The topological counterparts of non-Hermitian SSH models

The breakdown of the conventional bulk-boundary correspondence due to non-Hermitian skin effect leads to the non-Bloch bulk-boundary correspondence in the generalized Brillouin zone. Inspired by the case of the equivalence between the non-reciprocal hopping and imaginary gauge field, we propose a method to construct the topological equivalent models of the non-Hermitian dimerized lattices with the similarity transformations. The idea of the constructions is from that the imaginary magnetic flux vanishes under the open boundary condition and the period boundary spectra can be well approximated by open boundary spectra. As an illustration, we apply this approach to several representative non-Hermitian SSH models, efficiently obtaining topological invariants in analytic form defined in the conventional Bloch bands. The method gives an alternative way to study the topological properties of non-Hermitian system.

It is well known that there is a Hermitian counterpart for a non-Hermitian Hamiltonian within the symmetry-unbroken region, in which the two Hamiltonians have an identical fully real spectrum [36][37][38]. This allows us to find the nontrivial states of the non-Hermitian Hamiltonian with the open boundary condition from its Hermitian counterpart. Here, a natural question that arises in this topics is whether there is a partner of non-Hermitian system with non-Hermitian skin effect that share the same topological phase diagrams. As such, the topological invariant of the original model can be obtained from its partner, which can be calculated in an easier way. In particular, the topological invariant of their partner may have the analytical form which is important to understand the whole properties of non-Hermitian system.
In general, the non-Hermitian skin effect comes from the non-reciprocal hopping of the lattice. This effect can be realized experimentally in atomic systems by laser assisted spinselective dissipations [39,40]. As the case studied in Ref. [14] and [17], the model with balanced hopping and only gains and * csliu@ysu.edu.cn losses is mathematically equivalent to the non-Hermitian Su-Schrieffer-Heeger (SSH) model with the imbalance hopping when Pauli matrix σ z is replaced by σ y although the original model can not be interpreted by SSH model. The recent study provides the conditions under which on-site dissipations can induce non-Hermitian skin modes [34].
The asymmetric couplings are equivalent to an imaginary gauge field applying to the lattice [10,31]. Under the periodical boundary condition, the imaginary gauge field enclosed in an area induces a nonzero imaginary magnetic flux, which is non-Hermitian Aharonov-Bohm effect as the Hermitian case [31]. As pioneered by Yang and Lee in 1952, quantum phase transition can be driven by a complex external parameter [41,42]. The nonzero imaginary magnetic flux breaks the conventional bulk-boundary correspondence and leads to a topological phase transition. Under the open boundary condition (OBC) however, the imaginary gauge field is not enclosed in an area and the imaginary magnetic flux vanishes [31]. Although the OBC breaks the translational symmetry, according to the bulk-boundary correspondence in the long chain case, the boundary scattering can still be regarded as a perturbation whether in Hermit case or not due to the introduction of the GBZ. This gives a strong hint that the Bloch Hamiltonian can be well approximated by the bulk Hamiltonian under the OBC. Following this route, non-Hermitian Bulk-boundary correspondence of the non-Hermitian SSH model without the t 3 was recovered after getting rid of the effective imaginary gauge [31]. A key step is transforming the non-Hermitian terms to the Bloch phase factor with a similarity transformation. This step can also be realized under the OBC through another similarity transformation [24]. After the similarity transformation, the non-Hermitian SSH model is transformed to its topological equivalent model which leads to understand the nontrivial topological phase easily.
