the winding of the spectrum 
under periodic boundary condition (PBC) with respect to the OBC spectrum defined on the GBZ <cit.>. Recently, Yao et al. <cit.> demonstrated that the bulk topological invariants performed on the GBZ can precisely feature the existence of EMs in the non-Hermitian systems, marking a reestablishment of BBC. 

In Hermitian topological systems, such as the one-dimensional Su-Schrieffer-Heeger (SSH) model with intercell coupling larger than intracell coupling <cit.>, pairs of EMs are separately distributed at the two ends of systems <cit.>. In non-Hermitian systems, the interplay between topological localization and non-Hermiticity localization gives rise to diverse patterns of EMs <cit.>. For example, the balance between these two localization mechanisms allows the delocalization of topological EMs, a phenomenon observed in experiments <cit.>. This offers potential pathways for manipulating the EMs and designing “extended state in a localized continuum” <cit.>. While in the NHSE-dominant region, two EMs are localized at the same ends together <cit.>. Several topological invariants, such as the winding number based on the evolution of pseudomagnetic fields or spectrum, have been proposed for building the corresponding BBC <cit.>, for the domain of EMs and these invariants appears to be consistent. However, this approach does not provide the wavefunction forms of EMs.
Additionally, the reasons why such a BBC works well and the relationships between these topological invariants remain not entirely clear.


In this work, by considering EMs as special solutions distinct from bulk state solutions of eigenequation, we derive the analytical solutions for the EMs in a generalized non-Hermitian SSH model. According to the existence condition and behavior of this analytical solutions, we inversely construct a topological winding tuple { w_  GBZ,w_  BZ} for non-Hermitian topological EMs. A nontrivial w_  GBZ defined on the GBZ characterizes the presence of EMs and w_  BZ=0,±1 defined on the BZ corresponds to two EMs being separately distributed at two ends, or both localized together at the left (right) end, respectively. This winding tuple is exactly a combination of previously defined topological invariants from the literature and is applicable to systems without sublattice symmetry (SLS). Topological invariants concerning spectrum derived from w_  BZ has the form of the product of energies of different bands, revealing the connection between energy subbands and potentially advancing the spectral winding number of multi-band systems. Our results provide a comprehensive understanding of topological EMs and the associated BBC in non-Hermitian systems.  
 
Model.—We consider a two-band non-Hermitian SSH model, as illustrated in Fig. <ref>(a), with a Hamiltonian defined in momentum space

    H(β)=[  i γ+i λ(1/ β -β)  t_1 e^-iθ+t_2 /β;    t_1 e^iθ+t_2 β -i γ-i λ( 1/β -β) ].

Here H(β) represents the Bloch Hamiltonian for β∈BZ or the non-Bloch Hamiltonian for β∈GBZ. Here, ± i γ represents the on-site gain and loss at the A/B sites, which are the only non-Hermitian term in this model; t_1 (t_2) and λ are conjugated intracell (intercell) coupling and neighbor A-to-A or B-to-B sites. Without loss of generality, we only consider {t_1, t_2, λ}≥ 0. Additional phase ±θ and ±π/2 are introduced in t_1 and λ, respectively, which leads to magnetic fluxes of Φ_±=π/2±θ in the model's triangular loops as shown in Fig. <ref>(a). Such an SSH model enables gain (loss)-controlled and flux-controlled NHSE <cit.> as shown in Fig. <ref>(b), where the beige (blue) region represents the pile-up of bulk states at the left (right) boundary (see Fig. <ref>(c-d)). Such skewness corresponds to the positive (negative) winding of the PBC spectrum with respect to OBC spectrum <cit.>, respectively, as shown in insets of Fig. <ref>(c-d). 




This model could support paired EMs, as shown in Fig. <ref>(c-d), with their energy symmetrically distributed with respect to the real axis. The EMs can be either localized with skin modes in Fig. <ref>(c) or one of them can be isolated at the left end as depicted in Fig. <ref>(d). The positive spectral winding number for EMs and their localization in the same direction in Fig. <ref>(c) appears to stick to conventional spectral BBC <cit.>. However, for separately distributed EMs in Fig. <ref>(d), the winding number is trivial.

