Gauge-dependent topology in non-reciprocal hopping systems with pseudo-Hermitian symmetry

Energy conservation is not valid in non-Hermitian systems with gain/loss or non-reciprocity, which leads to various extraordinary resonant characteristics. Compared with Hermitian systems, the intersection of non-Hermitian physics and topology generates new phases that have not been observed in condensed-matter systems before. Here, utilizing the designed two-dimensional periodical model with non-reciprocal hopping terms, we show how to obtain both the ellipse-like or hyperbolic-like spectral degeneracy, the topological boundary modes and the bulk-boundary correspondence by the protection of time-reversal symmetry and pseudo-Hermitian symmetry. Notably, the boundary modes and bulk-boundary correspondence can simultaneously appear only for specific selection of the primitive cell, and we explored the analytical solution to verify such gauge-dependent topological behaviors. Our topolectrical circuit simulation provides a flexible approach to confirm the designed properties and clarify the crucial role of pseudo-Hermiticity on the stability of a practical system. In a broader view, our findings can be compared to other platforms such as meta-surface or photonic crystals, for the purpose on the control of resonant frequency and localization properties. Non-Hermitian physics, an active topic in photonics, is also being increasingly extended to investigate the band topologies of condensed-matter systems. Here, the authors report a 2D non-Hermitian model exhibiting exceptional rings and topological boundary modes in the spectral degeneracy, they propose how to realise these features using topolectrical circuits.

N on-Hermitian systems which possess gain/loss or nonreciprocal hopping terms, have attracted great attention due to their fantastic resonant characteristics 1,2 . For example, the spectral degeneracy is always along with the reduction of eigenstates, which is the so-called exceptional point (EP) [3][4][5][6][7] , distinctive from the Dirac point or Weyl point in Hermitian systems. The intersection of non-Hermitian physics and topology can also lead to novel phenomenon, such as the non-Hermitian skin effect that all the delocalized eigenstates become localized at the boundary or the corner when the periodical boundary condition (PBC) changes to the open boundary condition (OBC) [8][9][10][11][12] , resulting in the breakdown of bulk-boundary correspondence 13 . These properties, realized in diverse systems such as mechanical metamaterials 14 , quantum walk systems 15 , photonic systems 7,16 , acoustic systems 17 , and electrical circuits [18][19][20][21][22][23] , have created opportunities to achieve significant enhancement of the sensitivity for sensors 24 ,or perfect absorption of the input wave 25 .
To investigate the spectral degeneracy characteristic in the higher dimension, recent works have extended the exceptional point to exceptional ring (ER) [26][27][28][29][30] or exceptional surface (ES) 31,32 , theoretically. In addition, when combining the non-Hermitian systems with topological behaviors, the topological invariants defined in Hermitian systems cannot predict the existence of the boundary modes any more 33 . Several methods have been proposed to solve this issue 10,[33][34][35][36][37] , but the limitation manifests as the particular precondition that point gap or line gap should exist under PBC 38 .
In this study, we utilized an uncomplicated two-dimensional (2D) non-reciprocal hopping model, and extended the spectral degeneracy characteristic into exceptional elliptical rings and exceptional hyperbolic lines. We show how to design the exceptional rings, the topological boundary modes and the bulkboundary correspondence, by the protection of time-reversal symmetry and pseudo-Hermitian symmetry. We found that the selection of the primitive cell influences the appearance of boundary modes, which is the so-called gauge-dependent topological behaviors, and can be predicted by the well defined topological invariant successfully even for the condition that no gap exists under PBC. Attentively, the skin effect and the breakdown of bulk-boundary correspondence cannot occur in our model due to the pseudo-Hermitian symmetry [38][39][40] , verified by the transfer-matrix method 10 as well. Furthermore, to extend the theoretical model in a practical system easily, we designed a 2D non-reciprocal hopping topolectrical circuit lattice, and confirmed the spectral degeneracy characteristic and topological behaviors through circuit simulation, which even can be compared to other platforms such as photonics, acoustics, mechanics and meta-surface.

