Two-dimensional non-Hermitian topological phases induced by asymmetric hopping in a one-dimensional superlattice

TL;DR

This paper demonstrates how asymmetric hopping in a 1D superlattice can induce two distinct classes of 2D non-Hermitian topological phases, revealing new topological states with potential experimental realizations.

Contribution

It introduces a method to realize non-Hermitian 2D topological phases using 1D superlattices with asymmetric hopping, expanding the understanding of non-Hermitian topological matter.

Findings

Realization of non-Hermitian Chern insulator and topological semimetal phases.

Support for real edge modes due to $ ext{PT}$-like symmetry.

Potential for experimental implementation in photonic or atomic systems.

Abstract

Non-Hermitian systems can host topological states with novel topological invariants and bulk-edge correspondences that are distinct from conventional Hermitian systems. Here we show that two unique classes of non-Hermitian 2D topological phases, a 2$\mathbb{Z}$ non-Hermitian Chern insulator and a $\mathbb{Z}_{2}$ topological semimetal, can be realized by tuning staggered asymmetric hopping strengths in a 1D superlattice. These non-Hermitian topological phases support real edge modes due to robust $\mathcal{PT}$-symmetric-like spectra and can coexist in certain parameter regime. The proposed phases can be experimentally realized in photonic or atomic systems and may open an avenue for exploring novel classes of non-Hermitian topological phases with 1D superlattices.

Equations

$$C^{ab}=\frac{1}{2\pi }\int dkd{\varphi }_L({\partial }_k{\mathcal{A}}_{{\varphi }_L}^{ab}-{\partial }_{{\varphi }_L}{\mathcal{A}}_k^{ab}),$$

$$H(\mathbf{k})|\psi (\mathbf{k}){⟩}_R$$

$$H(\mathbf{k}{)}^{†}|\psi (\mathbf{k}){⟩}_L$$

$${\mathcal{A}}_{\mathbf{k}}^n=\frac{1}{2}\left({\mathcal{A}}_{\mathbf{k}}^{RL}+{\mathcal{A}}_{\mathbf{k}}^{LR}\right),$$

$$C^n$$

$$H$$

$$E_{\pm ,\pm }=\pm \sqrt{2-{\lambda }^2\pm \sqrt{\frac{1}{2}(8-8{\lambda }^2+{\lambda }^4(1-\cos (4{\varphi }_L)))}}$$

$$t_{i+1,i}=1-\lambda \cos (2\pi \alpha i+{\varphi }_L+\delta {\varphi }_L),\delta {\varphi }_L\in \mathbb{R}$$

$$H_{PT}=t\sum _i\left({\hat{c}}_i^{†}{\hat{c}}_{i+1}+h.c\right)+(-1{)}^i\gamma {\hat{c}}_i^{†}{\hat{c}}_i,$$

Full text

I Introduction

In the past few decades, topological states of matter have been extensively studied in various physical systems because of their unusual properties and significant applications in quantum devices and information processing XiaoD2010 ; ChiuCK2016 . While the study has mainly focused on solid-state materials WangJ2017 , ultra-cold atomic gases provide another platform for realizing topological states with high tunability and controllability AidelsburgerM2013 ; AtalaM2013 ; AidelsburgerM2014 ; JotzuG2014 ; LohseM2015 ; NakajimaS2016 ; GoldmanN2016 ; CooperNR2019 . Furthermore, the concept of topology has been extended to classic systems govern by wave equations such as photonics HaldaneFDM2008 ; WangZ2009 ; LuL2010 ; LuL2014 ; KhanikaevAB2017 ; SirokiG2017 ; BahariB2017 , acoustics HeC2016 ; HeH2018 , and electric circuits ImhofS2018 , yielding many interesting topological states.

One significant feature of ultra-cold atomic gases and these classical systems comparing to quantum materials is their capability of controllably inducing non-Hermiticity (e.g., gain and loss), which makes them excellent platforms for exploring novel non-Hermitian physics, such as unidirectional transportation WuJH2014 , spontaneous $\mathcal{PT}$ symmetry breaking with exceptional points KreibichM2014 ; LiJ2019 , fast eigenstate transition MostafaviF2019 and novel superfluidity PanL2019 ; YamamotoK2019 , etc. In particular, significant effects and applications have been proposed or demonstrated in non-Hermitian photonics MakrisKG2008 ; RegensburgerA2012 ; PengB2014 ; ZeunerJM2015 ; FengL2017 ; HouJ2019 .