A natural issue is whether the method can be extended to construct the partner of the general non-Hermitian model with the asymmetric coupling terms. Motivated by the above considerations, we develop this method to construct the partners of several non-Hermitian SSH models. As is shown in the following studies, the central tenet of the constructing is building the relationship between non-Hermitian skin effect and imaginary equivalent gauge field. The topological invariants based on band-theory are effective to predict the topological nontrivial states. The remainder of this paper is organized as follows. In Sec. II, we present the Hamiltonians of non-Hermitian SSH model and its various equivalent models. We show how the non-Hermitian SSH model is transformed to its equivalent models by the similarity transformations and how the non-Hermitian skin effect is eliminated due to the offset of the imaginary gauge field. In subsection III A, the effectively gauge field is find to construct a partner model of the non-Hermitian SSH model by a similarity transformation. Due to the topologically equivalence of the two models, we study the topological phase transitions with the partner models in subsection III B. Inspirited by the consistence of the QPTs with the numerical method, we further apply the method to study the non-Hermitian SSH model with spin-orbit coupling in Sec. IV A and the topological defect states of the non-Hermitian SSH model in Sec. IV B. Finally, a summary and discussion are given in Sec. V.

II. THE NON-HERMITIAN SSH MODELS
The non-Hermitian SSH model is described Bloch Hamil- where σ = σ x , σ y , σ z is the Pauli matrix for spin-1/2 and d x = t 1 + (t 2 + t 3 ) cos k, d y = (t 2 − t 3 ) sin k [43,44]. The Numb wavefunction . When Pauli matrix σ y is replaced by σ z and setting t 1 = M + 4B, t 2 + t 3 = 2B and t 2 − t 3 = 2A, the generalized SSH model h k can be mapped to 1D Creutz-type model [45] which is the dimensional reduced BHZ model [46,47]. When Pauli matrix σ x is replaced by σ z and replacing t 1 = −µ, t 2 + t 3 = −2t and t 2 − t 3 = −2∆, the generalized SSH model h k can be mapped to 1D Kitaev model model [47,48]. The non-Hermiticity comes from the term Γ. When Γ = (0, 0, γ/2), the non-Hermiticity doesn't change the topological phase transition and the existence of the edge state due to the pseudoanti-Hermiticity protection [19,49,50]. If the imaginary part Γ = (γ/2, 0, 0), the inversion symmetry of h k ensures nonexistence of the non-Hermitian skin effect and the validity of conventional bulk-boundary correspondence. The nontrivial topology is established from the linking geometry of the vector fields [44]. We study the case of Γ = (0, γ/2, 0). The non-Hermitian SSH model and its equivalent two-leg ladder model are pictorially shown in Fig. 1 (a) and (b). The Bloch Hamiltonian reads where σ x,y are the Pauli matrices The asymmetric intracell coupling amplitude t 1 ± γ/2 can be realized in open classical and quantum systems with gain and loss [39,40,51,52]. The model has a chiral symmetry σ −1 z h k σ z = −h k , which ensures that the eigenvalues appear in (E, −E) pairs.
There are various equivalent models for this model. For example, a mathematically equivalent model is studied in Ref. [11]. This model can be obtained by a similarity transforma-tionH = S −1 1 HS 1 with a diagonal matrix S 1 whose diagonal elements are judiciously chosen as [24]  here Under the similarity transformation, the non-Hermiticity in t 1 term is transformed to t 3 term where t 3 term becomes t 3 r 2 1 and t 3 /r 2 1 . The wave- The extended non-Hermitian SSH discussed in Ref. [53] can be obtained by a similarity transformationH = S −1 2H S 2 with a diagonal matrix here r 2 = r −2 1 . After this transformation shown in Fig. 1 (c), the non-Hermiticity is transformed from t 3 term to t 2 term where t 2 term becomes t 2 /r 2 and t 2 r 2 . The wavefunction |ψ = S −1 2 |ψ and the intracell hoppingt 1 remains unchanged. By relabeling the sites A→D and B→C, the model can be mapped to an extended non-Hermitian SSH model. The Hamiltonian in momentum space takes the form In the mapping, the we have replaced ∆ = t 3 , t ′ =t 1 , t − δ = t 2 /r 2 and t + δ = t 2 r 2 .
The non-Hermitian SSH model in Eq. (1) is also topological equivalence to the model discussed in Ref. [26] where the non-Hermiticity occurs in t 1 , t 2 and t 3 terms. One can first do the inverse similarity transformation in Eq. (3). The non-Hermiticity in t 2 term is transferred to t 3 term. Then further doing the inverse similarity transformation in Eq. (2), the non-Hermiticity in t 3 term is transferred to t 1 term. After doing the two similarity transformations, the non-Hermiticity in t 2 and t 3 terms disappear and the asymmetric intercell coupling is transformed to symmetrical. The intercell coupling relating to t 1 term remains asymmetrical.

