Bulk-boundary correspondence in topological systems with the momentum dependent energy shift

Topological nontrivial materials featuring isolated in-gap edge modes are gaining wide applications in various platforms, such as cold atom systems [1–7] and optical waveguide arrays [8–17]. One of the most typical ways to characterize topological nontrivial states of matter is achieved via the bulk topological invariants obtained with periodical boundary conditions. Conventionally, preserved bulk-boundary correspondence (BBC) suggests that non-vanishing bulk topological invariants indicate the presence of nontrivial edge modes in open boundary conditions [18–21].

However, with the recent development of non-Hermitian physics, it has been found that not only the edge modes but all the bulk states can pile up at the edges of the systems as well, and such phenomenon is dubbed as noptn-Hermitian skin effects [22–35]. In these cases, the bulk boundary modes can be extremely sensitive to local perturbations and the conventional BBC can be broken [36, 37]. Meanwhile, non-Hermitian topological edge modes may not remain degenerate but collapse to a single one, becoming exceptional for Hamiltonian of not full rank [38–44]. Consequently, the original topological invariants may fail to describe the non-Hermitian nontrivial edge modes. To retain the proper topological characterization, the normal Bloch wave vector has to be deformed to be complex and the generalized Brillouin zone is named accordingly [45–47].

Despite the achieved results so far, it has been discovered that BBC still shares profound features beyond the existing framework. In this work, we fill in this blank by demonstrating how BBC behaves when the Hamiltonian contains terms of the form $d_0(k)I$ in both Hermitian and non-Hermitian cases. Such structures can be deemed as the momentum dependent energy shifts which usually appear in systems possessing the inter-cell tunneling or the site-dependent chemical potential. First, for Hermitian cases, we present that topological phase transitions can take place with the upper and the lower bulk bands kept untouched. At this stage, the conventional gap closing points may not characterize the emergence of topological edge modes properly. Indeed, the momentum dependent energy shift can force the system to be metallic phases and topological features can only be captured by the information of the indirect band gap, which measures the gap amplitude between the minimum of the upper band and the maximum of the lower band.

For non-Hermitian systems, we exhibit that the complex $d_0(k)I$ term can facilitate the presence of discontinuous skin effects which coexist with the preserved BBC. Specifically, in this process, the determinant of transfer matrices remains identity, and bulk states are localized at different ends of the lattice. Here, it shall be pointed out that such skin effects differ from the Z₂ skin effects by the absence of Kramer's pair considering the symmetry constraints. Meanwhile, we reveal the physical origin for the phenomenon above lying in that $d_0(k)I$ can dramatically influence the point gap topology, which manipulates the shape of the generalized Brillouin zone and shifts the energy spectrum at the same time. Remarkably, these results can be led a further step to the non-intuitive phenomenon that the modified BBC in Hermitian systems with metallic phases can be restored to be conventional by introducing extra non-Hermiticity. As a detailed illustration, we consider a concrete example taking the form of the modified Su–Scrieffer–Heeger (SSH) chain and demonstrate the bulk invariants characterizing topological phase transitions. Finally, the experimental simulations to illustrate topological edge modes with preserved BBC are proposed via the platform of electric circuits [48–52].

First, we are to demonstrate how BBC shall be constructed when coming across the purely real $d_0(k)I$ inpt Hermitian systems. As a concrete example, we consider the following modified SSH chain

$$\begin{eqnarray} \begin{aligned} H = &\sum^{N}_{i = 1} t_1\left(c^{\dagger}_{i,A}c_{i,B}+c^{\dagger}_{i,B}c_{i,A}\right)+\sum^{N-1}_{i = 1}\Big[t_2 \left(c^{\dagger}_{i,B}c_{i+1,A}+c^{\dagger}_{i+1,A}c_{i,B}\right)\\ &+iJ \left(c^{\dagger}_{i,A}c_{i+1,A}-c^{\dagger}_{i+1,A}c_{i,A}\right)+iJ\left(c^{\dagger}_{i,B}c_{i+1,B}-c^{\dagger}_{i+1,B}c_{i,B}\right)\Big], \end{aligned} \end{eqnarray} \tag{ 1 }$$

where $J,t_{2}\in \mathrm{R}$ denote the inter-cell tunneling between the same and different types of sublattices respectively. t₁ describes the intra-cell tunneling. N is the length of the chain. The momentum space Hamiltonian can be obtained by applying the spatial Fourier transformation $c_{k,A(B)} = \frac{1}{\sqrt{N}}\sum_i c_{i,A(B)} \mathrm{e}^{-ikR_{i,A(B)}}$ and takes the form $\hat{H}(k) = \sum_k \Psi^{\dagger}_k H(k) \Psi_k, \Psi^{\dagger}_k = (c^{\dagger}_{k,A},c^{\dagger}_{k,B})$. H(k) is given as

