Multiple skin transitions in two-band non-Hermitian systems with long-range nonreciprocal hopping

Non-Hermitian skin effect (NHSE) is a prominent feature in non-Hermitian physics, leading to novel topological properties and expanding the traditional energy band theories. In this paper, we investigate a two-band non-Hermitian system in which multiple skin transitions are induced by long-range nonreciprocal hopping. The spectral winding number under periodic boundary conditions reveals the localization directions of skin states. Further, we present the analytical solution of transition points by tracing the self-intersecting points on the complex plane. Interestingly, the current system exhibits the abundant NHSEs, including the normal, W-shaped, and bipolar localization properties, which the eigenstate distributions and the generalized Brillouin zone can clearly illustrate. We also provide a phase diagram to represent the skin transition properties of the system comprehensively. Further, we demonstrate that the multimer non-Hermitian lattices also present the anomalous skin effect and multiple transitions, which occur in the region of the bulk band touching, the same as the two-band lattice. Moreover, a feasible scheme is proposed to realize the current non-Hermitian system with long-range nonreciprocal hopping by a topoelectrical circuit. This work further supplies the content of skin transitions and may help us explore more plentiful localization features in the two-band non-Hermitian systems.

A primary challenge for developing non-Hermitian band topology is the NHSE, which refers to topological edge states that will be hidden in the bulk states under open boundary conditions (OBCs) and the breakdown of the traditional bulk-boundary correspondence (BBC).Previous studies have proposed the non-Bloch topological invariants defined on the generalized Brillouin zone (GBZ) [2,5] and the biorthogonal polarization [22] on which a non-Bloch BBC is established for exploring topological edge states in non-Hermitian systems.Very recently, the unique point-gap band topology in the non-Hermitian systems is regarded as the topological origin of NHSE [4,23,24].The point gaps define energy band structures that form a closed loop with nonzero area on the complex plane.The closed loops support nonzero windings topology, and a similar BBC correspondence relationship between the emergence of skin states under OBC and the complex energy winding under periodic boundary conditions (PBCs) was established.In addition, the NHSE has been successfully observed in various experimental platforms, such as optical ring resonators [25], superconducting quantum system [26], and topoelectrical circuits [27][28][29][30].
Over an extended period, one simply believes that the localization direction of skin states is determined by the strength of nonreciprocal hopping [31][32][33][34].However, the emergence of bipolar [35,36] and V-shaped NHSE [37] vividly defy this intuitive notion, which direction is according to the complex nontrivial point-gap topology.These interesting point-gap structures lead to multiple phase transitions of NHSE and anomalous localization features, enabled by introducing long-range hopping.Until now, most studies on multiple skin transitions have focused on one-band systems with long-range or next-nearest-neighbor hopping [37,38].However, this transition in two-band non-Hermitian systems remains largely unexplored.
In this paper, we investigate the multiple skin transitions in a two-band non-Hermitian system with long-range nonreciprocal hopping.As the hopping parameter changes, the current system displays rich NHSEs, including the normal, W-shaped, and bipolar NHSE, where the localization direction of the skin states can be revealed by complex point-gap band topology.Furthermore, we analytically determine the critical point of the phase transition by tracing the self-intersecting points on the complex plane.The eigenenergy spectrum with a directional inverse participation ratio shows consistency with the results of multiple transitions, we employ the eigenstate distributions and the GBZ method to characterize these intriguing phenomena.Moreover, we identify the anomalous NHSE in multimer non-Hermitian lattices and demonstrate that multiple skin transitions still exist.Especially, such transitions always occur in the contact regions between bulk bands.Finally, we present a feasible method to simulate the current non-Hermitian system with long-range nonreciprocal hopping by a topoelectric circuit composed of inductors, capacitors, and operational amplifiers.This work further supplies the content of skin transitions and may help us explore novel localization features in the two-band non-Hermitian systems.
This paper is organized as follows: In section 2, we give the model and Hamiltonian of the two-band non-Hermitian system.In section 3, we investigate the skin transitions induced by the long-range nonreciprocal and present the analytical solutions of transition points.In section 4, we analyze the normal and anomalous skin effect of the system and present an NHSE phase diagram.Furthermore, we investigate the multiple skin transitions in multimer lattice.In section 5, we propose a feasible method to construct a two-band non-Hermitian system by using topoelectric circuit.Finally, a conclusion is given in section 6.

