Topological phase in a nonreciprocal Kitaev chain

We systematically investigate the nonreciprocal Kitaev chain, where the nonreciprocity arises from the hopping amplitude and pairing strength. By studying the Hamiltonians under three different bases, we reveal that the nonreciprocal hopping amplitude cannot induce a topological phase transition, but can result in the complex energy spectrum and non-Hermitian skin effect. Moreover, the Majorana zero energy edge modes, which are robust against the nonreciprocal hopping amplitude, exist stably in the topologically nontrivial phase. On the other hand, the nonreciprocal pairing strength can trigger a topological phase transition, which is associated with the pseudo-Hermitian symmetry breaking. More interestingly, we observe that the exceptional points independent of the topological phase can be determined by the dispersion relation, and there is no non-Hermitian skin effect in the system. Furthermore, we calculate the topological invariant to demonstrate the validity of the bulk-edge correspondence in the pseudo-Hermitian symmetry-unbroken region. Our investigation provides a path to explore the fundamental physics pertaining to the interplay between nonreciprocity and topology in the non-Hermitian topological superconductors.


Introduction
The simplest model of topological superconductors, introduced first by Kitaev [1], is the mean Bogoliubov-de Gennes (BdG) Hamiltonian of spinless fermions in one dimension (1D).In general, spinless fermions are regarded as a toy model for the more complicated spinful case or fermions spin-polarized fully.Topological superconductors, as a nascent state of matter, have significantly captured the interest of researchers in recent years, notably the Kitaev model.The open chain of this model exhibits two physical regimes, topologically trivial and nontrivial phases, which are determined by the absence and presence of Majorana zero energy modes localized at the edges of chain [2][3][4].In addition to the original Kitaev model, alternative versions such as bosonic and ladder models have been proposed [5][6][7][8], revealing a range of intriguing physical phenomena.On the other hand, the robustness of the Majorana zero energy modes against disorders and perturbations theoretically makes them crucial in the topological quantum fault-tolerant calculations [9][10][11].Moreover, the experimental observation of Majorana zero energy modes in some materials [12][13][14][15][16][17][18] has further highlighted the potential for practical applications in topological superconductors.
Due to dual focus on topological superconductors and non-Hermiticity, there has been significant interest in the non-Hermitian Kitaev models [61][62][63][64][65][66].The non-Hermiticity of these models can be fully added to the on-site chemical potential in various ways.Zeng et al [64] demonstrated that topologically nontrivial phase region becomes narrower when imaginary potentials are added to all lattice sites.However, the Majorana bound states always exist stably when imaginary potentials are only added to the two ends of the chain.In the case of the non-Hermitian Kitaev model with PT -symmetry, a topologically trivial region and two different nontrivial regions are observed [65].It is also clarified that PT -symmetry breaking is independent of the topological phase [66].The nonreciprocal Kitaev chain with nonreciprocal hopping amplitude and pairing strength is an interesting system to study its non-Hermitian and nonreciprocal nature.It is expected that the nonreciprocal hopping amplitude and pairing strength will break the symmetries and lead to a topological phase transition.However, the topological properties are not yet fully understood.One possible proposal to studying the topological properties of a Kitaev chain with imbalanced pairing terms [67] is to use the extended Zak phase, which has been defined in terms of the biorthonormal inner product under the influence of the conditional imbalance.Another proposal is to investigate the defective Majorana zero modes and number-anomalous bulk-boundary correspondence in the the dimerized non-Hermitian Kitaev chain [68].Overall, the nonreciprocal Kitaev chain with nonreciprocal hopping amplitude and pairing strength is promising system for studying the interplay between nonreciprocity and topology.Further research is necessary to fully comprehend the topological properties of the system and its potential applications in topological quantum computing.
In this paper, we focus on unraveling the effects of nonreciprocal hopping amplitude and pairing strength on the Kitaev chain.We find that the energy spectrum of the system become complex upon the introduction of nonreciprocal hopping amplitude, which is also readily obtained from the dispersion relation.Remarkably, we highlight the presence of a non-Hermitian skin effect and absence of any topological phase transition as the nonreciprocal hopping amplitude increases.Moreover, the Majorana zero energy edge modes, which are robust to the nonreciprocal hopping amplitude, exist stably in the topologically nontrivial region.On the other hand, within the nonreciprocal Kitaev chain featuring exclusively nonreciprocal pairing strength, we observe that the system undergoes a topological phase transition, accompanied by the breakdown of the pseudo-Hermitian symmetry.Notably, the breakdown of the pseudo-Hermitian symmetry is independent of the topological phase, where corresponding exceptional points can be deduced from the dispersion relation.Furthermore, it is essential to highlight that the non-Hermitian skin effect is invalid due to the inherent equilibrium within the overall system under the Majorana basis.In this framework, the next-nearest-neighbor hopping amplitudes between two distinct types of Majorana fermions are opposite, resulting in the absence of any generated magnetic flux.In the meanwhile, we observe that the Majorana zero energy edge modes also exist in an open chain, and the corresponding topological invariant is calculated in the pseudo-Hermitian symmetry-unbroken region, demonstrating the bulk-edge correspondence relationship in such a nonreciprocal Kitaev chain.
The paper is organized as follows: in section 2, we derive the Hamiltonian of the nonreciprocal Kitaev chain with nonreciprocal hopping amplitude and pairing strength, obtain the dispersion relation of the energy spectrum, analyze the symmetries of the system, and illustrate the schematic diagram in the Majorana basis.In section 3, we first briefly review the concepts of the inverse participation ratio (IPR) and fractal dimension, and then display the effects of the nonreciprocal nearest-neighbor hopping amplitude and pairing strength on the nonreciprocal Kitaev chain, respectively.In section 4, we calculate the topological invariant in the pseudo-Hermitian symmetry-unbroken region, i.e. the winding number.Finally, a conclusion is given in section 5.