The combination of topology and non-Hermiticity leads to the emergence of novel topological effects, such as anomalous edge states, non-Bloch waves, non-Hermitian skin effects, etc LeeTE2016 ; KunstFK2018 ; YaoS2018N ; YaoS2018 ; LieuS2018 ; ShenH2018 ; KawabataK2019 ; JinL2019 ; LinS2019 . In experiments, photonic MalzardS2015 ; StJeanP2017 ; BandresMA2018 ; PartoM2018 ; TakataK2018 ; ZhouH2018 ; CerjanA2018 ; HouJ2018 ; LuoXW2019 and atomic XuY2017 ; LiuT2019 systems are leading platforms for realizing non-Hermitian topological states. Although in most work the non-trivial topology is attributed to the Hermitian part of the Hamiltonian, it has been recently proposed that topological states may be solely induced by non-Hermiticity TakataK2018 with a focus on higher-order topological insulators LiuT2019 ; LuoXW2019 ; LeeChH2018 . The classification of non-Hermitian topological phases has been proposed through a reduction from AZ classes GongZ2018 , which remains incomplete because a random non-Hermitian matrix belongs to a broader BL class BernardD2001 . More recently, a comprehensive classification of non-Hermitian systems has been achieved through introducing the complex AZ*†* class KawabataKSymmetry2019 .

In Hermitian systems, it is known that 2D topological phases, such as a Chern insulator, can be simulated using a 1D lattice with staggered hopping or on-site potential LangLJ2012 characterized by an additional periodic parameter. The experimental realization of such 2D topological phases in momentum-parameter space could be significantly simpler, comparing to their 2D lattice counterparts. In this context, two natural questions arise for non-Hermitian systems: i) Can 2D non-Hermitian topological phases be realized using 1D lattices with solely non-Hermitian effects? 2) If so, is there any unique class of non-Hermitian topological phases that are difficult to realize in Hermitian systems?

In this work, we address these two important questions by showing that two unique symmetry-protected non-Hermitian topological phases can be realized in a 1D superlattice with staggered asymmetric hopping. Our main results are:

i) The 1D superlattice hosts a 2D topological insulator phase characterized by a $2\mathbb{Z}$ Chern number, which is protected by certain symmetry that is unique to non-Hermitian systems BernardD2001 . Such a phase is difficult to access in Hermitian systems because it is absent in the AZ classification and requires a crystalline symmetry as well as a global $\mathbb{Z}_{2}$ symmetry ShiozakiK2014 .

ii) A topological semimetal phase hosting complex Dirac points and zero-energy modes is found to be associated with a $\mathbb{Z}_{2}$ invariant, which is extracted from the normalized Berry phase for non-Hermitian systems.

iii) The system may support edge states with real energy due to a robust $\mathcal{PT}$-symmetric-like phase. In most non-Hermitian systems with on-site gain and loss, the $\mathcal{PT}$-symmetric phase becomes fragile with increasing system size. In contrast, the $\mathcal{PT}$-symmetric-like phase here is robust to varying chain length. Furthermore, these two topological phases can coexist in proper parameter spaces.

II 2D non-Hermitian topological phases in 1D superlattice with

asymmetric hopping

II.1 Model Hamiltonian

We consider a lattice model with nearest neighbor hopping, which can be described by a tight-binding Hamiltonian $H_{r}=\sum_{i}\left(t_{i,i+1}\hat{c}_{i}^{\dagger}\hat{c}_{i+1}+t_{i+1,i}\hat{c}_{i+1}^{\dagger}\hat{c}_{i}\right)+V_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}$. Here $\hat{c}_{i}^{\dagger}$($\hat{c}_{i}$) is the creation (annihilation) operator of local modes at site $i$, and the hopping term is non-Hermitian $t_{i,i+1}\neq t_{i+1,i}^{\ast}$. For simplicity of the presentation, we assume uniform on-site potential and staggered hopping terms $t_{i,i+1}=1+\lambda\cos(2\pi\alpha i+\phi_{L})$ and $t_{i+1,i}=1-\lambda\cos(2\pi\alpha i+\phi_{L})$, where $\lambda,\alpha\in\mathbb{R}$. We consider rational $\alpha$ that can be written as the quotient of two relatively prime integers $\alpha=p/q$, $p,q\in\mathbb{Z}$. Without loss of generality, $p,q$ are assumed to be positive and $p<\lceil q/2\rceil$. The corresponding Bloch Hamiltonian $H(k,\phi_{L})$ for the superlattice is a $q\times q$ matrix in the Brillouin zone $|k|\leq\pi/q$ with non-zero entries $H_{j,j}=V_{j}$, $H_{j,j+1}=t_{j,j+1}$, $H_{j+1,j}=t_{j+1,j}$ for $j+1<q$, $H_{1,q}=t_{1,q}e^{-iqk}$ and $H_{q,1}=t_{q,1}e^{iqk}$.