$$\begin{eqnarray} \begin{aligned} H\left(k\right)& = -2J\sin{\left(k\right)}I\!+\!\left[t_1\!+\!t_2\cos{\left(k\right)}\right]\sigma_x\!+\!t_2\sin{\left(k\right)}\sigma_y\\ & = d_0\left(k\right)I+d_x\left(k\right)\sigma_x+d_y\left(k\right)\sigma_y, \end{aligned} \end{eqnarray} \tag{ 2 }$$

where I and $\sigma_{x,y,z}$ are the identity matrix and Pauli matrices. The first term on the right hand side, $d_0(k) I$, can shift the energy spectrum momentum dependently and force the system to include metallic phases. Equation (2) indicates the absence of the time-reversal symmetry (TRS). Nevertheless, the particle-hole symmetry (PHS) is preserved as

$$\begin{equation} \Xi H^{*}\left(k\right) \Xi^{\dagger} = -H\left(-k\right), \Xi = \sigma_z. \end{equation} \tag{ 3 }$$

Hence, our one-dimensional quantum chain belongs to the topological class D and the reasonable topological invariants are the Z₂ indexes. Meanwhile, it shall be noticed that when $k = 0,\pi$, the $d_0(k),d_y(k)$ terms are vanished and only the $d_x(k)$ term survives. In consequence, the Z₂ topological invariants are defined as

$$\begin{equation} Q = sgn\left[d_x\left(k = 0\right)d_x\left(k = \pi\right)\right], \end{equation} \tag{ 4 }$$

where topological nontrivial regions are supposed to be signified by $Q = -1$ and $|t_1|\lt|t_2|$.

One illustration of BBC manifests that the critical value of parameters, where the real space topological edge modes merge to the bulk, should be consistent with the values that the momentum space Hamiltonian becomes gapless. For our system schematically shown in figure 1(a) with $t_1 = 1.0$, when the inter-cell tunneling $J\lt0.5 = t_1/2$, the system appears to be SSH alike (SSHA) and topological nontrivial edge modes lie in the regions $\!|t_1|\lt|t_2|\!$ (see figure 1(b)). At this stage, the conventional BBC is fulfilled (see figures 1(b), (c) and 2(a)), where the gapped topological insulator (TI) phases are marked by $Q = -1$, and the topological trivial phases depict the normal insulator (NI).

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However, as the inter-cell tunneling J is further enlarged, the system exhibits distinguishable topological properties compared to the original SSH model. In detail, the topological nontrivial region appears to be located at $|t_2|\gt|max\{t_1,2J\}|$ and there exist topological trivial regions satisfying $|t_2|\gt|t_1|$ for $|2J|\gt|t_1|$ (see figure 1(d)). At this stage, the bulk invariant Q shall fail to characterize the topological features. Specifically, from the perspective of the momentum space energy spectrum, it can be observed that topological phase transition can still take place while the upper bands do not touch the lower bands (see figure 2(b)). We tend to term such cases to be non-SSHA.

Numerically, the results in figure 1(e) suggest that the proper modified BBC should be reconstructed by including the definition of the indirect band gap δ, which measures the energy difference between the minimum of the upper band [$E^{u}_{\mathrm{min}}(k_1)$] and the maximum of the lower band [$E^{l}_{\mathrm{max}}(k_2)$], $\delta = E^{u}_{\mathrm{min}}(k_1)-E^{l}_{\mathrm{max}}(k_2)$. Here, we need to point out that δ has taken the momentum dependence of energy shift into consideration and as long as $k_1\neq k_2$, the indirect band gap differs from the direct band gap Δ. Meanwhile, δ < 0 signifies the metal phase rather than the NI phase. Consequently, transitions between the metal and TI can properly be described by the gap closing points of the indirect band gap. The topological phase diagram as a function of the inter-cell tunneling t₂ and J is shown in figure 1(f).

The previous discussion demonstrates how the BBC behaves when the Hermitian systems consist of metallic phases induced by $d_0(k) I$. For open quantum systems with asymmetric tunneling or on-site gain-loss, the Hamiltonian may not remain Hermitian and can feature the skin effects. Conventionally, the presence of non-Hermitian skin effects can be identified in two ways: (a) the generalized Brillouin zone does not coincide with the original Brillouin zone and the momentum space energy spectrum $\{\mathrm{Re}(E_k),\mathrm{Im}(E_k)\}$ encircles non-zero area with the momentum k running over the whole Brillouin zone; (b) the determinant of transfer matrix is not identify. In the following, we are to present how the BBC will be affected by the complex $d_0(k) I$ terms.