It should be noted that establishing the BBC for EMs in our model may be challenging. One reason is that Eq. (<ref>) does not respect SLS and another reason is that diagonal elements in Eq. (<ref>) depend on wavevector. Thus, conventional approaches, such as defining topological invariant using the “Q matrix” or calculating the winding number of non-diagonal elements in Hamiltonian, may not be applicable <cit.>. Below, we attempt to establish the BBC for these EMs in non-Hermitian systems.

Analytic solutions for EMs—We begin by analytically calculating the wavefunction of EMs with details available in the Supplementary Materials. By substituting the non-Bloch wavefunction in Eq. (<ref>), the eigenequation yields four roots for each energy E, denoted as {β_1,β_2,β_3,β_4} sorted by their moduli. Then the wavefunction can be constructed as

    [ ψ_A(n); ψ_B(n) ]
    = ∑_i=1^4
    [ ϕ_Ai; ϕ_Bi ]β_i^n,

where ϕ_Ai =α(β_i) ϕ_Bi with α(β_i)= t_1 e^- i θ + t_2/β_i/E - q(β_i) and q(β_i)=iγ +  i λ(1/β_i -β_i). 
For the boundary conditions of N-site-lattice, the coefficient matrix satisfies Mϕ=0, where ϕ=[ϕ_B1,ϕ_B2,ϕ_B3,ϕ_B4]^𝖳 and

    M=α(β_1)    α(β_2)    α(β_3)    α(β_4) 
    
    1     1     1     1 
    α(β_1) β_1^N+1   α(β_2) β_2^N+1   α(β_3) β_3^N+1   α(β_4) β_4^N+1
    β_1^N+1   β_2^N+1   β_3^N+1   β_4^N+1.

The solutions exist when det(M)=0. For bulk states, this condition becomes the roots of characteristic function fulfilling the GBZ condition |β_2|=|β_3| in the thermodynamic limit <cit.>. The existence of EMs implies that special solutions can be found, which satisfy det(M)=0 but with |β_2| < |β_3|. By dividing the third and fourth rows of M by β_3^N+1 and taking the thermodynamic limit N→∞, M becomes a block upper triangular matrix

    M=M_I    M_III
    0    M_IV,

where each element represents a 2 × 2 submatrix. Now, det(M)=det(M_I)×det(M_IV). The existence of special solutions requires either det(M_I) or det(M_IV) to be zero. Let C_e1 = α(β_1) = α(β_2) to ensure det(M_I)=0, where the subscript e1 denotes one EM if special solutions exist. Then we have 


    f_e1(β,E_e1) =    - iλ C_e1β + (t_2 +  iλ C_e1)/β
       + (t_1 e^- iθ + iγ C_e1 - C_e1 E_e1)=0.

Here, E_e1 is the energy of this EM. According to Eq. (<ref>), -E_e1 should be the energy of another EM f_e2(β,E_e2)=0 with E_e2=-E_e1 and β_3,4 being associated with E_e2. Same results would be obtained if we first let det(M_IV)=0. By simultaneously solving the eigenequation of Eq. (<ref>) and Eq. (<ref>), the characteristic equation can be derived

    Γ_e(β) =    (-t_2C_e^2-2 iλ C_e)β+(2 iλ C_e+t_2)/β
       +(-t_1 e^iθ C_e^2+t_1e^-iθ+2 iγ C_e)=0,

with index e={e1;e2}. Here C_e^2 - i t_2 /λ C_e + 1=0 (details available in Supplementary Materials). With four roots solved, we further obtain the energies of EMs


    E_e=±γ t_2/λ+2it_1sinθ/C_e1-C_e2.

and the wavefunction for two EMs on each site 

    ψ_e1(n) = [ C_e1β_1^n - C_e1β_2^n;         β_1^n - β_2^n ]
       ψ_e2(n) = [ C_e2β_3^n-(N+1) - C_e2β_4^n-(N+1);         β_3^n-(N+1) - β_4^n-(N+1) ].