The resulting Bloch Hamiltonian $H(k,\phi_{L})$ preserves time-reversal symmetry $\mathcal{T}_{k}H(k,\phi_{L})\mathcal{T}_{k}^{-1}=H^{\ast}(k,\phi_{L})=H(-k,\phi_{L})$ with $\mathcal{T}_{k}=K$ and $K$ the complex conjugate, therefore the real (complex) part of the band is symmetric (antisymmetric) about $k=0$. Particle-hole symmetry yields $\mathcal{P}_{\phi_{L}}H(k,\phi_{L})\mathcal{P}_{\phi_{L}}^{-1}=H^{T}(k,-\phi_{L})$, where $\mathcal{P}_{\phi_{L}}$ has a permutation representation $(q,1)(q-1,2)(q-2,3)...$ in 2-cycle forms.

II.2 $2\mathbb{Z}$ Chern insulator

We start with the simplest case $\alpha=1/3$, which represents a Chern insulator, as shown in Fig. 1(a). The Bloch band is gapped by its real part in the entire momentum space for any $\phi_{L}$ and exhibits the symmetry $\omega(k)=\omega(-k)^{\ast}$. We also plot the open-boundary spectrum with varying $\phi_{L}$ in Fig. 1(b). Since the open-boundary Hamiltonian $H_{O}(\phi_{L})=H_{O}(\phi_{L}+\pi)^{T}$, the spectrum is symmetric about $\phi_{L}=\pi$. When the lattice size satisfies $N_{L}/q\in\mathbb{Z}$, the Hamiltonian on a cylindrical geometry enjoys a chiral symmetry $\mathcal{C}_{O}H_{O}(\phi_{L})\mathcal{C}_{O}^{-1}=H_{O}(\phi_{L})$ with $\mathcal{C}_{O}=I_{N_{L}/q}\otimes\mathcal{C}$, where $I_{n}$ is a $n\times n$ identity matrix and $\sigma_{i}$ are Pauli matrices. Thus the open-boundary spectrum is symmetric about $\omega=0$, as shown in Figs. 1(b) and (c). However, the spectra shown in Figs. 1(b) and (c) may not obey these symmetries exactly due to numeric errors, which are significantly enhanced when diagonalize a non-Hermitian matrix.

Two pairs of “chiral” surface waves emerge in each gap [Fig. 1(b)] for varying $\phi_{L}$. When $\alpha=1/5$, similar results with two or four pairs of edge modes in any of the four gaps are observed. In general, the number of the pair of edge states is always even, as expected for a $2\mathbb{Z}$ Chern insulator phase. The Hamiltonian satisfies $H(k,\phi_{L})=H^{T}(-k,\phi_{L}+\pi)$, which leads to $H(k,\phi_{L})=H^{\dagger}(k,\phi_{L}+\pi)$ when combined with time-reversal symmetry, yielding the $2\mathbb{Z}$ Chern insulator phase. When an extra term $\delta\phi_{L}\in\mathbb{R}$ is added in the asymmetric hopping $t_{i+1,i}=1-\lambda\cos(2\pi\alpha i+\phi_{L}+\delta\phi_{L})$ that breaks this symmetry, only one pair of surface waves are observed in each gap, as shown in Fig. 1(c).