One way to induce non-Hermiticity in our modified SSH chain can be achieved via changing the inter-cell tunneling between the same type of sublattices to be purely imaginary $J\rightarrow -iJ$ and the system can be redescribed as

$$\begin{equation} H_{nh}\left(k\right) = 2iJ\sin{\left(k\right)} I+\left[t_1+t_2\cos\left(k\right)\right]\sigma_x+t_2\sin\left(k\right)\sigma_y. \end{equation} \tag{ 5 }$$

Equation (5) suggests that the imaginary part of the energy spectrum is only decided by the first term on the right hand side. Hence, $\mathrm{Im} (E_k)$ should always be gapless as long as J ≠ 0. Considering the real part of the eigenenergy, $\mathrm{Re} (E_k)$ is exactly the same as that of the SSH chain and should be gapped as long as $|t_1|\neq |t_2|$. Hence, if the BBC is fulfilled, topological edge modes shown by the absolute energy spectrum $|E|$ will appear (or merge to the bulk) at $|t_2| = |t_1|$.

Numerically, the energy spectrum with open boundary conditions of the model in equation (5) is illustrated by figure 3(a). Such results approved the preserved BBC and topological phase transitions can also be depicted by the bulk invariant Q in equation (4). However, in figure 3(b), we notice that the real space energy spectrum $\{\mathrm{Re}(E), \mathrm{Im}(E)\}$ obtained with open boundary conditions (red dashed line) does not coincide with that obtained by periodic boundary conditions (blue dashed line) in the large size limit, fixing $t_2 = -0.5, J = 1$. Meanwhile, in figure 3(c), the momentum space energy spectrum $\{\mathrm{Re}[E(k)], \mathrm{Im} [E(k)]\}$ encircles the non-zero area with k running over the Brillouin zone $[-\frac{\pi}{a}, \frac{\pi}{a}]$, of which the results, according to [28], suggest the presence of non-Hermitian skin effects.

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Typically, non-Hermitian skin effects tend to result in the broken BBC and the discussions above are noticeably against this. To further identify the localization of all eigenstates, we apply the transfer matrix approach, of which the method is entirely based on the real space Hamiltonian. Here, we assume the boundary modes to be $\psi = \sum_{p}\varphi^{N}_{p}, p = A,B, \varphi^{N}_{P} = \beta^{N} \varphi^{0}_p$. The absolute value of the determinant of transfer matrix is equal to $|\beta|$ and describes how the norm of states are changed during the wave propagating. A demonstration of how the transfer matrix approach succeeds in identifying skin effects in the non-Hermitian SSH (NH-SSH) chain is presented in appendix A. Specific to our model, we present that the determinant of transfer matrix is identity. In detail, given the basis $\Psi^{\dagger}_i = (c^{\dagger}_{i,A},c^{\dagger}_{i,B})$, the Hamiltonian can be depicted by

$$\begin{equation} H_{nh} = \sum_i \Psi^{\dagger}_i A \Psi_i+\Psi^{\dagger}_i B \Psi_{i+1}+\Psi^{\dagger}_{i+1} C \Psi_i, \end{equation} \tag{ 6 }$$

$$\begin{equation} \begin{split} A = \left( \begin{smallmatrix} 0, & t_1 \\ t_1, & 0 \end{smallmatrix} \right), B = \left( \begin{smallmatrix} J, & 0 \\ t_2, & J \end{smallmatrix} \right),C = \left( \begin{smallmatrix} -J, & t_2 \\ 0, & -J \end{smallmatrix} \right). \end{split} \end{equation} \tag{ 7 }$$

Given the single particle state $|\Phi\rangle = \zeta_i \Psi^{\dagger}_i |0\rangle$, the Schrodinger equation can lead to the recursion function

$$\begin{equation} A\zeta_i+B\zeta_{i-1}+C\zeta_{i+1} = \epsilon \zeta_i. \end{equation} \tag{ 8 }$$

Correspondingly, the transfer matrix can be introduced via

$$\begin{equation} \begin{split} \left( \begin{smallmatrix} \zeta_{i+1} \\ \zeta_{i} \end{smallmatrix} \right) = T \left( \begin{smallmatrix} \zeta_{i} \\ \zeta_{i-1} \end{smallmatrix} \right) \end{split}, T = \left( \begin{smallmatrix} -C^{-1}\left(A-\epsilon I\right), & -C^{-1}B \\ I, & 0 \end{smallmatrix} \right), \end{equation} \tag{ 9 }$$

and the determinant of transfer matrix is

$$\begin{equation} det\left(T\right) = det\left(C^{-1}\left(A-\epsilon I\right)\right)*det\left(\left(A-\epsilon I\right)^{-1}B\right) = 1. \end{equation} \tag{ 10 }$$

Intuitively, the transfer matrix approach indicates that the bulk states should neither be boosted nor suppressed with the growing lattice site index and non-Hermitian skin effects can not be recognized.