III. THE PARTNER MODEL AND TOPOLOGICAL INVARIANT
A. The partner model The asymmetric couplings in t 1 term of Eq. (1) can be expressed as a symmetric couplingt 1 with phase factor of amplification/attenuation e ±φ , i.e.
the Hamiltonian H k can be rewritten in the form of With the similarity transformation, the AB sublattice becomes A ′ B sublattice and the asymmetric coupling is transformed from t 1 term to t 2 and t 3 terms shown in Fig. 2 (a). Here, the site label A is replaced by A ′ which describes the annihilation operator a of A site is transformed to a ′ . Assuming t 1 , t 2 , t 3 and φ to be greater than zero, the non-Hermitian skin effect in model (5) can been analysed in real space as follows. When t 2 0 and t 3 = 0 shown in 2 (b), the model (5) is reduced to the general non-Hermitian SSH model. The hopping t 2 e φ from B sites to A ′ sites is larger than that t 2 e −φ from A ′ to B sites. The asymmetry hopping leads to the particles tend to right side. However, in the case of t 3 0 and t 2 = 0 shown in 2 (c), the model (5) is also reduced to the general non-Hermitian SSH model where the hopping t 3 e φ from A ′ sites to B sites is larger than that t 3 e −φ from B to A ′ sites which induces the particles tend to left side. For the case t 2 = t 3 0, the Hamiltonian in Eq. (5) has the PT symmetry and the two non-Hermitian skin effects cancel. Therefore, we conclude that there exist a non-Hermitian skin effect when t 2 t 3 . When t 2 > t 3 , the non-Hermitian skin effect is governed by the t 2 term and all the states are localized in the right edge. When t 2 < t 3 , t 3 term dominates the non-Hermitian skin effect which leads the all the states localized in the right edge.
The amplification and attenuation factors e ±φ in Eq. (5) indicate an imaginary gauge field φ applying to the lattice. The effective imaginary gauge field can be analysed in moment space as follows. In the unit cell shown in Fig. 2 (d), two channels are provided for a particle tunneling from A ′ to B with different hopping amplitude. It indicates that a particle tunneling from A ′ site to B site will obtain a e φ phase for the t 2 and t 3 channels. The product of the above two terms contribute the overall accumulated phase factor e 2φ which suggests the enclosed imaginary gauge field be 2φ.
We therefore rewriteĥ k in Eq. (5) aš where σ ± = σ x ± iσ y . Considering the terms t 2 e −i(k+2iφ) and t 3 e i(k+2iφ) in Eq. (6), they are equivalent to an imaginary magnetic field applying to the A ′ B lattice. We take a complexvalued wave vectork → k + 2iφ to describe open-boundary eigenstates. In this replacement, the Hamiltonian (6) can be treated as parameter ofk which a deformation of the standard Brillouin zone. Accordingly, we define a partner Hamiltonian as follows:ȟ˜k which is also non-Hermitian Hamiltonian. h + (k) and θ + (k) are the module and angle of complex functiont 1 + t 2 e −ik e −φ + t 3 e ik e 3φ . h − (k) and θ − (k) are the module and angle of complex functiont 1 + t 2 e ik e φ + t 3 e −ik e −3φ . The partner model is still a non-Hermitian model. The non-Hermitian t 2 and t 3 terms lead to the accumulation of real phase factor which suggests the enclosed imaginary gauge field be 2φ. The 2φ imaginary gauge field suggests that the wave vector should bek →k + 2iφ. The model in Eq. (7) is transformed back to the model in Eq. (5).
