Topological semimetal phase in non-Hermitian Su–Schrieffer–Heeger model

We explore the non-Hermitian Su–Schrieffer–Heeger model with long-range hopping and off-diagonal disorders. In the non-Hermitian clean limit, we find that the phase diagram holds topological semimetal phase with exceptional points except the normal insulator phase and the topological insulator phase. Interestingly, it is found that the topological semimetal phase is induced by long-range nonreciprocal term when the long-range hopping is not equal to the intercell hopping. Especially, we show the existence of topological semimetal phase with exceptional points and determine the transition point analytically and numerically under the Hermitian clean limit when the long-range hopping is equal to the intercell hopping. Furthermore, we also investigate the effects of the disorders on topological semimetal phase, and show that the disorders can enhance the region of topological semimetal phase in contrast to the case of non-Hermitian clean limit, indicating that it is beneficial to topological semimetal phase whether there is one disorder or two disorders in the system, that is, the topological semimetal phase is stable against the disorders in this one-dimensional non-Hermitian system. Our work provides an alternative avenue for studying topological semimetal phase in non-Hermitian lattice systems.


Introduction
As a state of matter with novel quantum properties, topological insulators (TIs) [1][2][3][4][5] have been widely explored in condensed-matter physics and various engineered systems [6][7][8][9][10][11][12].Topological insulators are generally topologically characterized by periodical bulk topological invariant and gapless edge states under open boundary conditions (OBCs).Most of the earlier study on TIs are mainly focused on Hermitian system.Recently, the non-Hermitian system has gradually come to focus for their intriguing properties and potential applications [13][14][15][16][17][18].The Su-Schrieffer-Heeger (SSH) model system [19] which has simplest structure but rich topological properties provides a possible platform for the study of TIs and has been explored widely in the non-Hermitian systems [20][21][22][23][24][25][26].Different from Hermitian systems, non-Hermitian systems exhibit some peculiar phenomena.The most important is the breakdown of conventional bulk-boundary correspondence, i.e. edge states can no longer be characterized by the topological invariants defined by the Bloch Hamiltonian.To be more precise, the systems under periodic and open boundary conditions display remarkably different eigenspectra and eigenstates [21].Actually, this difference can be attributed to the non-Hermitian skin effect, which sketches the originally extended bulk states are now localized toward one of the system boundary [21,22].To address this issue, the generalized Brillouin zone (GBZ) is brought up to recover the bulk boundary correspondence in the non-Hermitian systems [21][22][23].
And it has been find out that there are non-Hermitian skin effect and the failure of conventional bulk-boundary correspondence as long as GBZ and Brillouin zone (BZ) do not coincide [21,25].In addition, the topological semimetal (TSM) phase with exceptional points, in one-dimensional (1D) non-Hermitian system, is caused by the unique characteristics of the GBZ [27].
Besides the non-Hermitian topological lattice systems, the non-Hermitian disordered systems also have drawn significant interests recently.Different from the non-Hermitian topological lattice system with the well-defined invariant in GBZ, the periodical invariant in non-Hermitian disordered topological system is usually ill-defined.And the topological invariants defined in real space provides the available pathway for non-Hermitian disordered system [28].Generally, disorders have a complicated influence on the topological systems [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47], e.g.disorder can cause metal-insulator transition and create the extended edge state [29], strong disorder can cause a sharp change of winding number ν and it marks a critical point where the localization length diverges [33], and a topological phase which is dubbed non-Hermitian topological Anderson insulators induced by the combination of non-Hermiticity and disorders [46].It is noted that the investigations mainly focus on the influence of disorder on the TI phase.However, the TSM phase in 1D non-Hermitian disordered systems and the effects of different kinds of disorders on the TSM phase are rarely explored.
In this paper, we investigate the TSM phase in a 1D non-Hermitian SSH model with long-range hopping and off-diagonal disorders.We employ the real-space winding number and the topological invariant with respect to the GBZ to characterize its topology.In the non-Hermitian clean limit, the phase diagram is composed by the normal insulator (NI) phase, TI phase, and TSM phase with exceptional points.We reveal that the TSM phase is caused by the long-range nonreciprocal term when the long-range hopping is not equal to the intercell hopping.Furthermore, we also demonstrate the presence of TSM phase and determine the transition points analytically and numerically when the long-range hopping is equal to the intercell hopping in the Hermitian clean limit.Surprisingly, we find that the disorders can enlarge the region of the TSM phase in contrast to the case of non-Hermitian clean limit, and it is conducive to the TSM phase whether there is a single disorder or two disorders in the system.The region of the TSM phase increases with the increase of the disorder strength, showing that the TSM phase is stable against the disorders in this 1D non-Hermitian system.
The paper is organized as follows: in section 2, we describe the non-Hermitian SSH model and introduce the topological invariant.In section 3, we find that the TSM phase is the phase possessing exceptional points and is induced by long-range nonreciprocal term in the non-Hermitian clean limit.Moreover, we observe the presence of the TSM phase with exceptional points in the Hermitian case and determine the transition point analytically and numerically.Furthermore, we study the effect of disorders on the TSM phase in section 4. Finally, a conclusion is given in section 5.