Results and discussion
Non-reciprocity and spectral degeneracy. Compared with onsite gain/loss systems, systems with non-reciprocal hopping terms provide more freedom on the flexible realization of different types of spectral degeneracy in 2D non-Hermitian systems. For simplicity and without loss of generality, we started with a 2 × 2 tightbinding model described by σ x,y,z are Pauli matrices for the spin degree of freedom, and σ 0 represents the identity matrix. According to the real-space Hamiltonian described in Supplementary Note 1, ε 0 represents the on-site energy, t 1α/2α represents the reciprocal part of the hopping terms, while γ 0 , γ 1α/2α represent the non-reciprocity of intra-cell interaction and inter-cell interaction between the nearest cells, respectively. The existence of exceptional ring relies on a sufficient condition that t 2α = 0 and γ 1α = 0, as proved in Supplementary Note 2, thus the resulted Hamiltonian is protected by timereversal symmetry and pseudo-Hermitian symmetry, as follows, where T is a unitary operator, and η is a Hermitian invertible operator 41 , which is system-specific in the non-Hermitian context (see Supplementary Note 3). These two symmetries restrict that, if Ψ þ ¼ ðΨ A ; Ψ B Þ T is an eigenvector of the Hamiltonian with energy ε 0 + ε 1 , then Ψ À ¼ ðΨ A ; ÀΨ B Þ T or ðÀΨ A ; Ψ B Þ T is an eigenvector with energy ε 0 − ε 1 (ε 1 2 R or iR). Therefore, spectral degeneracy characteristic occur with coalescent eigenvalues E ± = ε 0 and coalescent eigenstates Ψ ± = (1, 0) T or (0, 1) T , if and only if The solutions to Eq. (3) give various spectral degeneracy behaviors, i.e., hyperbolic lines or elliptical rings, relying on whether the coefficients of cos k x and cos k y have opposite signs.
For example, as shown in Fig. 1a, two exceptional rings cross with each other, while as shown in Fig. 1b, a pair of exceptional hyperbolic lines intersect with an exceptional ring.
For the purpose to reveal the interesting physics of spectral degeneracy, we simplified the model by only considering the nonreciprocity of the intra-cell interaction and assumed that t 1x = t 1y = t 1 . The resulted Hamiltonian can be written as, Without loss of generality, assuming t 0 /t 1 and γ 0 /t 1 are positive, we classified the parameter space into four phases by the distinct forms of ER, and the detailed range is illustrated in Table 1. As shown in Fig. 1c-f, phase I and phase II both possess a single ER locating at cos k x þ cos k y ¼ Àðt 0 À γ 0 Þ=t 1 with Ψ ± = (1, 0) T , but encircle (π, π) and (0, 0) respectively. Phase III and phase IV both possess two ERs, locating at cos k x þ cos k y ¼ Àðt 0 ± γ 0 Þ=t 1 with Ψ ± = (0, 1) T and (1, 0) T , where the former encircle (0, 0) and (π, π), while the latter encircle (π, π) and (π, π).
Gauge-dependent non-Hermitian topology. As we know, the behavior of real systems is often sensitive to the boundary conditions, and the open boundary condition sometimes introduces the localized modes at the boundary or the corner. The intersection of non-Hermitian physics and topology also brings out new phases. For simplicity, we imposed OBC along x direction and PBC along y direction in the lattice model, and the system can be understood as the effective one-dimensional chain parametrized by the transverse momentum k y (see Supplementary Note 4.1). Two kinds of OBC are proposed for comparison. OBC 1 requires that the hopping terms between the first and the last cells are removed from the periodic lattice, as shown in Fig. 2a. For OBC 2 shown in Fig. 2b, the difference from OBC 1 is the selection of the primitive cell, which can be regarded as the gauge transformation that H 0 ðk x ; k y Þ ¼ S À1 Hðk x ; k y ÞS with S ¼ diagðe ik x ; 1Þ, which possesses the same symmetry and spectrum as H(k x , k y ) under PBC, but leads to the distinct difference in topological behaviors. The numerical results for OBC 1 and OBC 2 spectra are shown in Fig. S3 in Supplementary Note 4.1 and Fig. 3a, d respectively, which reveal that the former is trivial, while the latter generates topological boundary modes at the on-site energy ε 0 , which is the so-called topological non-triviality. For example, using the parameters belonging to phase I, the OBC 2 spectrum is shown in Fig. 3a, where the bulk modes are denoted by gray lines, and the topological boundary modes denoted by orange lines can exist for any k y ∈ (0, 2π). Correspondingly, the normalized amplitude of the localized wave function is shown in Fig. 3b.