System and Hamiltonian
We explore a 1D p-wave superconducting Kitaev chain with nonreciprocal hopping amplitude and pairing strength, as illustrated in figure 1(a).The fermionic Hamiltonian is given by (h = 1) where a † j (a j ) is the creation (annihilation) operator of the spinless fermion at the jth lattice site, t 1,2 (∆ 1,2 ) describes the nearest-neighbor nonreciprocal hopping amplitude (p-wave superconducting pairing strength), µ is the on-site chemical potential, and N is the size of the system.We define a N+1 = 0 (a N+1 = a 1 ) for an open (a closed) chain.For simplicity, we set describes the constant nearest-neighbor hopping amplitude (pairing strength) and is set as the unit of energy.η and γ denote the nonreciprocity of the nearest-neighbor hopping amplitude and p-wave superconducting pairing strength.Obviously, the non-Hermiticity of the system arises from both the hopping amplitude and pairing strength rather than the on-site chemical potential.Without loss of generality, the parameters involved are real numbers.The Hamiltonian in equation ( 1) can be physically implemented via the optomechanical system [69,70].
Due to the particle-hole symmetry (PHS), the Hamiltonian (1) can be diagonalized by the BdG transformation where ψ † n denotes the BdG operator, n (n = 1, 2, • • • , N) is the index of energy level, and two complex coefficients u n,j and v n,j are the components of wave functions.The BdG Hamiltonian in the Nambu space can be written as where the matrix form of the core Hamiltonian H BdG is given by with and for the system under periodic boundary condition (PBC), while for the system under open boundary condition (OBC), Obviously, H BdG is a 2N × 2N matrix.Besides, we introduce the wave function in the nth energy band index to derive the bulk BdG equations Therefore, the energy spectrum and state distribution at the jth lattice site of the system can be directly obtained through the above equations.Furthermore, we derive the Hamiltonian in the momentum space to obtain the dispersion relation of the energy spectrum and boundary conditions of the topological phase transition.By performing the Fourier transformations we can acquire the BdG Hamiltonian T is the basis in the Nambu space and the core Hamiltonian H(k) is presented as with According to the secular equation of the system, the dispersion relation can be expressed as Now we focus on the phase boundary in the thermodynamic limit, which is determined by the equation The energy spectrum is complex if η ̸ = 0, γ = 0, and then becomes purely real or imaginary when η = 0, γ ̸ = 0.For the more general case when neither η nor γ is zero, the energy spectrum can exhibit purely real, purely imaginary, zero, and complex values (see appendix for details).
Before discussing reduced model in the Majorana basis, we first focus on the symmetry of the system according to [38,39,46].The Hamiltonian in equation ( 11) can be written in the compact form where σ 0 is the identity matrix and σ x,y,z denote the Pauli matrices.The coefficients satisfy h 0 = −2iη sin k, h x = −2iγ sin k, h y = 2∆ sin k, and h z = 2t cos k + µ.The system has the time-reversal symmetry (TRS) defined as a complex conjugation operator T H(k . Also, the system naturally possesses the chiral symmetry (CS) S = T C. Thus, the topological invariant of the system is characterized by Z index.We further introduce the Majorana basis to rewrite equation (1).In general, the Dirac fermion operator a j can be written as the combination of two Majorana fermion operators b j and c j , which is true since the antiparticle of the Majorana fermion is itself.The Majorana fermion satisfies the anticommutation relation {α j , β j ′ } = 2δ α,β δ j,j ′ (α, β = b/c).Then the Hamiltonian in equation ( 1) is rewritten as The schematic diagram according to equation ( 15) is shown in figure 1(b).The nonreciprocal Kitaev chain can be considered as two coupled Su-Schrieffer-Heeger (SSH) chains [71,72] with the next-nearest-neighbor hopping amplitude.The system features two different types of Majorana fermions b and c, represented by orange and red circles, respectively.The next-nearest-neighbor hopping amplitudes between Majorana fermions of the same type (orange for b, red for c) are denoted by η + γ (for b) and η − γ (for c), which are shown as solid orange and red lines, respectively.The alternating nearest-neighbor hopping amplitudes between different types of Majorana fermions, i.e. between b and c for different sites, are shown as black solid and green dashed lines, corresponding to i(t − ∆) and i(t + ∆), respectively.Finally, the next-nearest-neighbor hopping amplitudes, denoted by iµ, are shown as solid blue lines for the same site.