At this stage, we tend to numerically demonstrate the density profiles of the representative bulk states. In figures 3(d)–(f), $|\psi^l_j|$ is plotted for $l = 1,10,20,50,51,100,91,81$, where $\psi^l_j$ denotes the lth eigenstate of the modified NH-SSH chain, j being the site index. It is illustrated that the bulk states $|\psi^l_j|$ are not equally extended, but are localized at the right and left end of the lattice respectively, which suggests the presence of non-Hermitian skin effects. Meanwhile, it shall be noticed that our system is in sharp contrast to the known NH-SSH chain in [26], where for $t_1+\frac{r}{2}\gt (\lt) t_1-\frac{r}{2}$, all the bulk states are singly localized at one end of the lattice. Besides, in the NH-SSH chain, the two zero energy end states collapse to the same one and become exceptional, exhibiting the single end boundary localization. Differently, specific to our modified model, it can be observed that $\psi^{50(51)}_j$ is the combination of the two zero energy edge states and has nearly equal localization on both ends of the lattice. Hence, it can be identified that non-Hermitian skin effects are absent for the zero energy states.

Given the results above, we sum up that the modified NH-SSH chain possesses the non-Hermitian skin effect, featuring the bulk states piled up at different ends of the lattice. Meanwhile, such non-Hermitian skin effects are not continuous at zero energy. We tend to term this phenomenon as the discontinuous double-side non-Hermitian skin effects (DDS-NHSE), which is the critical physical origin for the phenomenon above. Remarkably, the DDS-NHSE can coexist with the preserved BBC.

Conventionally, non-Hermitian skin effects can also be decided by properties of the generalized Brillouin zone. In detail, the normal Bloch wave vector is replaced in the way that $e^{ik}\Leftrightarrow \beta$, through which the bulk iteration function of our modified NH-SSH chain can be depicted as

$$\begin{eqnarray} \begin{aligned} &J\left(\beta-\beta^{-1}\right)\varphi_A+\left(t_2\beta^{-1}+t_1\right)\varphi_B = E \varphi_A, \\ &J\left(\beta-\beta^{-1}\right)\varphi_B+\left(t_2\beta+t_1\right)\varphi_A = E \varphi_B, \end{aligned} \end{eqnarray} \tag{ 11 }$$

where the cell wave function of the nth site is assumed to be $\Psi_{n,A(B)} = \sum_j \beta^{n}_j \varphi^{j}_{A(B)}$. The topological edge modes are supposed to be boundary modes of zero energies, which will lead equation (11) to the following characteristic function

$$\begin{equation} J^2 \beta^4-t_1t_2\beta^3-\left(2J^2+t^2_1+t^2_2\right)\beta^2-t_1t_2\beta +J^2 = 0. \end{equation} \tag{ 12 }$$

Equation (12) is quartic and has four roots $\beta_j, j = 1,2,3,4$. Meanwhile, the boundary cell wave functions satisfy

$$\begin{eqnarray} \mathrm{left\,end}\left\{ \begin{aligned} &t_1\Psi_{1,B}+J\Psi_{2,A} = E\Psi_{1,A} \\ &t_1\Psi_{1,A}+t_2\Psi_{2,A}+J\Psi_{2,B} = E\Psi_{1,B}. \end{aligned} \right. \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} \mathrm{right\,end}\left\{ \begin{aligned} &-J\Psi_{N-1,B}+t_1\Psi_{N,A} = E\Psi_{N,B} \\ &-J\Psi_{N-1,A}\!+\!t_2\Psi_{N-1,B}\!+\!t_1\Psi_{N,B}\! = \!E\Psi_{N,A}. \end{aligned} \right. \end{eqnarray} \tag{ 14 }$$

Through the equations above and according to the theory of generalized Brillouin zone, in the large size limit, two of the roots have to fulfill $|\beta^{E\rightarrow0}_i| = |\beta^{E\rightarrow0}_j|, i\neq j$, which will lead to $|\beta^{E\rightarrow0}| = 1$ for $|t_1| = |t_2|$, suggesting the preserved BBC. Such results also exhibit a sound agreement with figure 3(a). In appendix B, we present the detailed shape of the generalized Brillouin zone, which crosses the unit circle at E = 0. Meanwhile, regions satisfying $|\beta^{E \gt(\lt) 0}_i| = |\beta^{E\gt (\lt)0}_j|\neq 1$ correspond to the non-Hermitian skin effects for states away from zero energy. The results above are consistent with the analysis of DDS-NHSE.