To investigate the underlying mechanism of the existence or non-existence of topological boundary modes, we followed a convenient criterion applied for one-dimensional systems 35 . Generally, by polynomial factorization and rescaling the overall constants to unity without changing the topology, the Hamiltonian can be rewritten as, where the on-site energy is set to zero, and z ¼ e ik x , k x 2 C. q a , q b count the numbers of poles locating at z = 0, and fa 1 ; :::; a p a g,fb 1 ; :::; b p b g are the complex roots of a(z) = 0, b(z) = 0 respectively. The topological boundary modes exist if and only if 9 , and the red lines denote the spectral degeneracy that E + = E − = ε 0 . For simplicity and without loss of generality, the on-site energy ε 0 is set as zero, and the hopping parameters are set as t 1x = t 1y = t 1 . a, b The energy spectra defined by , which can exhibit exceptional rings and hyperbolic lines respectively. c-f The energy spectra defined by . The parameter space is classified into phase I-IV by the distinct forms of the exceptional rings. c Phase I with t 0 /t 1 Table 1 Illustration to the phase diagram of the tight-binding model. The parameter space (t 0 , t 1 , γ 0 ) is classified into phase I-IV according to the distinct locations and centers of the exceptional ring (ER) in the momentum space (k x , k y ). Phase I and phase II both possess a single exceptional ring, but encircle (π, π) and (0, 0) respectively. Phase III and phase IV both possess two exceptional rings, where the former encircle (π, π) and (0, 0), while the latter encircle (π, π) and (π, π).

Phase
Parameters Location of ER Center of ER cos k x þ cos k y ¼ Àðt 0 þ γ 0 Þ=t 1 (π, π) cos k x þ cos k y ¼ Àðt 0 À γ 0 Þ=t 1 (π, π) principles that, ðiÞ for OBC 1 ; maxfja 1 j; ja 2 jg< minfjb 1 j; jb 2 jgor maxfjb 1 j; jb 2 jg< minfja 1 j; ja 2 jg; ðiiÞ for OBC 2 ; minfjb 1 j; jb 2 jg< maxfja 1 j; ja 2 jg; where {a 1 , a 2 , b 1 , b 2 } are the solutions to Det HðzÞ ¼ 0 for both OBC 1 and OBC 2 . For example, for the case belonging to phase I, {|a 1 |, |a 2 |, |b 1 |, |b 2 |} curves are shown in Fig. 3c, which demonstrates that, criterion (i) is impossible to be satisfied, verifying the non-existence of topological boundary modes for OBC 1 . However, criterion (ii) can be satisfied for OBC 2 , where the colored regions satisfy the condition that W a (R) and W b (R) have opposite signs, confirming the existence of topological boundary modes. The winding number W(R) is defined as [W a (R) − W b (R)]/2, and the zero value of this topological invariant represents the nonexistence of topological boundary modes. The case for phase IV is shown in Fig. 3d-f, where the topological boundary modes can appear only when cos k y <1 À ðt 0 þ γ 0 Þ=t 1 , which also can be verified by the criterion(ii) in Eq. (6). We also explored the distribution properties of the topological eigenstates, in accordance with the biorthogonal bulk-boundary correspondence theory 13,42-44 (see Supplementary Note 5). For example, as shown in Fig. 3b, when |a 2 | < 1, |b 2 | < 1, two topological eigenstates are localized at both ends along x direction with penetration length À1=ln jb 2 j (n = 1, on A sites) and À1=ln ja 2 j (n = N, on B sites). In addition, when |a 2 | < 1, |b 2 | = 1, two topological modes are identical to each other and both localized at n = N along x direction, on B sites, with penetration length À1=ln ja 2 j. Furthermore, the relevant biorthogonal polarization was calculated to predict the appearance of topological modes, consistent with the results derived from winding numbers.