The above 1D physics mimic the 2D integer quantum Hall effect in the $\left(k,\phi_{L}\right)$ momentum-parameter space. The band topology can be characterized by the Chern number

[TABLE]

where $a,b=L,R$, the Berry connections are defined as $\mathcal{A}_{k}^{ab}=-i_{a}\langle\psi(k,\phi_{L})|\partial_{k}\psi(k,\phi_{L})\rangle_{b}$ and $\mathcal{A}_{\phi_{L}}^{ab}=-i_{a}\langle\psi(k,\phi_{L})|\partial_{\phi_{L}}\psi(k,\phi_{L})\rangle_{b}$, and the right and left eigenstates are defined as $H(k,\phi_{L})|\psi(k,\phi_{L})\rangle_{R}=\omega(k,\phi_{L})|\psi(k,\phi_{L})\rangle_{R}$ and $H(k,\phi_{L})^{\dagger}|\psi(k,\phi_{L})\rangle_{L}=\omega^{\ast}(k,\phi_{L})|\psi(k,\phi_{L})\rangle_{L}$. Here the four Chern numbers $C^{ab}$ are equivalent ShenH2018 and $C^{RL}$ is numerically found to be $-2$, $4$ and $-2$ from bottom to top bands in Fig. 1(a). Thus the bulk topological invariant agrees well with the edge states through usual bulk-edge correspondence. The Chern numbers are also consistent with the 2$\mathbb{Z}$ Chern insulator and the detailed proof of exact 2$\mathbb{Z}$ quantization of Chern number can be found in Appendix A.

The edge states are also closely related to the 1D topology. When combine the particle-hole symmetry with $H(k,\phi_{L})=H^{T}(-k,\phi_{L}+\pi)$, we obtain an inversion (glide) symmetry along $k$ ($\phi_{L}$), $\mathcal{I}_{k}H(k,\phi_{L})\mathcal{I}_{k}^{-1}=H(-k,-\phi_{L}+\pi)$. Such a symmetry leads to high-symmetry points $\phi_{L}=\frac{\pi}{2}$ and $\frac{3}{2}\pi$, at which the edge states are degenerate and the corresponding 1D normalized Berry phase $\gamma_{n}(\phi_{L})=\frac{1}{\pi}\oint dk\mathcal{A}_{k}^{n}$ is quantized. Here $\mathcal{A}_{k}^{n}=\frac{1}{2}\left(\mathcal{A}_{k}^{RL}+\mathcal{A}_{k}^{LR}\right)$ and $n$ represents band index (see Appendix A for details). We plot the Berry phase at different $\phi_{L}$ in Fig. 1(d) and the color for each band is the same as in Fig. 1(a). The top (green) and bottom (blue) bands have the same normalized Berry phases, so only the green one is visible.

We note that the normalized Berry phase is also quantized at $\phi_{L}=0$ and $\phi_{L}=\pi$, which is not a coincidence, but a result of a symmetry $\mathcal{Q}H(k,\phi_{L})\mathcal{Q}^{-1}=H^{\dagger}(-k,-\phi_{L})$ with $\mathcal{Q}=\mathcal{P}_{\phi_{L}}\mathcal{T}_{k}$ (see Appendix A for details). A close observation reveals that $\gamma_{n}(\phi_{L})$ shows a period $\pi$ and is anti-symmetric to $\phi_{L}=\pi$. Note that Fig. 1(d) also demonstrates a charge pumping process with respect to $\phi_{L}$, where the accumulation of Berry phases from $\phi_{L}=0$ to $2\pi$ gives the Chern numbers of three bands. This is consistent with the $2\mathbb{Z}$ Chern number.

Finally, we remark that the summation of the normalized Berry phase always vanishes (for both odd and even $q$), which suggests that the global Berry phase used in studying similar 1D non-Hermitian systems LiangSD2012 ; TakataK2018 does not apply here. The topological invariant (Chern number and the following $\mathbb{Z}_{2}$ invariant) always sums to zero as the non-Hermiticity here does not affect the additivity in sum rule HatsugaiY2004 . This is a well-defined 2D non-Hermitian topological insulator characterized by a 2$\mathbb{Z}$ Chern number and could be realized in a 1D lattice with only non-Hermitian modulations.

II.3 $\mathbb{Z}_{2}$ topological semimetal

We now consider $\alpha=1/4$ and the corresponding 2D band structure in momentum-parameter space is shown in Fig. 2(a). The spectra obeys the symmetry $\omega(k)=\omega(-k)^{\ast}$ and is symmetric about $\omega=0$. The latter results from an additional chiral symmetry $\mathcal{C}=I_{q/2}\otimes\sigma_{z}$ with $\mathcal{C}^{-1}H(k,\phi_{L})\mathcal{C}=-H(k,\phi_{L})$ when $q$ is even. Such a symmetry allows a finite band gap at zero energy and is well-known for supporting zero-energy modes in 1D Hermitian systems. The spectra is gapped everywhere in the momentum-parameter space except at a few high-symmetry points, at which emerge Dirac fermions. This naturally makes the system a semimetal. These complex Dirac points are nontrivial and can be characterized by the normalized Berry phase $\frac{1}{\pi}\oint_{L}dkd\phi_{L}\mathcal{A}_{k,\phi_{L}}^{n}=1$, where $L$ is a loop encircling the complex Dirac point.