In the previous discussion, it is shown that non-Hermitian skin effects can not guarantee the broken BBC. Now, we are to extend this result further and demonstrate the remarkable phenomenon that the modified BBC in Hermitian systems can be restored to be conventional by introducing external non-Hermiticity. For a detailed illustration, we still consider the modified NH-SSH chain and tune the purely imaginary inter-cell tunneling, iJ, to the general form Je^iφ , where φ is an arbitrary phase. Correspondingly, the real and imaginary parts of the energy spectrum are depicted as

$$\begin{eqnarray} &&\begin{aligned} \mathrm{Re}\left[E\left(k\right)\right] = -2J\sin{\phi}\sin{\left(k\right)}\pm \sqrt{\left[t_1+t_2\cos{\left(k\right)}\right]^2+\left[t_2\sin{\left(k\right)}\right]^2}. \end{aligned} \end{eqnarray} \tag{ 15 }$$

$$\begin{equation} \mathrm{Im}\left[E\left(k\right)\right] = 2J\cos{\phi}\sin{\left(k\right)}. \end{equation} \tag{ 16 }$$

For $\phi\neq N\pi$, $-2J\sin{\phi}\sin{(k)}$ in equation (15) can serve as $d_0(k) I$ and force the system to include metallic phases. Consequently, based on the discussions in Hermitian cases, only the indirect band gap, which closes at $t_2 = \pm 1.6$ for $t_1 = J = 1,\phi = 0.3\pi$, can properly describe the emergence of topological edge modes in open boundary conditions (see figure 4(a)). Meanwhile, it can be observed that the minimum value of the indirect band gap amplitude changes as a function of φ, the variation of which can frequently manipulate the topological phase transition points (see figure 4(d)).

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To proceed, we notice that the imaginary part of the energy spectrum is gapless for all values of k, which can lead to the intuitive hypothesis that topological nontrivial regions shown by the absolute energy spectrum should coincide with those given by the real energy spectrum, both exhibiting the modified BBC in part 2.

However, in figure 4(b), it is exhibited that the topological nontrivial edge modes presented by the absolute energy spectrum merge to the bulk at $|t_2| = |t_1|$, where the momentum space bulk bands also just become gapless (see figure 4(c)). Such non-intuitive results are equivalent to announce that non-Hermiticities are capable of refixing conventional BBC in Hermitian systems with metallic phases.

To illustrate the physical origin, it should be kept in mind that the characterization of topological edge modes can also be based on the information of line gaps. For traditional Hermitian systems, the energy spectrum can be flattened and projected to the real axis. In such cases, the line gap lies exactly on the imaginary axis. Considering the topological trivial metallic phases, different bands cross the line gap multiple times (see figure 5(a)). As $t_2/t_1$ is enlarged to the topological transition point, each band touches the line gap only once (see figure 5(b)). Nevertheless, when the external non-Hermiticity is included, accompanied by the non-zero imaginary eigenenergy, the real line gap shall be changed to be complex and lies at other positions. Specific to our model, the localization of the line gap should vary as a function of φ (see figures 5(c)–(d)). Importantly, it needs to be noticed that as long as the line gap is not located at the imaginary axis (i.e. any other positions), all two bands will touch the line gap only once with $|t_2| = |t_1|$ regardless of φ, suggesting the preserved BBC. In other words, by including non-Hermiticity, line gaps can be tuned away from the imaginary axis, which can usually be achieved by considering arbitrary $\phi,\phi\neq \frac{(2p+1)\pi}{2},p\in Z$, and any other general positions of the line gap are apt to restore the conventional BBC.

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Not all types of $d_0(k) I$ can result in nontrivial point gap topology, which fosters the non-Hermitian skin effects, and keep the topological edge mode at the same time. In the following, we focus on the discussions of the constraints. First, $d_0(k) I$ can hardly be compatible with the conventional chiral symmetry (CS$^\dagger$)[53]. Since $d_0(k) I$ will cause an energy shift to the upper and the lower band in the same manner, the chiral partner state of $|u_n(k)\rangle$ with energy $-E_n(k)$ is not contained in the Hilbert space. However, with the consideration of 38 fold symmetry classification in non-Hermitian systems, purely imaginary $d_0(k) I$ can support CS symmetry, $\Gamma H^{\dagger} (k) \Gamma^{-1} = -H(k)$. In correspondence, we have eigenenergies satisfying $E_n(k) = id_0(k)\pm \sqrt{d_x^2(k)+ d_y^2(k)+ d_z^2(k)}$ and the real (imaginary) line gap exists for $d_x^2(k)+ d_y^2(k)+ d_z^2(k) \gt(\lt)0$. For the latter case, the system can be purely anti-Hermitian and the BBC is supposed to be reconstructed in a similar way as those shown in part 2.