Both OBC 1 and OBC 2 systems exhibit the conventional bulkboundary correspondence and non-existence of skin effect, where the bulk modes under OBC are consistent with those under PBC. It should be noted that, symmetry plays a significant role in our non-reciprocal hopping model. Restricted by the pseudo-Hermiticity, the bulk Hamiltonian is generally insensitive to the boundary conditions, because the wave numbers are real even in the generalized Brillouin zone 38 , thus the bulk modes are delocalized. To further investigate the connection between the non-existence of the non-Hermitian skin effect and the restoration of the bulk-boundary correspondence, we followed the transfer-matrix method 10 , and found that j det Tj ¼ 1 for our model (see Supplementary Note 6).
Topolectrical circuit lattice. To clarify the previous results and explore the crucial role of pseudo-Hermitian symmetry in a practical system, we designed a 2D topolectrical circuit to map the tight-binding model. As shown in Fig. 4a, following the method described by Yu et al. 20 , the A/B site in the tight-binding model corresponds to the A/B junction in the circuit lattice, and the reciprocal hopping term between sites corresponds to the capacitor which connects the junctions, while the non-reciprocal interaction is realized by the operational amplifier which operates as a voltage follower. Furthermore, the 2D circuit lattice contains 80 sub-circuits, and each sub-circuit contains 20 unit cells arranged along y direction. The connection between the two subcircuits is illustrated in Fig. 4b. Followed by the Kirchhoff's Laws described in Methods, the effective Hamiltonian H e for the circuit lattice is given by where C m = − (t 0 − γ 0 ), C t = − t 1 /2 and C γ = − 2γ 0 and C A + C γ = C B , by comparing with the model Hamiltonian. The resonant frequency ω corresponds to the eigenvalue E with the relation of , where L represents the inductor value. Therefore, various phases can be realized in topolectrical circuits through tuning the values of capacitors. The voltage signals V(r, t) of the 2N 2 (N = 40) junctions can be extracted after the transient analysis of the circuit simulation, and then through Fourier transformation, the amplitude of eigenstates V(f) in the frequency domain can be calculated.
We carried out the simulation to realize the frequency response of phase I and phase IV for both PBC and OBC 2 , because (t 0 − γ 0 )/ t 1 > 0 should be satisfied to avoid negative values of capacitors. As shown in Fig. 5a, for phase I under PBC, the simulated frequency response colored by gray is consistent with the ideal spectra, and the single ER which locates at cos k x þ cos k y ¼ ÀC m =ð2C t Þ ¼ À1:6, is shown in Fig. 5b. The results of the OBC system are shown in Fig. 5c, d, which reveals that, when stimulated by the on-site frequency, both of the bulk modes and the topological boundary modes exist, and the combined modes are localized at the left side of the circuit lattice, if the source is only placed at the left boundary to avoid signal divergence. Similarly, as shown in Fig. 5e-h, the ERs of phase IV locate at cos k x þ cos k y ¼ ÀC m =ð2C t Þ ¼ À0:5 and cos k x þ cos k y ¼ ÀðC m þ C γ Þ=ð2C t Þ ¼ À1, and the combined modes stimulated by the on-site frequency are localized at the left side as well. Owing to the pseudo-Hermitian symmetry, in spite of the negative imaginary part of the frequency, the system is still stable in time domain, which can be used as reference for further experimental maneuverability. As mentioned above, the eigenstates Ψ þ ¼ ðV A ðkÞ; V B ðkÞÞ T is always paired with Ψ À ¼ ðÀV A ðkÞ; V B ðkÞÞ T or ðV A ðkÞ; ÀV B ðkÞÞ T under PBC, thus the voltage signal is approximately equal to jΨ þ je ÀIm½ωt e iRe½ωt . Therefore, the amplitude of the voltage cannot diverge seriously within several time periods.