The open-boundary energy levels with respect to $\phi_{L}$ are given in Figs. 2(b) and (c). The spectrum is real for a weak modulation strength $\lambda$ and the two-fold degenerate edge modes have exact zero and purely real energy despite the underlying non-Hermiticity of the system. The gap closing points at certain inversion-symmetric points of $\phi_{L}$ and $k=0$ are the complex Dirac point in Fig. 2(a) and represents topological phase transition in the 1D picture.

The topological invariant associated with these zero modes comes from the normalized Berry phases $\gamma_{n}(\phi_{L})$ that are computed numerically in Fig. 2(d). Only the quantization of Berry phase at inversion-symmetric points survives because it does not require a gapped bulk (see Appendix A for details). The two bands $\Re(\omega)>0$ (the particle branch) have the same Berry phases as the two with $\Re(\omega)<0$ (the hole branch). We define the $\mathbb{Z}_{2}$ invariant $\gamma_{s}(\phi_{L})=\sum_{j}\gamma_{n,j}(\phi_{L})\mod 2$, where the summation runs over the hole branch and $\gamma_{s}(\phi_{L})=1/0$ corresponds to topological/trivial phase. We find $\gamma_{s}(\phi_{L})=1$ in the region $\frac{1}{4}\pi<\phi_{L}<\frac{3}{4}\pi$ and $\frac{5}{4}\pi<\phi_{L}<\frac{7}{4}\pi$, and 0 otherwise [see Fig. 2(e)], which agree with the edge states and gap closing in Figs. 2(a-c). Since the bulk bands could be degenerate at high-symmetric points, we apply an infinitesimal $\delta\phi_{L}$ in the hopping term $t_{i+1,i}$ to break the degeneracy so that both the Berry phases and $\mathbb{Z}_{2}$ invariant can be well defined (see Appendix B). Such a small perturbation does not break the chiral symmetry therefore it does not affect the topological properties discussed in this section.

Finally, we remark that an alternative definition of the $\mathbb{Z}_{2}$ invariant is to use the normalized non-Abelian Berry phase and take partial trace over the hole branch. This does not require the gap-lifting term but leads to the same result in Fig. 2(e).

II.4 Robust $\mathcal{PT}$-symmetric-like and mixing phases

The open-boundary spectrum is surprisingly real when the Bloch bands in the Brillouin zone are imaginary, which is very different from most non-Hermitian systems and is a result of the robust $\mathcal{PT}$-symmetric-like phase. The $\mathcal{PT}$ symmetry is generally fragile in the sense that the critical value of driving term becomes very small when the system size is very large YuceC2014 (see also Appendix C). In contrast, the $\mathcal{PT}$-symmetric-like phase here is robust and cannot be spontaneously broken when $\lambda<1$ regardless of the chain length $N_{L}$. As an example, we show the spectrum of an open chain with different length $N_{L}$ in Fig. 3(a) and (b). The spectrum becomes complex at a fixed modulation strength $\lambda=1$ for both $N_{L}=10$ and $N_{L}=120$. This notable feature of robust $PT$-symmetric-like phases ensures the reality of both bulk spectrum and edge modes in an extended parameter space.

To get insights of the $\mathcal{PT}$-symmetric-like phase, we consider two-level systems with the following two non-Hermitian Hamiltonians $H_{\text{gl}}=\sigma_{y}+i\lambda_{\gamma}\sigma_{z},\lambda_{\gamma}\in\mathbb{R}$ and $H_{\text{ah}}=\sigma_{x}+i\lambda_{\mu}\sigma_{y},\lambda_{\mu}\in\mathbb{R}$. The former one defines the simplest $\mathcal{PT}$-symmetric model with balanced gain and loss while the latter one corresponds to the asymmetric hopping discussed in this work. It is obvious that $H_{\text{ah}}$ is equivalent to $H_{\text{gl}}$ through basis rotation $\sigma_{x}\rightarrow\sigma_{z}$, $\sigma_{y}\rightarrow\sigma_{x}$ and $\sigma_{z}\rightarrow\sigma_{y}$, which explains the $\mathcal{PT}$-symmetric-like phase in our system.