The non-Hermitian skin effect in model (7) can also been analysed in real space. In the unit cell shown in Fig. 2(e), ±φ in the asymmetry hopping t 2 term is equivalent to an imaginary magnetic field applied to the lattice with an imaginary magnetic vector potential iφ alone x direction. For the asymmetry hopping t 3 term, ±3φ can be written as ±φ(3a) here the lattice constant a = 1. 3a indicates three sites have been crossed when a particle hopping from B site to A ′ through the t 3 channel. It is also equivalent to an imaginary magnetic field applied to the lattice with an imaginary magnetic vector potential iφ. However, the imaginary magnetic vector potential is alone the −x direction. The two imaginary magnetic vector potentials are cancelled and the imaginary magnetic flux does not exist under the periodical boundary condition. No non-Hermitian skin effect occurs in the model [Eq. (6)] and wave vector is still a good quantum number.
The partner model in Eq. (7) is also effective to the case t 2 0 and t 3 = 0. Replacingk →k − iφ, the partner model in Eq. (7) is transformed the standard SSH model with the phase transition pointt 1 = t 2 . When t 3 0 and t 2 = 0, we can replacek →k + 3iφ. The partner model in Eq. (7) is also transformed the standard SSH model with the phase transition pointt 1 = t 3 . The above cases have been studied in Ref. [24] and [31].
Under the OBC, the partner Hamilton in Eq. (7) can also be obtained from the model in Eq. (1) by a similarity trans-formationH = S −1 3 HS 3 directly. S 3 = S 1 S 2 is also a diagonal matrix where S 1 and S 2 are given in Eq. (2) and (3) with r 1 = r 2 = e φ . The similarity transformation changes the eigenstates and don't change the its eigen-energy. So the models in Eq. (1) Eq. (5) and Eq. (7) are topological equvalent. As shown in Subsec. III B, the topological nontrivial phases can be charactered by the winding number based on the Eq. (7).

B. The topological invariant
According to the usual bulk-boundary-correspondence scenario, the chiral edge states of an 1D open-boundary system should be determined by the winding numbers which are closely related to the Zak phase across the Brillouin zone [54]. For the non-Hermitian system with chiral symmetry and the complex eigenvalues, the winding number of energy is defined as a topological invariant [15,16,49]. Ask goes across the generalized Brillouin zone, the winding number of energy ν E is defined as where the integral is also taken along a loop withk from 0 to 2π. The eigenvalues of Hamiltonian (7) are which are smoothly continuous withk. ν E is summation of winding numbers of two winding vectors h + k e iθ +(k) and h − k e iθ − (k) . In Hermitian systems, ν E is always zero due to the real energy E 1,2 .
The non-Bloch winding number can also be introduced with the "Q matrix" [19,24,29,55]. The Q matrix is defined by where the right vector |u R and left vector |u L are defined by |ũ R ≡ σ z |u R and |ũ L ≡ σ z |u L are also right and left eigenvectors, with eigenvalues −E and −E * due to the chiral symmetry. The normalization conditions are u L |u R = ũ L |ũ R = 1 and u L |ũ R = ũ L |u R = 0. The "Q matrix is off-diagonal, namely Q = qσ + + q −1 σ − where q = h 2 k /h 1 k exp i θ 2 k − θ 1 k /2 . The non-Bloch winding number is given by which is difference of winding numbers of two winding vectors h + k e iθ +(k) and h − k e iθ −(k) . According to the geometry of two winding vectors h + k e iθ + (k) and h − k e iθ −(k) , at the phase transitions points, the quantums must meet the relationship: Taking t 2 = 1 and γ = 4/3 for example, the phase transition points are t 1 = 1.5660 and t 1 = 1.7050 according to the Eq. (10). At t 1 = 1.5660, the system transforms firstly from topological nontrivial phase to topological trivial phase which is the result in Ref. [24].
To summarize the approach: With a similarity transformation, the non-Hermitcity of the model is transformed from t 1 term to t 2 and t 3 terms. According the non-Hermitcity of the equivalent model, the imaginary gauge field Φ is the obtained. Using the imaginary gauge field, the partner model is constructed with the Peierls replacementk → k + iΦ. At last, the non-Hermitian winding number is solved.