Model and Hamiltonian
We consider a non-Hermitian Su-Schrieffer-Heeger (SSH) model with long-range nonreciprocal hoppings t 3 ± γ, while owning the reciprocal intracell hopping t 1 and intercell hopping t 2 , as shown in figure 1(a).Each unit cell contains two sublattices, denoted as a and b.By deformation of the graphical non-Hermitian SSH model, which can be equal to a two-leg ladder model, as shown in figure 1(b).Under PBC, the model can be described by the following Bloch Hamiltonian where σ x,y,z is the Pauli operator, γ denotes the nonreciprocal hopping parameter, and k is the Bloch wave vector.Obviously, the Hamiltonian possesses the chiral symmetry From equation (1), solving the eigenvalue equation H(k)ψ = Eψ gives the distributions of E in the entire first Brillouin zone with k ∈ [0, 2π).
According to the non-Bloch band theory, the generalized Bloch Hamiltonian H(β) can be obtained by replacing the Brillouin zone (BZ) plane e ik as the GBZ β, which is given by where The   As shown in figure 2, we plot the energy spectrum of the system as a function of t 1 under OBC with t 2 = 1, t 3 = 0.4, γ = 0.15, and L = 80, respectively.For simplicity, we take t 2 = 1 as the energy unit throughout this paper.The zero-energy edge states can be identified in the spectrum within −1.47 < t 1 < 1.47.The topologically trivial or nontrivial properties of the current system can be characterized by non-Bloch topological invariants, as reported in [35].This indicates that the BBC has been established, namely, w corresponds to the appear or not of the edge states under OBC.We know that the NHSE is a unique feature of the non-Hermitian system with nonreciprocal hopping, which refers to the localization of bulk states at the boundary of the system under OBC.To characterize the localization properties of the eigenstates, we adopt the method of the directional inverse participation ratio (dIPR), which is defined as [39,40] dIPR with where the parameter δ ∈ (0, 0.5).In this way, the positive and negative dIPR values represent the eigenstates localized at the right and left boundaries of the system, while they become extended when dIPR is close to 0. Generally, for the simplest non-Hermitian SSH model (without long-range nonreciprocal hopping), the NHSE represents that all the eigenstates are localized at the same boundary of the system, and the localization direction tends to the stronger hoppings strength.In the current system, the NHSE is no longer straightforward but exhibits anomalous behavior different from normal NHSE caused by the introduction of long-range nonreciprocal hopping.It should be noted that the nonreciprocal terms can simultaneously exist in intercell and long-range hopping, while still owning the anomalous NHSE.Therefore, the t 2 and t 3 terms are equivalent for this phenomenon.In figure 2(a), as the chiral symmetry of the eigenenergies, we consider the E > 0 side for simplicity (similar arguments apply on the E < 0 side).When , the bulk eigenstates with E ⩽ 0.58 (E ⩾ 0.58) are localized at the right (left) boundary of the system.Notably, the red dot-dashed lines (E = 0.58) in the real part divide the skin states at opposite boundaries.Here, the eigenenergies with extended eigenstates are dubbed the 'Bloch points' [35], and these lines defined by Bloch points are referred to as the 'non-Hermitian skin effect edge' [40].The direction of the skin effect reverses when the eigenenergy traverses the NHSE edge, and the Bloch energies can be described by the following equation with the restriction | − t 1 /2t 2 | ⩽ 1.One can see that the above equation is independent of t 1 .Additionally, when t 1 ∈ [−0.8, −0.96] ∪ [0.8, 0.96], the right-localized skin states are sandwiched between left-localized skin states, and the direction of skin effect undergoes two reversals.