The $2\mathbb{Z}$ Chern insulator exists for any $q>2$, while the topological semimetal requires chiral symmetry and even $q$, which suggest that these two distinct phases can mix for a large even $q$. We consider $\alpha=1/8$ and the Bloch band is plotted in Fig. 3(c), which also exhibits flat band at high-symmetry points in the parameter space as shown in Fig. 3(d). The real open-boundary spectrum for the mixed phase is shown in Fig. 3(e). The topological properties can be similarly characterized by taking into account of band degeneracy (see Appendix D for details).

III Discussion and conclusion

Our model uses non-Hermitian nearest-neighbor hoppings on a lattice structure with uniform on-site potential. This could be realized in platforms like cold atoms in optical lattice or array of coupled waveguides. Many schemes have been proposed to realize asymmetric hoppings in both platforms LonghiS2015 ; LiuT2019 . A few sites with small period $q=2$ or $4$ is sufficient for versatile topological phases (see Appendix B) and it is readily accessible in current experiments. The chiral edge waves and zero-energy modes have real energies in broad parameter spaces, therefore they are free of dissipations/amplifications over time and are easier to observe in experiments comparing to imaginary edge modes in many other models YuceC2014 ; TakataK2018 .

There are many other non-Hermitian topological phases that could be explored in this system by considering, for instance, other types of non-Hermitian hoppings, staggered $V_{i}$, tilted lattice, and long-range hopping. These new ingredients could bring richer physics as demonstrated already in certain bipartite superlattice LieuS2018 ; LeeTE2016 ; YaoS2018 .

In conclusion, we show that two unique non-Hermitian 2D topological phases – a 2$\mathbb{Z}$ Chern insulator and a $\mathbb{Z}_{2}$ topological semimetal – can be realized in a 1D lattice with staggered non-Hermitian hoppings. These phases can be experimentally realized in photonic or atomic systems and may open an avenue for exploring novel classes of non-Hermitian topological phases with 1D superlattices.

Note added. During the finalization of this work arxivversion , we notice a recent paper studying non-Hermitian Aubry-André-Harper models ZengTopological2020 . We would like to point out that the topological properties of our model originate only from the non-Hermitian effects and thus require $\lambda\neq 0$, yet their model is topological even when the non-Hermitian effects vanish $\lambda=0$. Our work reveals that even pure non-Hermiticity in 1D could generate nontrivial topological phases in 2D.

Acknowledgements.

This work is supported by Air Force Office of Scientific Research (FA9550-16-1-0387), National Science Foundation (PHY-1505496, PHY-1806227), and Army Research Office (W911NF-17-1-0128). This work is also supported in part by NSFC under the grant No. 11504285 and the Scientific Research Program Funded by Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JQ1058).

Appendix A Proof of Berry phase periodicity/quantization and the 2$\mathbb{Z}$

Chern number

In the main text, a few statements are made based upon physical considerations and are supported with numerics. Here, more rigorous proofs are provided for some important statements. In this section, we denote $k_{y}\equiv\phi_{L}$ to write the momentum-parameter space $(k,\phi_{L})$ as $\mathbf{k}=(k_{x},k_{y})$ for easy notation.

The right and left eigenvectors are defined as

[TABLE]

with the normalization condition ${}_{L}\langle\psi(\mathbf{k})|\psi(\mathbf{k})\rangle_{R}=1$. There are four different definitions of Berry connection $\mathcal{A}_{\mathbf{k}}^{ab}=-i_{a}\langle\psi|\partial_{\mathbf{k}}\psi\rangle_{b}$, where $a,b=L,R$. A natural generalization from Hermitian systems would be $\mathcal{A}_{\mathbf{k}}^{LR}$, which is not generally correct. While $\mathcal{A}_{\mathbf{k}}^{aa}$ is always real because ${}_{a}\langle\psi|\partial_{\mathbf{k}}\psi\rangle_{a}$ is purely imaginary due to the normalization condition, $\mathcal{A}_{\mathbf{k}}^{ab},a\neq b$ could be any complex number since the normalization condition only restricts ${}_{a}\langle\psi|\partial_{\mathbf{k}}\psi\rangle_{b}+{}_{a}\langle\partial_{\mathbf{k}}\psi|\psi\rangle_{b}=0$. To resolve this, we consider a normalized Berry connection