Model and topological invariant
We consider a usual SSH model formed by two sublattices A and B in each unit cell, as shown in figure 1.The Hamiltonian of the system is written as where ) creates a particle on the sublattice site A (B) in the n-th lattice cell, and C A,n (C B,n ) is the corresponding annihilation operator.T n 1 , T n 2 , and T n 3 characterize the intracell, intercell, and long-range hopping strengths, respectively.γ 1 , γ 2 , and γ 3 are the nonreciprocal terms.The Hamiltonian H in equation ( 1) fulfills the chiral symmetry with VHV −1 = −H, where the chiral operator is defined as We consider the hopping terms as where W 1 , W 2 , and W 3 represent the corresponding disorder strengths, ω n 1 , ω n 2 , and ω n 3 are the uncorrelated random numbers chosen uniformly in the range [−1/2, 1/2].In the clean limit, we can characterize the topological phase by using non-Bloch topological invariant and the corresponding zero energy edge modes in this SSH model.Moreover, the translation symmetry and the periodicity of the lattice system are broken due to the presence of disorder.In this case, the non-Bloch topological invariant is inapplicable.Therefore, we can take the real-space winding number [28] to investigate the topological properties.Given a disorder configuration denoted by s, we can obtain Q s matrix, which is expressed as where ∑ n is the sum over the eigenstates in the bulk continuous spectrum except for the edge modes.It is to say that the chain length L will be divided into three intervals with length l, L ′ , l, and ∑ n only stands the sum over the middle interval with length Then the topological invariant in real space can be defined as the following form [28]: where X is the coordinate operator and Tr ′ only means the trace over the middle interval with length L ′ .Given the disorder configuration number N s , the real-space disorder-averaged winding number for a non-Hermitian chiral-symmetric system can be expressed as