Non-Hermitian skin transitions
In this section, we discuss the skin transition of the system induced by the long-range nonreciprocal hopping and also give the analytic expression of the transition point by tracing the self-intersecting points on the complex plane in detail.As we know, NHSE is closely connected with the point-gap topology of a non-Hermitian system, which can be characterized by the spectral winding number, described as [4, 41] where E b is the base energy.Depending on the dynamical evolution for the closed loops, i.e. the directions in which k evolves from 0 to 2π, we can relate spectral winding number W to the existence or absence of skin states.W = −1 and W = 1 denote that the base energy E b is surrounded by the PBC energy spectrum in a clockwise and counterclockwise direction, corresponding to the right and left localization of skin states, respectively.If W = 0, the base energy is not surrounded, indicating extended bulk states without NHSE.Thus, the bulk topological invariants under PBC can accurately predict the NHSE under OBC.Next, we determine the position of self-intersections (marked by black circles in figure 3(a)), which must satisfy the condition with i = 1, 2, 3 representing the self-intersections index.These equations will give three groups of solutions, where the first solution is  3(a)).
As shown in figure 3, we present the complex energy spectrum colored solid lines under PBC with t 2 = 1, t 3 = 0.4, γ = 0.15, where the black arrows and circles denote the winding direction and self-intersecting points, respectively.In figure 3(a), when t 1 = 0.7, one can see that closed curves evolve counterclockwise with k increasing from 0 to 2π on the complex plane.Obviously, the interior of the upper or lower energy band is divided into four loops, where three peripheral loops (II, III, IV) correspond to W = 1, and center loop I corresponds to W = 2, indicates that counterclockwise winding once and twice, respectively.As shown in figure 3(b), when t 1 continues increasing and attains t 1 = 0.8, the three self-intersecting points in the PBC spectrum meet together at one Bloch point denoted by a red circle.The three loops still maintain counterclockwise winding with W = 1, which implies that the skin states are localized at the left boundary of the system.When t 1 = 0.85, the two complex conjugate self-intersecting points separate, and the interior of the energy spectrum under PBC is divided again into four loops, but three peripheral loops (II, III, IV) and center loop I with W = 1 and W = −1, as shown in figure 3(c).This indicates that the skin states corresponding to the center loop transition from left to right localization.Thus, the current system undergoes a transition of NHSE, which is topological as the winding number of the PBC spectrum changes sign.The critical points for the first transition corresponds to the merging point of the self-intersections in the PBC spectrum.By solving equation E 1 = E 2 = E 3 , we obtain analytic expression as with the restriction of |t 3 /t 2 | ⩽ 1. Upon further increase of t 1 , one can see that the center loop I with W = −1 continuously expands, while the smaller peripheral loops II and III with W = 1 are gradually suppressed in figure 3(d).The critical points for the second transition corresponds to the disappearance of conjugate self-intersections points.By calculating the restriction conditions of E 2 and E 3 , we obtain analytic expression as In figure 3(f), such a structure divides the upper or lower energy band into two loops with opposite winding numbers, where the skin states are localized at both ends of the system.As shown in figures 3(g)-(i), with t 1 further increases, the clockwise winding loop I contracts and only one loop IV with counterclockwise winding remains.The critical points for the third transition corresponds to the disappearance of Bloch points.By calculating the restriction conditions of the Bloch energies, we obtain analytic expression as Notably, the Bloch point keeps stationary during evolution.To sum up, we elucidate the phase transition of NHSE and analytically determine the critical point for the current two-band non-Hermitian system.Furthermore, in order to compare the salient features of the different kinds of NHSEs, we provide spectral topology information in table 1.