A. The non-Hermitian SSH model with spin-orbit coupling
When spin-orbit coupling is taking into account, the non-Hermitian version of SSH model can be written by the Hamiltonian H = H SSH + H SOC [56] where where a † n,σ (a n,σ ) and b † n,σ (b n,σ ) are the electron creation (annihilation) operators with spin σ = (↑ or ↓) on the sublattices A and B of the nth unit cell, respectively. The non-Hermiticity of the Hamiltonian is due to the introduction of δ. The spin-orbit coupling Hamiltonian is described by where λ and λ ′ denote the spin-orbit coupling amplitudes in the unit cell and between two adjacent unit cells, respectively. When the spin-orbit coupling is Dresshaus type, the coupling amplitudes λ and λ ′ are real value. Otherwise the coupling amplitudes are imaginary value when the spinorbit coupling is Rashba type. Adopting periodic boundary conditions and Fourier transforming, the Hamiltonian H = H SSH + H SOC can be easily written as with ζ k = t + t ′ e −ik and ς k = λ − λ ′ e −ik . The non-Hermitian skin effect of the model can be analysed as follow. When the spin-orbit coupling effect is negligible, the Hamiltonian in Eq. (11) is the direct-sum of two non-Hermitian SSH Hamiltonians. The non-Hermitian skin effect exists in the system. When the spin-orbit coupling effect dominates the system, the Hamiltonian (11) is a Hermitian. Therefore, the non-Hermitian skin effect exists in system. Considering the spin-up and spin-down of the particles, two channels have been provided for the particles tunneling from A sites to B sites of the AB sublattice. The spin-orbit coupling effect provides a new channel which induces the different spin particles tunneling from A sites to B sites.
Referring the non-Hermitian terms t 1 ± δ =t 1 e ±φ , the asymmetric hopping is equivalent to an imaginary magnetic field applying with the imaginary vector potential −iφ. Considering the two channels, the effective imaginary field with vector potential −2iφ is applying to the lattice. The −2iφ vector potential suggests the wave vector should be with the Peierls replacementk → k + 2iφ.
To finish this replacement, we do the similarity transformation to the Hamiltonian in Eq. (11)h k = S −1 4 h k S 4 with a diagonal matrix S whose diagonal elements are e −2φ , 1, e −2φ , 1 .
The wave function becomes ψ † k = e 2φ a k,↑ , b k,↑ , e 2φ a k,↓ , b k,↓ † . The partner is still an non-Hermitian model. Under the OBC, the partner can be obtained by a similarity transformatioñ H = S −1 1 HS 1 with a diagonal matrix S 1 given in Eq. (2), where r 1 = |(t 1 − δ) / (t 1 + δ)|. Under the similarity transformation, the non-Hermitcity is transformed to t and λ terms and the t 2 term remains unchanged. The wavefunction |ψ = (a 1 , b 1 , a 2 , b 2 , · · · , a L , b L ) T becomes |ψ = S −1 1 |ψ . For the non-Hermitian terms λe ±2φ , the asymmetric hoppings are equivalent to an imaginary magnetic field applying to the sublattice with the imaginary vector potential 2iφ. Considering the spin-up and spin-down channels, the effective imaginary field with vector potential is −2iφ applying the lattice. The effective imaginary fields are canceled. It indicates that the non-Herimitian skin effect disappears and the wave vector is a good quantum number in the partner model in Eq. (12).