The IPR and fractal dimension
Before discussing the topological properties of the system under different regimes, we briefly review the concepts of the IPR and fractal dimension of the wave function [73][74][75].It is well known that the IPR and fractal dimension are usually used to quantify the localization-delocalization transition.Here we utilize these diagnostic tools to intuitively distinguish whether the non-Hermitian skin effect exists or not.For any given wave function of the system, the IPR of the nth eigenstate is defined as which measures the reciprocal of the number of sites occupied by particles, rather than the number of sites occupied by particles.For a localized state, the IPR approaches 1 since only finite number of sites are occupied, whereas it goes to 0 for an extended state in thermodynamic limit.Another tool, the fractal dimension of the wave function, is described as Contrary to the IPR, the fractal dimension Γ approaches 1 for an extended state, while Γ goes to 0 for a localized state in thermodynamic limit.

The nonreciprocal Kitaev model with only nonreciprocal hopping amplitude
We numerically solve equation ( 4) to obtain the topological properties of the nonreciprocal Kitaev chain when η ̸ = 0 and γ = 0, as shown in figure 2. Firstly, we present the real and imaginary parts of the energy spectrum, as well as the fractal dimension Γ as a function of µ when η = 1 in figures 2(a) and (b).We find that the present system supports topologically nontrivial and trivial phases, separated by two topological phase transition points at µ = ±2, as shown in figure 2(a).Notably, except for µ = 0 and two topological phase transition points, we observe that both the edge and bulk states are localized due to the fractal dimension Γ → 0, which reflects the non-Hermitian skin effect unique to the non-Hermitian system.Moreover, the imaginary part of the energy spectrum always exists throughout the entire interval of µ, as depicted in figure 2(b).In figures 2(c) and (d), we illustrate the real and imaginary parts of the energy spectrum, as well as the fractal dimension Γ as a function of η in the topologically nontrivial region (µ = 0.5).It is indicated that the degenerate Majorana zero energy modes persist across the entire range of η, as shown in figure 2(c), implying that the nonreciprocal hopping amplitude η cannot induce a topological phase transition.Furthermore, as illustrated in figure 2(d), the imaginary part of the energy spectrum appears immediately as long as η ̸ = 0, which aligns with the previous analytic discussions on equation (12).
To further demonstrate the fate of Majorana zero energy modes and topological phase transition, we present the minimum absolute eigenvalues of the energy spectrum under OBC and the energy band gap under PBC in the µ-η plane, as shown in figures 2(e) and (f), respectively.We observe that the degenerate Majorana zero energy edge modes always appear (disappear) in the topologically nontrivial (trivial) region between the two white dashed lines, independent of the nonreciprocal hopping amplitude η, as shown in figure 2(e).This provides further evidence that η cannot induce a topological phase transition.In figure 2(f), one can see that there is an opening-closing-reopening process of the energy band gap marked by the two white dashed lines at µ = ±2 for all η, which indicates that the system undergoes a topological phase transition from the topologically trivial phase to nontrivial phase, and then back to the trivial phase.In other  words, the nonreciprocal hopping amplitude η does not alter the topological phase or phase transition points of the system.
To gain a deeper understanding of the physical mechanisms, we also analyze the real and imaginary parts of the energy spectrum in the momentum space, as depicted in figures 3(a) and (b).We observe that the energy spectrum is purely real when either η = 0 or k = mπ, and complex otherwise, which is in agreement with equation (12).Furthermore, the schematic diagram of the nonreciprocal Kitaev chain in the Majorana basis is exhibited in figure 3(c), which can be regarded as two SSH lattice chains coupled to each other with the next-nearest-neighbor hopping amplitude η.This coupling makes the whole system unbalanced, resulting in the non-zero magnetic flux, which leads to the appearance of the non-Hermitian skin effect and complex energy spectrum.Interestingly, we note that the appearance of the exceptional points is independent of the topological phase, and is solely determined by the dispersion relation of equation (12).Moreover, we observe that the system does not exhibit the non-Hermitian skin effect due to fractal dimension of the bulk state Γ → 1, which can be explained by the symmetries and Majorana picture without considering it temporarily.We plot the minimum absolute eigenvalues of the energy spectrum under OBC in the µ-γ plane in figure 4(e) and show that the system supports stable degenerate Majorana zero energy modes within the white dashed box when |γ| ⩽ √ ∆ 2 + µ 2 /4 and |µ| ⩽ 2. Furthermore, as depicted in figure 4(f), we present the energy band gap under PBC in the µ-γ plane, which shows the topological phase transitions occurring at µ = ±2 and γ = ± √ ∆ 2 + µ 2 /4 due to the fact that the energy band gap marked by white dashed box undergoes a opening-closing-reopening process.Therefore, we conclude that the nonreciprocal pairing strength γ can induce a topological phase transition, and despite the non-Hermitian nature of the current system, it does not exhibit the non-Hermitian skin effect.
In addition, we calculate the real and imaginary parts of the energy spectrum in the momentum space, as shown in figures 5(a) and (b).It is worth noting that the energy spectrum in the present system is purely real or imaginary, owing to presence of the pseudo-Hermitian symmetries.Specifically, we find that the purely real energy spectrum occurs either at k = mπ or when |γ| ⩽ √ ∆ 2 + µ 2 /4, and the purely imaginary energy spectrum otherwise.To understand the results more intuitively, we illustrate the schematic diagram in the Majorana basis, as depicted in figure 5(c).Here, the next-nearest-neighbor hopping amplitudes between one type of Majorana fermions b are represented by gray solid lines with γ, while those between the other type of Majorana fermions c are denoted by gray dashed lines with −γ.We consider a unit cell consisting of four Majorana fermions, which is highlighted by the red dotted box.This configuration leads to zero magnetic flux in the panel, similar to a PT -symmetric system, indicating that the present system is balanced and does not exhibit non-Hermitian skin effect.