[TABLE]

which is purely real because \mathcal{A}_{\mathbf{k}}^{n}-\mathcal{A}_{\mathbf{k}}^{n\ast}=-\frac{i}{2}\big{(}{}_{R}\langle\psi|\partial_{\mathbf{k}}\psi\rangle_{L}+{}_{L}\langle\psi|\partial_{\mathbf{k}}\psi\rangle_{R}+{}_{L}\langle\partial_{\mathbf{k}}\psi|\psi\rangle_{R}+{}_{R}\langle\partial_{\mathbf{k}}\psi|\psi\rangle_{L}\big{)}=-\frac{i}{2}\partial_{\mathbf{k}}\big{(}{}_{R}\langle\psi|\psi\rangle_{L}+{}_{L}\langle\psi|\psi\rangle_{R}\big{)}=0 or simply $\mathcal{A}_{\mathbf{k}}^{LR}=\left(\mathcal{A}_{\mathbf{k}}^{RL}\right)^{*}$. Such a well-defined Berry connection yields a (normalized) Berry phase $\gamma_{n}=\frac{1}{\pi}\oint d\mathbf{k\cdot}\mathcal{A}_{\mathbf{k}}^{n}$, which is real and quantized along a closed loop for any gapped non-Hermitian systems.

We now show that such a Berry phase $\gamma_{n}(k_{y})=\frac{1}{\pi}\oint dk_{x}\mathcal{A}_{k_{x}}^{n}(k_{y})$ would have a period $T$ if $H(k_{x},k_{y})=H^{\dagger}(k_{x},k_{y}+T)$ and the bands are real and gapped (a weaker condition is that the bands are separable by their real parts). Starting with the definition of left eigenvector in Equ. 2 and applying the aforementioned conditions, we have $H(k_{x},k_{y})|\psi(k_{x},k_{y}+T)\rangle_{L}=\omega^{\ast}(k_{x},k_{y}+T)|\psi(k_{x},k_{y}+T)\rangle_{L}$ so that $\mathcal{A}_{k_{x}}^{RL}(k_{y})=\mathcal{A}_{k_{x}}^{LR}(k_{y}+T)$ and vice versa. Now, we realize that $\mathcal{A}_{k_{x}}^{n}(k_{y})=\frac{1}{2}\left(\mathcal{A}_{k_{x}}^{RL}(k_{y})+\mathcal{A}_{k_{x}}^{LR}(k_{y})\right)=\frac{1}{2}\left(\mathcal{A}_{k_{x}}^{LR}(k_{y}+T)+\mathcal{A}_{k_{x}}^{RL}(k_{y}+T)\right)=\mathcal{A}_{k_{x}}^{n}(k_{y}+T)$. Integrating along $k_{x}$, we find the Berry phase has a similar periodicity $\gamma_{n}(k_{y})=\gamma_{n}(k_{y}+T)$. Another consequence here is that the Chern number will be the multiple of some integers due to the periodicity of Berry connection. We could also define a normalized Chern number $C^{n}$ based on the normalized Berry connection so that

[TABLE]

As C_{T}^{n}=\frac{1}{2\pi}\oint dk_{x}\int_{k_{y}}^{k_{y}+T}dk_{y}\big{(}\partial_{k_{x}}\mathcal{A}_{k_{y}}^{n}-\partial_{k_{y}}\mathcal{A}_{k_{x}}^{n}\big{)} must be quantized due to the quantized charge pumping along $k_{y}$ demanded by the periodicity of the Berry phase, it’s easy to see $C^{n}=N_{y}C_{T}^{n}$ belongs to a $N_{y}\mathbb{Z}$ class if $k_{y}$ has the period $k_{y}=k_{y}+N_{y}T$.