For systems possessing TRS (TRS$^{\dagger}$) or PHS (PHS$^{\dagger}$ ) symmetry, which can be unified to the form $A H^{T}(k) A^{-1} = e^{i\Theta} H(-k)$ [or $A H^{*}(k) A^{-1} = e^{i\Theta} H(-k)$] and $A A^{\dagger} = 1$, it is supposed to be fulfilled that $d_0(k) = e^{i\Theta} d_0(-k)$ [or $d^{*}_0(k) = e^{i\Theta} d_0(-k)$]. Since Θ is irrelevant to k, the most common cases are $\Theta = 2p\pi, (2p+1)\pi, p\in Z$, and the latter case shall force the spectrum to be central symmetric around E = 0. Meanwhile, the condition $A A^{*} = -1$ can guarantee the non-Hermitian Kramer's pairs and enable the possibilities of Z₂ skin effect. However, such conditions do not apply to the Hamiltonian in equation (5) and hence the DDS-NHSE differs from Z₂ skin effect.

In contrast to the line gap topology, the complex $d_0(k)I$ can greatly influence the point gap topology. As an illustration, considering a Hermitian Hamiltonian H' free from skin effect, the synthesized Hamiltonian H, where $H = H^{^{\prime}}(k)+e^{i\gamma_p}d_0(k+\gamma_q)I$, can exhibit the skin effect by properly choosing $\gamma_{p,q}$ to obtain a nontrivial phase difference. The Hamiltonian in equation (1) can serve as an example and another detailed demonstration is shown in appendix C. Indeed, the eigenenergy can be shifted in an artificial way by controlling the coefficients of $d_0(k)I$, and skin effects can be observed in a wide range of parameters. Therefore, it can be easily achieved $|\beta^{E\rightarrow0}| = 1$, which suggests the coexistence of skin effects and the preserved BBC.

To realize the modified NH-SSH chain, we propose to utilize the electrical circuits, which have been frequently applied in simulating diverse topological lattice models. Specifically, the experimental setups are shown in figure 6(a), where the blue and red dots distinguish the $A, B$ types of sublattices respectively, and the purple dashed line labels the unit cell. The current flowing through the jth node of the circuit lattice is governed by Kirchhoff's law

$$\begin{equation} I_j = \sum_i L_{i,j} V_j. \end{equation} \tag{ 17 }$$

Here, V_j depicts the voltage potential and $L_{i,j}$ is the net impedance of all the electrical elements linked to the node j, which is named the circuit Laplacian. Considering our systems, the current on $A,B$ sublattices of the jth node is characterized by

$$\begin{eqnarray} &&\begin{aligned} I^{A}_j = i\omega C_1\left[V^{A}_{j}-V^{B}_{j-1}\right]\!+\!i\omega C_2\left[V^A_{j}-V^B_{j}\right]+\frac{1}{i\omega L} V^{A}_{j}+ \frac{1}{R} \left(V^{A}_j-V^{A}_{j+1}\right), \end{aligned} \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} &&\begin{aligned} I^B_{j} = i\omega C_1\left[V^{B}_{j}-V^{A}_{j+1}\right]+i\omega C_2\left[V^B_{j}-V^A_{j}\right]+\frac{1}{i\omega L} V^{B}_{j}+\frac{1}{R} \left(V^{B}_j-V^{B}_{j+1}\right) , \end{aligned} \end{eqnarray} \tag{ 19 }$$

where $i\omega C_{1,2}$ are the impedances of the capacitors and $\frac{1}{i\omega L}$ is the impedance of the inductor. R measures the resistance of the operational amplifier, which serves as the negative impedance converter with current inversion, of which the detailed structure is shown in [50–52]. Meanwhile, for the current flowing towards the left side, $R = |R|$ is positive and for the rightward flowing current, $R = -|R|$ is negative. The matrix form of the circuit Laplacian can be written as

$$\begin{equation}L_{i,j} = \begin{pmatrix} S & -i\omega C_2 & R^{-1} & 0 & 0 & {\ldots} \\ -i \omega C_2 & S & -i\omega C_1& R^{-1} & 0 & {\ldots} \\ -R^{-1} & -i \omega C_1 & S & -i\omega C_2 & R^{-1} &{\ldots}\\ 0 & -R^{-1} & -i \omega C_2 & S & -i \omega C_1\\ {\ldots} \end{pmatrix}, \end{equation} \tag{ 20 }$$

where the constant terms proportional to the identity matrix I are of coefficient $S = i\omega(C_1+C_2-\frac{1}{\omega^2 L})$. It can be seen that the impedances are related to parameters of the modified NH-SSH in the following way

$$\begin{equation} t_1 = -i \omega C_2, t_2 = -i\omega C_1, J = R^{-1}. \end{equation} \tag{ 21 }$$