The topological semimetal phase with exceptional points under the clean limit
We first consider the case of the clean limit, i.e.W 1 = W 2 = W 3 = 0 and t 2 ̸ = t 3 .Here, the translation symmetry of the system is preserved and the real-space Hamiltonian is written as By applying the Fourier transformation, we can obtain the Hamiltonian in the momentum space as the following form: Here, the off-diagonal matrix form of generalized Bloch Hamiltonian can be expressed as with , and the trajectories of β 2 and β 3 constitute the GBZ C β .Then, the topological invariant [21,22] can be defined as From the definition in equation (11), the geometrical meaning of the topological invariant means the change of the phase of R ± (β) with β turning around the GBZ in the counterclockwise way.In addition, the chiral symmetry of the Hamiltonian is preserved in this case, and thus the real-space winding number can also be used to characterize the topology of the system according to In figure 2(a), it is surprising that the TSM phase (B region) emerges in the real-space phase diagram.Here, the real-space winding number W = 0 represents the NI phase, W = ±1 predicts the TI phase, and the gapless TSM phase is characterized by the abnormal but nonzero value.In order to demonstrate this phenomenon more clearly, we show the phase diagram in the momentum-space, including the NI phase ω = 0 (A region), the TI phase with ω = ±1 (C and D regions), and the TSM phase with the ill-defined non-Bloch winding number (B region), as shown in figure 2(b).Moreover, just as described in section 2, the Hamiltonian H in equation ( 1) has the chiral symmetry VHV −1 = −H, where the chiral operator is defined as That is to say, the chiral symmetry cannot be destroyed with the change of hopping terms or disorders strengths and the chiral symmetry VHV −1 = −H can always be satisfied.Thus, the TSM phase is protected by topology and symmetry.
The energy spectra with t 2 = 0.2, 0.75 are respectively shown in figures 2(c-1) and (c-2).The results indicate that, the touching regions of the energy bands in the spectra are exactly consistent with the phase  6) cannot be diagonalizable, thus such point called the exceptional point emerges in the TSM phase of the system, i.e. the TSM phase is the phase with the exceptional points for the system.
Additionally, based on the argument principle, the topological invariants in the momentum-space can also be expressed as [23] where It is found that the eigenenergies are doubly degenerate for the TSM phase.Furthermore, we also investigate the effect of long-range nonreciprocal term on the TSM phase.We show the phase diagram which only exists NI phase and TI phase for the case of t 3 = 0 and γ 1 = γ 2 = γ 3 = 0, as depicted in figure 4(a).Compared figure 4(b) with figure 4(a), there are still only TI phase and NI phase, however, the region of TI phase becomes larger when t 3 varies from 0 to 0.4.It is very intriguing that a rich topological phase diagram is exhibited when t 3 and γ 3 are nonzero, which contains the NI phase, TI phase, and TSM phase, as displayed in figure 4(c).In other word, the nonreciprocal long-range hopping term leads to the emergence of TSM phase when the intercell hopping is not equal to the long-range hopping.Moreover, the region of the TSM phase becomes larger when γ 3 changes from 0.1 -0.3, as shown in figures 4(c) and (d).In addition, we plot the phase diagram versus the t 2 -t 3 plane as shown in figures 4(e) and (f).The results indicate that the region of the TSM phase becomes larger when γ 3 changes from 0.1 to 0.3.
Specially, in the Hermitian case, we show the energy spectra under OBC and find that there are only TI phase and NI phase when the intercell hopping t 2 is not equal to the long-range hopping t 3 , which is coincident with figure 4(b), just as shown in figures 5(a) and (b).Meanwhile, we also find from figures 5(a) and (b) that the topological phase preserves in the region of t 2 + t 3 ⩽ t 1 when γ 1 = γ 2 = γ 3 = 0 and t 2 ̸ = t 3 , and it can be determined that the topological transition point is t 1 = t 2 + t 3 .Surprisingly, we find that the region of the TI phase increases and the bulk gap decreases gradually with the increase of the intercell hopping amplitude t 2 .Finally, the bulk gap closes when t 2 + t 3 ⩽ t 1 if t 2 = t 3 , as shown in figure 5(c).Moreover, we show that the region of the gapless phase can be enlarged when the non-reciprocal terms γ 2 = γ 3 , as shown in figure 5(d).
On the other hand, we also analytically calculate the topological transition point by the Hamiltonian in the momentum space due to the bulk-boundary correspondence.Firstly, the Hamiltonian of the system under OBC can be written as By applying Fourier transformation, the Hamiltonian in the momentum space can be described by where H k 1 can be written in the form of Pauli matrices, i.e. Here, x and σ y represent the Pauli matrices.Thus, the energy spectrum of the system can be obtained as follows One can see that the minimum value of E(k) is always 0 when t 2 + t 3 ⩽ t 1 , while not be 0 when t 2 + t 3 > t 1 .Therefore, the transition point can be determined as t 1 = t 2 + t 3 .The analytical result is consistent with the above numerical conclusion, which indicates the correctness of the bulk-boundary correspondence in the Hermitian system.Furthermore, we numerically calculate the bulk gap of the system E g = |E L+2 − E L−1 |, and the bulk gap as a function of t 1 for different long-range hopping amplitudes t 3 is displayed in figure 5(e).We find that the gapless phase remains when t 2 + t 3 ⩽ t 1 only if t 2 = t 3 .To further explore the bulk gap, we display the bulk gap E g as a function for hopping amplitudes t 3 and t 1 with t 2 = t 3 in figure 5(f).We observe that the bulk gap is close to 0 only if t 2 + t 3 ⩽ t 1 , and the white dotted line represents the critical line determined by equation t 1 = t 2 + t 3 .Similar to figure 2, the gap-closing points and BZ are shown in figure 5(g).One can find that the gap-closing points hold somewhere on BZ when the system The bulk gap Eg as a function of t1 for fixed t2 = 1 and t3 = 0.5, 0.8, 1, 1.5, 1.8 when γ1 = γ2 = γ3 = 0. (f) Eg as a function of the hopping terms t3 and t1 for t2 = t3.The white dotted line represents the critical line of the phase transition.Gap-closing points and Brillouin zone are shown in (g) with γ1 = γ2 = γ3 = 0. (g-1) t1 = 1.5, (g-2) t1 = 2.5.The red squares and asterisks (blue rhombuses and circles) express the gap-closing points.Here the lattice size is L = 160.remains gapless.That is, the TSM phase can also be characterized by exception points in the Hermitian clean limit case for this system.