A non-Hermitian SSH model in a domain-wall configuration on a ring has been investigated numerically in Ref. [65]. When the length of the two chains is large, the domain can be taken as a perturbation. In particular, the coupling of the domain states can also be neglected. The energy spectra of the domain-wall configuration on the ring can be well approximated by that on a chain. A non-Hermitian SSH model in a domain-wall configuration on a chain is pictorially shown in Fig. 3. The Hamiltonian can be written as where here α = (L, R) denotes the left or right bulk. a † j (a j ) and b † j (b j ) are the creation (annihilation) operators for the sublattice sites a and b on the j-th unit cell. The left and right bulks have different parameters and contain N L and N R unit cells respectively. The non-Hermicity of the system is from the difference intra-cell hopping with phase factor of amplification/attenuation e ±φ α . The non-Hermitian SSH model has also chiral symmetry ΓHΓ −1 = −H, with the chiral-symmetry operator Γ = N L +N R j=1 (a † j a j − b † j b j ). To understand the topological properties of the model, we do the following transformation to the Hamiltonian in Eq. (16): The transformation matrix S = S 1 S 2 . S 1 and S 2 are the diagonal matrices. The diagonal elements of S 1 are seted as 1, r, r, r 2 , r 2 , · · · , r N L −1 , r N L −1 , r N L , r N R , r N R −1 , r N R −1 , · · · , r 2 , r 2 , r, r, 1 with r = e (φ R −φ L )/2 . After the similarity transformation S 1 , we have the same amplification/attenuation e ±φ of phase factor in asymmetric couplings of the two bulks, here φ = (φ R + φ L ) /2. Then further doing the similarity transformation S 2 with a diagonal matrix whose diagonal elements are selected as 1, r, r, r 2 , r 2 , · · · , r N L , r N L , · · · , r N L +N R −1 , r N L +N R −1 , r N L +N R , we get the Hermitian Hamiltonian of the two bulks connecting in a chain configuration When the two bulks connecting in a ring, we get the partner model of non-Hermitian SSH model in domain configuration studied in Ref. [65]. The Bloch winding number of this model is the winding number different ν R − ν L of the two bulks. The transition points of the two bulks are t L = t 2 and t R = t 2 .
Another type of defect is impurity which can vary the forward and backward scattering amplitude of the continuous states and induce gap bound states. The gap bound states are exponentially localized at the impurity [66,67]. A hard-wall boundary is a special impurity which is equivalent to the infinity impurity strength and the forward scattering is forbidden. The single-impurity problem in the non-Hermitian SSH model is shown in Ref. [68].
According to bulk-boundary correspondence, when the lattice is long enough, the impurity state on the dimer ring can be approximated by that on the long chain since the boundary scattering can be regarded as a perturbation. The Hamiltonian of impurity problem is described by where v is the strength of the impurity potential and ψ † = a † −L/2 , b † −L/2 , a † −L/2+1 , b † −L/2+1 · ··, a † L/2 , b † L/2 with lattice length L. The non-Hermitian SSH Hamiltonian Here te ±φ is the right (left) intra-hopping amplitude and t ′ is the inter-hopping amplitude. The partner of the impurity model can be obtained by a conventional similarity transformationh ip = S −1 h ip S with a diagonal matrix S = diag{1, r, r, r 2 , r 2 , · · · , r L/2 , r L/2 }, here r = e φ . After this transformation. H NS S H becomes the standard SSH model with the intra-hopping amplitude and the inter-hopping amplitude t ′ . The wavefunction |ψ = S −1 |ψ and the the strength of the impurity potential v remains unchanged. The transition points are t = t ′ .

V. SUMMARY
In summary, we have proposed a way to construct a counterpart of non-Hermitian SSH model. The kernel of this method is to find the effectively imaginary gauge field referring to the non-Hermitian skin effect under PBC. Under the OBC, the imaginary magnetic flux vanishes and the OBC spectra is used to approximate the PBC spectra. The corre-sponding Hamiltonian is the counterpart of the original model. In fact, the partner model can be obtained by a similarity transformation. Due to the skin effect is eliminated from the wave vector, we can study the topological equivalent model in conventional Bloch space. Our work gives an alternating method to study the non-Hermitian SSH model with its counterpart. Several non-Hermitian SSH models are used to illustrate the method and phase transition points are given in analytic form. This method is expected to be used to construct the counterpart of a class of non-Hermitian model. In view of the nonreciprocal hopping, the similarity transformations and the effective imaginary gauge fields may depend on model details without any general rules. Not all of the non-Hermitian models may have the partner models constructed by this method, for example, the models proposed in Refs. [30,32,34]. Finding their counterparts is still a challenging subject.