Topological invariant
In this section, we focus on studying the nonreciprocal Kitaev chain with only the nonreciprocal nearest-neighbor pairing strength, and its corresponding Hamiltonian is given by The above equation satisfies TRS, PHS, and CS.Moreover, equation ( 18) also has the pseudo-Hermitian symmetry, defined as ξ = σ x T , which satisfies and possesses the following Hermitian counterpart where hy = 2 √ ∆ 2 − γ 2 sin k and hz = h z .It is convenient for computing the energy spectrum and topological invariant, we can convert it into an off-diagonal matrix form as by performing the unitary transformation with hx = − hz , hy = hy , and U is a unitary operator where Here, β plays the role of a momentum-like variable, but unlike conventional Bloch theory, it is not restricted to the Brillouin zone.In the non-Hermitian systems, the non-Bloch band theory and the generalized Brillouin zone (GBZ) are used to describe the topological properties of the systems.GBZ can be constructed by |β 2 (E)| = |β 3 (E)| for given E in the system, then the continuum bands can be calculated.The winding number [19,38,40,46,49] can be written as where l and E b are any closed loop and reference base energy independent of the energy spectrum of the system, respectively.The simplified expression of w can be described by where N z (±) and N p (±) represent the number of zero and pole point of R ± (β) inside the GBZ, respectively.It is indeed that number of pole point of R ± (β) is always 1 according to equation ( 21), i.e.N p (±) = 1.Therefore, the topological invariant of the system can be directly determined by solving the number of zero point of R ± (β), as depicted in figure 6.From figures 6(a) and (b), it is evident that the number of zero point of R + (β) is always 2, while of R − (β) is 0. Thus, the system belongs to the topologically nontrivial phase characterized by the winding number w = −1 in the pseudo-Hermitian symmetry-unbroken region, where purely real energy spectrum and two degenerate Majorana zero energy modes are observed.It indicates the preservation of the bulk-edge correspondence relation.Figure 6(c) illustrates that some eigenvalues of R ± (β) are real and all lie above the GBZ at the critical point.In the pseudo-Hermitian symmetry-broken region, the eigenvalues of R ± (β) are complex and also lie above the GBZ, as shown in figure 6(d).