The resonant frequency of the electrical circuits can be obtained as $\bar{\omega} = 1/\sqrt{L(C_1+ C_2)}$. Besides, the presence of topological edge modes can be identified via the peaks of the two-point impedances between the left end node (q₁) and the right end mode (q₂), which take the form $|Z^{\textrm {mod-SSH}}| = G(q_1,q_2)+G(q_2,q_1)$ $-G(q_1,q_1)-G(q_2,q_2)$ and G is the circuit Green function. In figure 6(b), it is demonstrated that with a fixed large $|R|$, only for $|C_1|\gt |C_2|$, the peaks of $|Z^{\textrm {mod-SSH}}|$ at $\omega = \bar{\omega}$ can be observed. Such results are consistent with the topological nontrivial region $|t_2|\gt|t_1|$.

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We have demonstrated how the BBC will be affected by the presence of momentum dependent energy shift, which takes the form of $d_0(k) I$. It is observed that the purely real $d_0(k) I$ in Hermitian systems can result in metallic phases and the modified BBC should be reconstructed based on the definition of indirect band gaps. In non-Hermitian cases, complex $d_0(k) I$ can greatly influence the point gap topology and facilitate the foundation of non-Hermitian skin effects. Remarkably, we propose the DDS-NHSE, which coexists with the preserved BBC. Meanwhile, contrary to the intuitive consideration, it is illustrated the modified BBC in Hermitian systems with metals can be restored to be conventional by introducing external non-Hermiticity. To the end, it shall be mentioned that $d_0(k) I$ can frequently appear in systems with the inter-cell tunneling between the same type of sublattices or systems possessing the site-dependent chemical potential. Our work reveals the unexpected features of BBC and has potential applications in constructing new types of topological materials.

This work is supported by XRC-23079.

All data that support the findings of this study are included within the article (and any supplementary files).

We tend to demonstrate how the transfer matrix approach succeeds in predicting the non-Hermitian skin effects via the example of the NH-SSH chain in [26]. In detail, the non-Hermiticities lie in the asymmetric intra-cell tunneling, $t_1-\frac{r}{2}, t_1+\frac{r}{2}$ and the inter-cell tunneling between the same type of sublattices is assumed to be absent. The Schrodinger equation of the bulk states corresponding to $A,B$ types of sublattices can be depicted by

$$\begin{eqnarray} \begin{aligned} &\left(t_1-\frac{r}{2}\right)c_{j,A}+t_2c_{j+1,A} = E c_{j,B},\\ &\left(t_1+\frac{r}{2}\right)c_{j,B}+t_2c_{j-1,B} = E c_{j,A}, \end{aligned} \end{eqnarray} \tag{ A.1 }$$

where $c_{j,A}$ presents annihilating an A type of particle at the jth site and t₂ is the inter-cell tunneling between different types of sublattices. $[t_1-(+)\ \frac{r}{2}]$ depicts the rightward (leftward) intra-cell tunneling. Through equation (A.1), the following iterative function can be obtained

$$\begin{eqnarray} &&\begin{aligned} t_2\left[\left(t_1-\frac{r}{2}\right)c_{j-1,B}+\left(t_1+\frac{r}{2}\right)c_{j+1,B}\right]+\left(t^2_1+t^2_2-\frac{r^2}{4}-E^2\right)c_{j,B} = 0, \end{aligned} \end{eqnarray} \tag{ A.2 }$$

and the transfer matrix is given as

$$\begin{equation} \begin{split} \left( \begin{matrix} c_{j+1,B} \\ c_{j,B} \end{matrix} \right) = T \left( \begin{matrix} c_{j,B} \\ c_{j-1,B} \end{matrix} \right), T\! = \! \left( \begin{matrix} a_{11}, & a_{12} \\ 1, & 0 \end{matrix} \right), \end{split} \end{equation} \tag{ A.3 }$$

$$\begin{equation} a_{11} = -\frac{t^2_1+t^2_2-\frac{r^2}{4}-E^2}{t_2\left(t_1+\frac{r}{2}\right)}, a_{12} = -\frac{t_1-\frac{r}{2}}{t_1+\frac{r}{2}}. \end{equation} \tag{ A.4 }$$