It can be observed that the ψ_e1(n) and ψ_e2(n) are predominantly governed by β_2 and β_3, respectively. 



For arbitrary given parameters, the EMs exist if both two roots associated with a specified E_e among four roots are larger (less) than the other two roots (also can see Fig. S1 in the Supplementary Materials). This allows us to map the system's phase diagram for EMs, as shown in Fig. <ref>(a), where the white (colored) region represents the absence (presence) of EMs. From the wavefunction Eq. (<ref>), it can be predicted that two EMs are localized at the left end when |β_2|<|β_3|<1 and right end when 1<|β_2|<|β_3|, and both ends when |β_2|<1<|β_3|, which corresponds to the beige, blue and green region in Fig. <ref>(a), respectively. These predictions are further verified and exemplified by numerical results in Fig. <ref>(b-d). The emergence of delocalized EM when |β_2|=1 or |β_3|=1, corresponding to a critical case between different colored regions such as point b in Fig. <ref>(a). An extended EM over bulk is shown in Fig. <ref>(b) with the associated energy located on the PBC spectrum. Thus, according to the root distribution but without the knowledge of any topological topological invariants, we derive the conditions for the existence of EMs and their spatial distribution. 

Inverse Design of Topological Winding Tuple—
To establish the BBC for topological EMs, we inversely design the corresponding topological invariants by the root distribution as follows. Note that the existence of EMs requires two largest (smallest) roots are associated with E_e2 (E_e1) among four roots. According to the GBZ theory, for an energy in our model that does not belong to OBC bulk spectrum, two roots must lie within the GBZ, while the other two roots must lie outside the GBZ <cit.>. In other words, the GBZ bisects the four roots. Considering that Γ_e(β) has a first-order pole at β=0, we can construct the winding number by Γ_e1;e2(β) (only the solutions for EMs satisfy Γ_e1;e2(β)=0) along the GBZ

    w_e1;e2 = ∮_ GBZ1/2π idlnΓ_e1;e2(β).

For the existence of EMs, w_e1=1 and w_e2=-1. A trivial w_e1=0 implies that β_1 and one of β_3 and β_4 are associated with E_e1. In this case, the corresponding w_e2=0, indicates the absence of EMs. Both nontrivial w_e1;e2 feature the existence of EMs, as does their combination in the form of 

    w_  GBZ = w_e1-w_e2/2.


Given the dominance of |β_2| and |β_3| in Eq. (<ref>), we naturally compare their magnitudes with 1 to determine the domain of EMs. Note that the BZ on the complex plane is a unit circle with a modulus of 1, which inspires us to define a winding number along the BZ

    w_e1;e2' = ∮_ BZ1/2π idlnΓ_e1;e2(β).

Here, w_e1'=1(0) implies that there are two (one) roots for E_e1 in BZ and corresponds to this EM at the left (right) end. For the other EM, a similar distribution is characterized by w_e2'=0(-1). The distribution of these two EMs can also be characterized by   

    w_  BZ = w_e1' + w_e2',

where w__BZ∈{-1, 0, 1}. When w__BZ = 1(-1), three (one)  of the four β_i of the EMs are within BZ, implying 
|β_2;3| < 1 (|β_2;3| > 1) and EMs are localized at the left end (right end). When w_1 = 0, |β_2| < 1 < |β_3|, two EMs are localized at opposite ends, a distribution analogous to the Hermitian scenario.







Thus, the topological EMs in non-Hermitian systems correspond to a winding tuple

    W = { w_  GBZ,w_  BZ}.

The first element defined on the GBZ determines the existence of EMs, while the second element defined on the BZ determines the localized nature of the EMs. 

Such a winding tuple is consistent with the phase diagram in Fig. <ref>(a), as illustrated in Fig. <ref>. Fig. <ref>(a-c) show four roots of EMs with respect to GBZ and BZ for three representative cases b, c,  d in Fig. <ref>(a), corresponding to w_  GBZ=1, W={1,1}, W={1,-1}.  For a specified case b in Fig. <ref>(a), w_  BZ is not well-defined as the evolution of Γ_e2(β) crosses the zero point as shown in Fig. <ref>(d). The phase transition in Fig. <ref>(a) by adjusting t_1e^iθ under fixed t_2 is captured by the variation of winding tuple as shown in Fig. <ref>(e). 