Conclusions
In conclusion, we have investigated the nonreciprocal Kitaev chain along with nearest-neighbor nonreciprocal hopping amplitude and pairing strength from different aspects.For the nonreciprocal Kitaev chain with only the nonreciprocal hopping amplitude, we observe the non-Hermitian skin effect, the stable existence of degenerate Majorana zero energy modes in topologically nontrivial region, and the absence of a topological phase transition as the nonreciprocity of hopping amplitude increases.Moreover, we also find that the energy spectrum immediately change from real to complex upon the introduction of nonreciprocal hopping amplitude.However, in the presence of nonreciprocal pairing strength, we uncover that the pseudo-Hermitian symmetry is broken during the topological phase transition, and the exceptional points determined by the dispersion relation are independent of the topological phase.Furthermore, we demonstrate the failure of the non-Hermitian skin effect, and calculate the topological invariant to illustrate the bulk-edge correspondence relation in the pseudo-Hermitian symmetry-unbroken region.Our work provides a more comprehensive exploration of the topological properties in the nonreciprocal Kitaev chain and enhances the understanding of the interplay between nonreciprocity and topological phases.

Figure 1 .
Figure 1.Schematic diagram of the 1D p-wave superconductor Kitaev chain with nonreciprocal hopping amplitude and pairing strength.(a) In real space.The pink circles denote spinless fermion with the chemical potential µ.The blue dashed (green solid) arrows indicate the rightward (leftward) nearest-neighbor interactions including hopping amplitude t1 (t2) and p-wave superconducting pairing strength ∆1 (∆2), respectively.(b) In the Majorana basis.The orange and red circles represent two different kinds of the Majorana fermions b and c, respectively.

Figure 2 .
Figure 2. (a) The real and (b) imaginary parts of the energy spectrum, as well as the fractal dimension Γ of the system as a function of µ when η = 1 under OBC.(c) The real and (d) imaginary parts of the energy spectrum, as well as the fractal dimension Γ of the system as a function of η in the topologically nontrivial region (µ = 0.5).(e) The minimum values of |E| under OBC and (f) the energy band gap under PBC in the µ-η plane.The white dashed lines represent the phase boundary.The rest parameters are set as t = 1, ∆ = 0.8, γ = 0, and N = 100.

Figure 3 .
Figure 3. (a) The real part of the energy spectrum as a function of k and η when γ = 0 and µ = 0.5 in the momentum space.(b) The imaginary part of the energy spectrum in the k-η plane.(c) The schematic diagram in the Majorana basis.The next-nearest-neighbor hopping amplitudes between the same types of Majorana fermions b or c both are η tagged as purple solid lines.The red dotted box represents a unit cell.

Figure 4 .
Figure 4. (a) The real and (b) imaginary parts of the energy spectrum, as well as the fractal dimension as a function of γ in topologically trivial region (µ = 2.5).(c) The real and (d) imaginary parts of the energy spectrum, as well as the fractal dimension of in the topologically nontrivial region (µ = 0.5).(e) The minimum values of |E| under OBC and (f) the energy band gap under PBC in the µ-γ plane.The white dashed box represents the phase boundary.The rest parameters are set as t = 1, ∆ = 0.8, η = 0, and N = 100.

3. 3 .
The nonreciprocal Kitaev model with only nonreciprocal pairing strength In this section, we present the numerical results for the nonreciprocal Kitaev chain with only nonreciprocal pairing strength, as illustrated in figure 4. In figures 4(a)-(d), we exhibit the real and imaginary parts of the energy spectrum and the fractal dimension Γ as a function of the nonreciprocal pairing strength γ in two different topological phases.In the topologically trivial regime (µ = 2.5), we observe two exceptional points at γ = ± √ 2.205, as depicted in figures 4(a) and (b).While in the topologically nontrivial regime (µ = 0.5), we find two exceptional points at γ = ± √ 2.81/2, as shown in figures 4(c) and (d).

Figure 5 .
Figure 5. (a) The real part of the energy spectrum as a function of k and γ when η = 0 and µ = 0.5 in the momentum space.(b) The imaginary part of the energy spectrum in the k-γ plane.(c) The schematic diagram of the nonreciprocal Kitaev model with only the nearest-neighbor nonreciprocal pairing strength in the Majorana basis.The next-nearest-neighbor hopping amplitudes between the same types of Majorana fermions b (c) are γ (−γ), which are marked by gray solid (dashed) lines.The red dotted box represents a unit cell.

Figure 7 .
Figure 7.The real and imaginary parts of the energy spectrum as a function of η and γ for different k in the momentum space.(a) and (b) k = 0. (c) and (d) k = 0.5π.The other parameters are set as t = 1, ∆ = 0.8, and µ = 0.5.