The absolute value of the determinant of transfer matrix takes the form $|det(T)| = |a_{12}| = |\frac{t_1-\frac{r}{2}}{t_1+\frac{r}{2}} |$. Hence, when the tunneling towards the right hand side is larger than the tunneling towards the left hand side, $|t_1-\frac{r}{2}|\gt|t_1+\frac{r}{2}|$, we have $|det(T)|\gt1$, and all the states are piled up at the right end of the lattice. Conversely, $|det(T)|\lt1$ contributes to the left end localization. To sum up, whenever $|det(T)|\neq 1$, there shall be non-Hermitian skin effects and all states exhibit the boundary localization. For the cases that $|det(T)| = 1$ and $|t_1-\frac{r}{2}| = |t_1+\frac{r}{2}|$, the system remains Hermitian and the non-Hermitian skin effects are vanished.

For our modified NH-SSH chain, non-Hermitian skin effects can force different parts of the bulk states to pile up at different boundaries. Such double-side localization phenomenon can be further identified via the shape of the generalized Brillouin zone, which is schematically shown in figure B1(a). In detail, the blue dashed points form the shape of the generalized Brillouin zone and the black dashed lines describe the unit circle, which characterizes the conventional Brillouin zone, $|e^{ik}| = 1, k\in [-\frac{\pi}{a}, \frac{\pi}{a}]$, in Hermitian systems. It can be seen that the generalized Brillouin zone consists of a partial outside circle with a radius larger than one, and the states corresponding to which should be localized at the right boundary. The numerical demonstration of the localization in figure 3(d) shows a sound agreement. Meanwhile, the generalized Brillouin zone also has a partial inside circle with a radius smaller than one, of which the states are piled up at the left boundary, being consistent with numerical results in figure 3(f). Interestingly, the generalized Brillouin zone will cross the unit circle at E = 0, suggesting that the zero energy edge states are not affected by the non-Hermitian skin effects. Indeed, Considering the NH-SSH chain in [26], the asymmetric intra-cell tunneling can force the edge states to collapse to the same one, exhibiting the single boundary localization, of which the center is decided by the generalized Brillouin zone (see figures B1(b) and (c)). Whereas the skin effect free edge modes in our model share totally different features and can be localized at both ends of the lattice simultaneously (see figure 3(e)).

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For the one-dimensional monatomic chain, the most typical way to achieve non-Hermitian skin effects comes via considering the Hatano-Nelson model, where the asymmetric leftward and rightward hopping ($|t_L|\neq |t_R|$) can push all states to one specific end. Here, we propose another one-dimensional structure with balanced hopping amplitude, and the presence of non-Hermitian skin effects can be artificially controlled. In detail, the Hamiltonian takes the form

$$\begin{equation} \tilde{H} = \sum_j \left[te^{i\gamma_t}\left(c^{\dagger}_{j+1}c_j+c^{\dagger}_jc_{j+1}\right)\!+\! C\left(c^{\dagger}_{j+p}c_j+e^{i\gamma_c}c^{\dagger}_jc_{j+p}\right)\right]. \end{equation} \tag{ C.1 }$$

$C,t \in \mathrm{R}$ denote the quantum tunneling and $\gamma_{t(c)}$ are the relevant phases, which can be manipulated by the techniques of laser-assisted tunneling. The energy spectrum can be obtained as

$$\begin{equation} E\left(k\right) = 2te^{i\gamma_t}\cos{k}+ C\left[\left(e^{i\gamma_c}+1\right)\cos{\left(kp\right)}+i\left(e^{i\gamma_c}-1\right)\sin{\left(kp\right)}\right]. \end{equation} \tag{ C.2 }$$

First, we set $\gamma_t = m\pi,m\in Z$. The former term in equation (C.2) is purely Hermitian and the latter term works in a similar way as the complex $d_0(k)I$. For $\gamma_c = (2n+1)\pi, n\in Z$, the imaginary spectrum only contains $-2iC\sin{(kp)}$ and there are no terms proportional to $\cos{(kp)}$. In this case, E(k) can form a closed loop in the complex plane regardless of the value of p, which can give rise to the skin effects. For $r_c = 2n\pi, n\in Z$, all terms are real and the skin effects are absent.

However, there exists one special case, $\gamma_t\neq m\pi, \gamma_c = 2n\pi$, where the real and imaginary parts can coexist in the energy spectrum. Skin effects are not necessarily to be retained. To be specific, when p = 1, the $\mathrm{Re}E (k)$ and $\mathrm{Im} E(k)$ are of equivalent phases, forming a line with zero encircled area in the complex plane. To form the non-equivalent phase and restore the skin effect, we shall set p ≠ 1. For this case, the skin effects are only decided by the length of effective tunneling, not the asymmetric tunneling strength.