Normal and anomalous NHSE
In this section, we further characterize the various skin states induced by the long-range nonreciprocal hopping in a two-band non-Hermitian system, including the normal, W-shaped, and bipolar NHSE.We present the exact phase diagram in the complete parameter space by identifying the critical point of the skin transitions.As shown in figure 4, we first demonstrate normal NHSE with t 2 = 1, t 3 = 0.4, and γ = 0.15.In figures 4(a) and (b), we plot the distribution of all eigenstates under OBC for t 1 = 0.7 and t 1 = 2.2, respectively.Clearly, one can observe the signatures of the NHSE, in which the bulk eigenstates are localized at the left boundary.Comparing the two figures, we verify that the current system possesses two topologically distinguishable phases.For t 1 = 0.7, the system exhibits a topologically nontrivial phase with zero-energy edge states.In contrast, the system is into a topologically trivial phase with t 1 = 2.2.
To further explain the NHSE, we calculate GBZ by non-Bloch Hamiltonian H(β).This method indicates that the skin effect can be determined by the size and location of GBZ relative to the BZ (unit circle).For instance, the skin states are localized at the right (left) boundary if the GBZ is outside (inside) the BZ, while the GBZ coincides with the BZ, implying that the bulk states are extended.As shown in figures 4(c) and (d), we follow the standard approach to calculate the GBZ for t 1 = 0.7 and t 1 = 2.2, respectively.We show the analytical solution of the GBZ (red curves) referred to as auxiliary generalized Brillouin zone (aGBZ) [42], and the numerical solution of the GBZ (black dots), with the BZ (blue circles) also present for comparison.For t 1 = 0.7 and t 1 = 2.2, the BZ fully encloses the GBZ (|β| < 1), leading to leftward amplification of wave propagation, the skin states show the left localization under OBC.It is worth noting that the continuity of GBZ changes saliently between these two values.The formation of cusps (green dots) in the GBZ is closely related to the non-Bloch PT symmetry, as reported in [43,44].Specifically, for the former one, the GBZ is irregular shapes at several cusps when PT symmetry is unbroken, while in the latter case, it becomes completely smooth when PT symmetry is broken.Next, we aim to analyze the anomalous localization feature of skin states with state distributions and GBZ information, as shown in figure 5. Obviously, one can see that the wave functions of skin states display a W-shaped form, implying that the direction of NHSE changes four times as eigenenergy varies in figure 5(a).On the other hand, as shown in figure 5(b), the skin states exhibit localizations from the left to the right and then back to the left boundary.This phenomenon is the so-called bipolar NHSE, where the direction of NHSE changes twice as eigenenergy varies.Figures 5(c) and (d) show the diagram of the GBZ for t 1 = 0.9 and t 1 = 1, respectively.It can be seen that the GBZ is intersected by BZ, leading to a partial region with cusps located outside the BZ (|β| > 1) and the rest inside (|β| < 1).More strikingly, there are several extended eigenstates, as if the Hermiticity is restored, which exactly corresponds to the intersections of the GBZ with the BZ (Bloch points).The result reveals that the behavior of skin states is no longer closely associated with the strength of nonreciprocal hopping.Meanwhile, the normal and anomalous (W-shaped and bipolar) NHSE can be observed in current two-band systems with long-range nonreciprocal hopping.Moreover, based on the analysis of the critical point in section 3, we exactly obtain the phase diagram on the t 1 − t 3 plane, divided into three different regions, as shown in figure 6(a).The orange, blue, and cyan regions correspond to the system owning the normal, bipolar, and W-shaped skin effect, respectively.In order to show the rich NHSEs more clearly, the partially enlarged view is shown in figure 6(b).One can see that, for |t 3 | < 1, the current system holding anomalous skin effects, including the W-shaped and bipolar localization properties.While for the case of |t 3 | > 1, a normal skin effect forms.Furthermore, the W-shaped skin effect only appears in the region of t 1 ∈ [−2, 2].Especially when t 3 = 0, the system only present the normal and bipolar skin effect.
In the standard SSH model each unit cell contains two sublattices.For comparison, for one class of extended SSH models, each unit cell has three or more sublattices, we called the SSH3 or SSHN models [45][46][47].We take non-Hermitian SSH3 model as an example to explore multiple skin phase transitions in multimer lattice.Under OBC, the real space Hamiltonian can be written as where a n , b n and c n are the annihilation operators for sublattices a, b and c in nth cell, respectively.The long-range nonreciprocal hopping is denoted as t 3 ± γ, while the reciprocal intracell and intercell hopping are represented by t 1 and t 2 , respectively.Applying the Fourier transformations, the Hamiltonian of trimer lattice in momentum space can be described as where h 1 = t 2 e −ik + (t 3 + γ)e ik and h 2 = t 2 e ik + (t 3 − γ)e −ik .The eigenvalue equation reads When k ∈ [0, 2π) ∈ R, equation ( 15) can give exactly the spectrum of Hamiltonian H SSH3 under the PBC.
As shown in figure 7, similar to the previous analysis, we first plot the energy spectrum of SSH3 model as a function of t 1 under OBC with t 2 = 1, t 3 = 0.4, γ = 0.2, and L = 40, respectively.Meanwhile, in figures 7(c) and (d), the energy spectrum of SSH4 model under the same parameters is also provided for comparison.One can see that the energy spectrum of both are clearly divided into several regions according to the positive or negative dIPR values of the eigenstates, which indicates the presence of multiple phase transitions and anomalous NHSE.Notably, we find a common feature between the trimer and tetramer lattices, where the emergence of anomalous NHSE is always accompanied by the mutual contact of energy bands.To further characterize these localized properties of skin states at different regimes, we choose t 1 = 1 and t 1 = 1.5 in SSH3 model, respectively.In figures 8(a) and (b), we show the complex energy spectrum under PBC.In figures 8(c) and (d), we plot the distribution of all eigenstates under OBC.The PBC spectrum forms closed loops with spectral winding number 1 (−1), corresponding to the OBC eigenstates localized at the left (right) end of the system.This indicates that the upper and lower energy bands show the multiple phase transitions similar to figure 3, while the middle energy band undergoes skin transition different from the former.It is important to note that such transition always occur in the contact regions between energy bands.Thus, in multimeric non-Hermitian lattices, the skin transitions will occur in the contact regions between each energy bands.
To sum up, we elucidate the multiple skin phase transitions in the multimeric non-Hermitian lattice.Such transitions are verified by other approaches including the spectral topology, eigenstates distribution and GBZ.However, we can find that SSH3 models, as well as from more complicated SSHN models, that information from the spectrum itself provides a simpler way to predict skin transitions.