The effect of disorders on the topological semimetal phase
In this section, we focus on studying the effect of disorders on the TSM phase in the non-Hermitian system.
Firstly, we only consider the long-range disorder, i.e.W 3 ̸ = 0, W 1 = W 2 = 0. Surprisingly, in addition to TI phase (disorder-averaged winding number ν = ±1) and NI phase (disorder-averaged winding number ν = 0), we also find the existence of TSM phase, and the region of TSM phase becomes a larger region when the long-range disorder strength W 3 changes from 0.5 to 3, as depicted in figures 6(a) and (b).In order to investigate the effect of long-range disorder strength W 3 on the TSM phase more clearly, the phase diagram with the variations of W 3 and t 2 is displayed in figure 6(c).We demonstrate that the TSM phase changes from a region t 2 ⩽ 0.4 to a larger one t 2 ⩽ 0.71 when W 3 varies from 0 to 3, which means that long-range disorder strength W 3 can increase the region of TSM phase.Specially, we also show the energy spectra with the fixed parameter t 3 = 0.4 and the disorder strengths W 3 = 0.5, 1.5, 2.5, respectively, as shown in Next we consider another case that the intracell disorder exists in the system, i.e.W 1 ̸ = 0, W 2 = W 3 = 0, as shown in figure 7.In figure 7(a), we show the phase diagram with the variations of intracell disorder strength W 1 and intercell hopping t 2 .One can clearly find that the region of TSM phase is enlarged when the intracell disorder strength W 1 gradually increases.
Meanwhile, we show the energy spectra with the fixed t 3 = 0.4 in a disorder configuration and with strengths W 1 = 1, 1.5, and 2.5, respectively, in figures 7(b)-(d).We find that the region of TSM phase is in agreement with the result presented in figure 7(a).Namely, the result is verified again that the intracell disorder strength W 1 can enlarge the region of TSM phase.In addition, we also can get the same results when there is only the intercell disorder in the system, i.e. the intercell disorder strength is also conducive to the TSM phase.
Finally, in figure 8, we explore the coexistent of two disorders including intercell disorder W 2 and long-range disorder W 3 .In figure 8(a), we find that the region of TSM phase changes from a region t 2 ⩽ 0.4 to a larger one t 2 ⩽ 0.98 when W 3 varies from 0 to 3 and W 2 = 1 2 W 3 .Meanwhile, compared figure 8(a) with figure 6(c), it can be seen that the additional intercell disorder can increase the region of TSM phase in contrast to the case of long-range disorder.We show the phase diagram with W 2 = W 3 in figure 8(b), and the differences between figures 8(a) and (b) indicate that the bigger intercell disorder W 2 can further enhance the region of TSM phase.We also can get the similar results for the existence of intracell disorder and intercell disorder (long-range disorder).Therefore, it is beneficial to the TSM phase whether there is only one disorder or two disorders in the present system.Moreover, the TSM phase can be enlarged due to that the non-reciprocity can be increased with increasing the disorder.Before concluding, we briefly discuss the experimental feasibility of the studied model in some artificial systems.Firstly, the SSH chains can be realized [48,49] and tunable hopping disorders can be added in electrical circuits.And a one-dimensional non-Hermitian electrical circuit lattice has been proposed [50], in which the long-range nonreciprocal hoppings can be easily achieved and the non-Hermitian effect is implemented by the operational amplifier.It is worth noting that the TSM phase can be realized in the electrical circuit lattice, i.e. the eigenstates gained from simulations are localized at the edge and remain continuous over the frequency range for the TSM phase.Moreover, the realization of non-Hermiticity and disorder have been come up in optical systems [32,51].It is found that non-reciprocity [52][53][54] and controllable disordered hoppings [10] can also be realized by cold atom systems and long-range interaction can achieved by Rydberg atoms [55].Therefore, the studied model is feasible under the current experimental conditions.

Conclusions
In conclusion, we have investigated a 1D non-Hermitian SSH model with off-diagonal disorder hopping terms.In the non-Hermitian clean limit, we find the existence of TSM phase in the phase diagram except the TI phase and NI phase, and the TSM phase can be described as the phase with exceptional points.Specifically, we demonstrate that the TSM phase is induced by long-range nonreciprocal term when the long-range hopping is not equal to the intercell hopping.We also observe the emergence of TSM phase which can also be depicted as the phase with exceptional points and determine the transition points analytically and numerically in the Hermitian clean limit case when the long-range hopping is equal to the intercell hopping.

Figure 1 .
Figure 1.Sketch of the non-Hermitian disordered SSH model.T n 1 , T n 2 , and T n 3 are hopping terms, γ1, γ2, and γ3 are the nonreciprocal terms.
and N p (R ± ) representing the number of zeros and poles of R ± (β) inside GBZ, respectively.According to equation (9), R ± (β) always possess one pole located at origin, i.e.N p (R + ) = 1, N p (R − ) = 1, which leads to that we can distinguish the topological invariant directly.From figure2(d-2) and equation(13), we can determine that the number of zeros of R + (β) and R − (β) inside GBZ is 1, i.e.N z (R + ) = N z (R − ) = 1, thus ω = 0. Similarly, according to equation (13) and figure2(g-2),N z (R + ) = 2, N z (R − ) = 0, ω = −1.Moreover, we plot the any four eigenstates as a function of the sites for the TSM phase under different parameters, which is corresponding exactly to the points d and e in figure2(b) , as shown in figures 3(a) and (b).