Cross-validation for topology—In this section, we further address the relationship between winding tuple Eq. (<ref>) and previously defined topological invariants <cit.>. The wavefunction of single EM in systems with SLS typically is located at the single site A or B. However, the wavefunction in Eq. (<ref>) is distributed on both sites, which inspires us to perform a similarity transformation H^'(β)=U^-1H(β)U with transformation matrix U=1/μ[ d_- d_+;   1   1 ]. Here, μ=√(d_–d_+) is normalization factor and d_-(d_+) is the ratio of amplitude between different sites. The new Hamiltonian reads 

    H^'(β) = 1/μ^2[ m_0 + m(β)     R_+(β);     R_-(β) -m_0 -m(β) ].

with the diagonal element including constant mass term 

    m_0=iγ(d_++d_-)-d_+ d_- t_1 e^iθ+t_1 e^-iθ;
 and β-dependent term 

    m(β) = iλ(d_+ + d_-)(1/β-β) -t_2 β + d_+d_-t_2 /β;
 
and the non-diagonal term 

    R_±(β)=   ±[(-t_2 d_±^2-2 iλ d_±) β + (2 iλ d_± +t_2)/β.
       .+(-t_1 e^iθd_±^2 +2iγ d±+t_1 e^-iθ)]
    .



To compare with the well-studied systems that have SLS, the selection of parameters should ensure the on-site mass term independent of β (m(β)=0), which requires d_+ + d_- = it_2/λ and d_+ d_- = 1. These conditions align with solutions for C_e, i. e., {d_+,d_-}={C_e2,C_e1} and thereby R_+(β)  =Γ_e2(β), R_-(β)  = -Γ_e1(β), and m_0^2/μ^4 = E_e^2. Since the constant on-site mass term only alters the energy of EMs but not the wavefunction <cit.>, the previously defined winding number on the GBZ <cit.> and on the BZ <cit.> return to Eq. <ref>
and Eq. <ref>, respectively. These results also cross-validate our proposed analytic solutions for EMs.



Considering that similarity transformation does not alter the eigenvalue 


    E(β)^2 =E_e^2 -Γ_e1(β)Γ_e2(β)/μ^4,

we can define a new topological invariant by spectrum equivalent to w_  BZ

    w_  BZ(E) =1/2π i∮_ BZdln[∏_i=1;2 E_i(β)-∏_i=1;2 E_ei],

where E_i(β) is the energy of different bands. Different from conventional spectral winding number <cit.> that refers to the winding of PBC spectrum related to reference energy, w_  BZ(E) take the form of the product of energies of different bands. 
As shown in Fig. <ref>, w_  BZ(E) fits well with the previous calculations. 
Two left-localized EMs in Fig. <ref> (c) and separately distributed EMs in Fig. <ref> (d) are capture by w_  BZ(E)=1 and w_  BZ(E)=0.  A critical case is shown in Fig. <ref>(c) where the product of two EMs falls in the ∏ E_i(β), corresponding to a delocalized state in  Fig. <ref>(b). 


Conclusions—In this study, we derive the analytic solutions for EMs and inversely construct the topological winding tuple W = { w_  GBZ,w_  BZ}, corresponding to the existence and localized nature of EMs, respectively.  We also define a novel spectral winding number based on the product of the energy of different bands, offering a new perspective for unveiling the extraordinary EMs in non-Hermitian systems. Our study not only advances the theoretical understanding of non-Hermitian topological EMs but also suggests potential avenues for manipulating and utilizing EMs for future experimental explorations aimed at harnessing the practical applications of topological phenomena. 

We thank Xiaoshang Jin for helpful discussions. This work is supported by the
Natural Science Foundation of Hunan Province (2024JJ6011) and Innovation Program for Quantum
Science and Technology (Grant No. 2021ZD0302300).