Two-band non-Hermitian topoelectrical circuit lattice
In the following, we construct a non-Hermitian two-band lattice system with long-range nonreciprocal hopping by employing topoelectrical circuits, as shown in figure 9(a).Usually, the electrical circuits can be mapped by the tight-binding Hamiltonian, where the hopping strength, wave function, and eigenvalue correspond to the capacitor, voltage, and resonance frequency.To simulate the nonreciprocal hopping in our non-Hermitian system, we need to use the negative impedance converter (NIC) with current inversion [48,49].The detailed structure of the NIC is presented in figure 9(b).
According to Kirchhoff 's current law, the currents flowing through the a n and b n nodes are given by For a circuit lattice under PBC, the voltage follows the Bloch theorem V(r + R) = e ikR V(r).Kirchhoff 's law I = JV requires the net current satisfying I a = I b = 0, where I is the input currents, V is the voltages, and J is the circuit Laplacian.Thus, we can be written the Hamiltonian by J = iωH, which is given by where C, L, and ω is capacitor, inductor and circuit frequency of the system, respectively.When performing the following settings of the resonant frequency with ω = 1/ √ L 1 (C 1 + C 2 + C 3 + C NIC ) and when the variable inductors satisfy L 2 /L 1 = (C 1 + C 2 + C 3 + C NIC )/(C 1 + C 2 + C 3 − C NIC ), we can get t 1 = C 1 , t 2 = C 2 , t 3 = C 3 , γ = C NIC .To sum up, the energy spectrum is obtained from the admittance spectrum of the circuit, and the NHSE is detected by measuring the resultant voltages.We can employ a topoelectrical circuit to investigate the two-band systems with long-range nonreciprocal hopping, which provides a platform for exploring multiple transitions and more plentiful localization properties of skin states.

Conclusions
In conclusion, we have investigated the multiple skin transitions induced by the long-range nonreciprocal hopping in a two-band non-Hermitian system.We have analyzed the localization directions of skin states by the PBC spectral winding number and analytically determined the critical point of skin transition by tracing the self-intersecting points on the complex plane.Interestingly, the system exhibits rich skin effects, including the normal, W-shaped, and bipolar NHSE, which can be clearly illustrated by the eigenstate distributions.Based on the analyses of GBZ, we have presented the physical understanding of these novel localization properties.Further, a phase diagram has been provided to comprehensively represent the skin transition properties of the system.In the multimer non-Hermitian lattice, we observe the anomalous NHSE and multiple skin transitions, and find that such transitions always occur in the contact regions between bulk bands.Finally, we have proposed a feasible scheme to realize the current non-Hermitian system with long-range nonreciprocal hopping by a topoelectrical circuit.Our work further supplies the content of skin transitions and may help us to explore novel localization features in the two-band non-Hermitian systems.

Figure 1 .
Figure 1.(a) Schematic diagram of non-Hermitian SSH model with long-range nonreciprocal hopping.(b) The equivalent non-Hermitian two-leg ladder model.The dotted box represents the unit cell.

Figure 2 .
Figure 2. (a) Real and (b) imaginary parts of the eigenenergies as a function of t1 under OBC.The parameters are taken as t2 = 1, t3 = 0.4, γ = 0.15, and L = 80, respectively.The colorbar of green (dIPR < 0) and brown (dIPR > 0) regions indicate eigenstate localization at the right and left boundary, respectively.The red lines indicate the NHSE edge.

Figure 6 .
Figure 6.Phase diagram on the t1 − t3 plane with t2 = 1 and γ = 0.15.The orange region represents the system owning normal NHSE, and the blue region represents the bipolar NHSE, while the cyan region represents W-shaped NHSE.The system size is L = 80.

Figure 7 .
Figure 7. (a) Real and (b) imaginary parts of the eigenenergies of the SSH3 model as a function of t1 under OBC.(c) Real and (d) imaginary parts of the eigenenergies of the SSH4 model as a function of t1 under OBC.The colorbar of green (dIPR < 0) and brown (dIPR > 0) regions indicate eigenstate localization at the right and left boundary, respectively.The other parameters are taken as t2 = 1, t3 = 0.4, γ = 0.2, and L = 40.

Figure 9 .
Figure 9. (a) Topoelectrical circuit realization of the two-band lattice system with long-range nonreciprocal hopping, which consists of inductors (L1, L2), capacitors (C1, C2, C3), and negative impedance converter CNIC.(b) Schematic diagram of the negative impedance converter consists of capacitors (Cp, Cq) and one operational amplifier, which mimic the nonreciprocal hopping of the present tight binding model.If the current runs from left to the right, the CNIC is negative, while if the current reverses, the CNIC becomes positive.

Table 1 .
Spectral winding number W of skin states.