Conclusions
In conclusion, we have proposed the general form of the twodimensional non-reciprocal hopping model protected by timereversal symmetry and pseudo-Hermitian symmetry, which exhibits exceptional elliptical rings or exceptional hyperbolic lines under PBC. The simplified form is mainly discussed, and the parameters space is classified into phases I-IV, according to the distinct forms of exceptional rings. Through the suitable selection of the primitive cell, the topological boundary modes can appear for all four phases, which is verified by the non-zero winding numbers, and the penetration lengths of the topological boundary modes are analyzed through the biorthogonal bulk-boundary correspondence method. In addition, in the presence of pseudo-Hermitian symmetry, our non-Hermitian model behaves like the Hermitian system under OBC, where the bulk-boundary correspondence exists and the non-Hermitian skin effect vanishes, verified by j det Tj ¼ 1. Finally, we designed the proposed phase I and phase IV in the topolectrical circuits, and the simulation results not only correspond to those in the model, but also demonstrate the crucial role of pseudo-Hermitian symmetry in a practical system. In a broader view, our findings can be compared to other platforms such as photonics, acoustics, mechanics or meta-surface, for the purpose on the control of frequency response and localization properties.  4 Schematic illustration of the two-dimensional topolectrical circuit. a Two unit cells in the circuit lattice. The reciprocal hopping terms are realized by capacitors C m , and the non-reciprocal hopping terms are realized by the operational amplifiers and capacitors C γ , where current flow can exist between A sites and the output of the amplifier, but cannot exist between B sites and the input of the amplifier. The LC-circuits which connect the A/B sites and the ground provide the on-site frequency response. b The 40 × 40 lattice contains 80 sub-circuits, and each sub-circuit contains 20 unit cells, arranged along y direction. The schematic view shows the connection between two sub-circuits along x direction, where A n , B n denote the sites of the nth unit cell inside the sub-circuit, and the connection between the unit cells is the same as (a). x direction. c, f |a 1 |, |a 2 |, |b 1 |, |b 2 | curves are colored by red, light red, dark green and light green, respectively, where a 1;2 ¼ À½ðt 0 þ γ 0 Þ=t 1 þ cos k y Ç ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðt 0 þ γ 0 Þ=t 1 þ cos k y 2 À 1 q and b 1;2 ¼ À½ðt 0 À γ 0 Þ=t 1 þ cos k y Ç ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðt 0 À γ 0 Þ=t 1 þ cos k y 2 À 1 q , and the colored regions marked by the non-zero winding numbers W(R) satisfy the condition of topological boundary modes.

Methods
Derivation of the effective Hamiltonian for circuit lattice. As shown in Fig. 4a, the non-reciprocal interaction is realized by the operational amplifier which operates as a voltage follower. Ideally, when working at the linear region, the operational amplifier can amplifies the difference of voltage between the two inputs, and the output voltage is given by V out = β(V + − V − ), where β is the open-loop gain, and V ± is the voltage of the non-inverting/inverting input. Therefore, if the inverting input is connected to the output and the non-inverting input is connected to junction B, which means Since β is generally large enough for amplifiers, the voltage of the output of amplifier equals to V B and no current flows through the two inputs.
Based on Kirchhoff's Law that, the sum of current into a junction equals the sum of current out of the junction, the currents through junction A and B can be written as Rewriting the equation above into a matrix form, the effective Schrödinger's equation for the periodic circuit lattice is given by, where the voltage V A /V B at A/B junctions corresponds to the wave function Ψ A /Ψ B , and the resonant frequency ω corresponds to the eigenvalue E with the relation of ω ¼ 1= ffiffiffiffiffi ffi EL p . The effective Hamiltonian H e for the circuit lattice is given by Eq. (7).
Simulation details. Transient analysis and Fourier transformation were carried out to simulate the frequency response of the designed topoelectrical circuit. First, the amplifiers, capacitors and inductors were selected as ideal elements, and the values of the components are chosen to limit that the resonant frequency ranges from 10 5 to 10 6 Hz, for further practical consideration. Secondly, the source of the lattice is set as pulse-excitation with 6 μs width, 0.1 μs rising edge and 0.2 μs falling edge. For OBC, the source was placed at the left of the circuit lattice, while for PBC, the source can be placed at any position of the lattice. Third, to satisfy the Nyquist sampling theorem, the total time of the transient analysis is set as 12 μs for both PBC and OBC with time step 10 ns, and the frequency step was set as 10 kHz when operating the Fourier transformation.

Data availability
The data that support the plots within this paper are available from the corresponding author on reasonable request.

Code availability
The computer codes used to generate the data presented in the manuscript are available from the corresponding author on reasonable request.