We also claim that the Berry phase is quantized at inversion-symmetric points. To show this, we consider the symmetry $\mathcal{I}_{k}H(k_{x},k_{y})\mathcal{I}_{k}^{-1}=H(-k_{x},-k_{y}+T)$. Such a symmetry dictates $H(k_{x},k_{y})|\mathcal{I}_{k}\psi(-k_{x},-k_{y}+T)\rangle_{R}=\omega(-k_{x},-k_{y}+T)|\mathcal{I}_{k}\psi(-k_{x},-k_{y}+T)\rangle_{R}$. When the system is gapped, we have $\mathcal{A}^{n}_{k_{x}}(k_{y})=\mathcal{A}^{n}_{-k_{x}}(-k_{y}+T)$, where the condition $\mathcal{I}_{k}^{2}=I$ is used. Then the Berry phase reads $\gamma_{n}(k_{y})=\frac{1}{\pi}\oint dk_{x}\mathcal{A}^{n}_{k_{x}}(k_{y})=\frac{1}{\pi}\oint dk_{x}\mathcal{A}^{n}_{-k_{x}}(-k_{y}+T)=-\frac{1}{\pi}\oint dk_{x}\mathcal{A}^{n}_{k_{x}}(-k_{y}+T)=-\gamma_{n}(-k_{y}+T)$. This implies that $\gamma_{n}(k_{y}^{\prime})=-\gamma_{n}(k_{y}^{\prime})$ if $k_{y}^{\prime}=k_{y}^{\prime}-T$ (i.e., the high-symmetry point), which means it can only take the quantized value 0 or 1.

We now go back to the specific model we discussed in the main text, where $T=\pi$ and $k_{y}=\phi_{L}$ has a period $2\pi$ ($N_{y}=2$). This means that the Berry phase satisfies $\gamma_{n}(\phi_{L})=\gamma_{n}(\phi_{L}\pm\pi)$ and the Chern insulator has a Chern number $2\mathbb{Z}$ which is determined by the symmetry $H(k,\phi_{L})=H^{\dagger}(k,\phi_{L}+\pi)$. Another constraint on Berry phase is $\gamma_{n}(\phi_{L})=-\gamma_{n}(-\phi_{L}\pm\pi)$ so that it is quantized at inversion-symmetric point $\phi_{L}=\pm\frac{\pi}{2}$. Combing the two conditions together, we find further $\gamma_{n}(\phi_{L})=\gamma_{n}(\phi_{L}+\pi)=-\gamma_{n}(-\phi_{L})$ and thus, the Berry phase is also quantized at $\phi_{L}=0$ and $\pi$. Note that the last relation is imposed by the $\mathcal{Q}$ symmetry, which can be similarly proved if the bulk band is gapped in the real part.

Appendix B Lifting band degeneracies in topological semimetal

In the topological semimetal phase, there could be band degeneracy in momentum space, which hinders the computation of Berry phase or $\mathbb{Z}_{2}$ invariant. In the following, we take the example $\alpha=1/4$ for illustration purpose. The Bloch Hamiltonian reads

[TABLE]

The bulk bands are

[TABLE]

at the high-symmetry point $k=0$. We notice that when $\cos(4\phi_{L})=-1$, i.e., $\phi_{L}=\frac{1}{4}\pi,\frac{3}{4}\pi,\frac{5}{4}\pi,\frac{7}{4}\pi$, the band gap closes in momentum space with $E_{+,-}=E_{-,+}=0$ and $E_{+,+}=-E_{-,-}=\sqrt{2(2-\lambda^{2})}$ when $\lambda<\sqrt{2}$, manifesting the topological phase transition that was discussed in the main text. There is always a degenerate point in the top $E_{+,\pm}=+\sqrt{2-\lambda^{2}}$ and bottom $E_{-,\pm}=-\sqrt{2-\lambda^{2}}$ two bands at $k=\frac{\pi}{4}$. Such a degeneracy prevents us from computing the topological invariant in momentum space, while it is irrelevant to the topological properties since it does not depend on $\phi_{L}$. To resolve this, we apply a perturbation term on the asymmetric hopping

[TABLE]

which perseveres the chiral symmetry so that it would not change the associated topological invariant. It breaks the degeneracy at $k=\pi/4$ but preserves the gap closings at topological phase transition points.

When a finite $\delta\phi$ is considered, a few interesting points raise. First, the topological semimetal phase can be realized in a 2-level system, i.e., $\alpha=1/2$ [see Fig. 4(a)], which represents some non-Hermitian SSH models that have been discussed in previous literatures LieuS2018 ; YaoS2018 . The $\mathbb{Z}_{2}$ index reduces to the 1D Zak phase of the ground state, which can correctly characterize the non-Hermitian SSH model with chiral symmetry.

A finite $\delta\phi$ would spoil the 2$\mathbb{Z}$ Chern number and thus, renders a $\mathbb{Z}$-type Chern insulator. Moreover, since it breaks the degeneracy within the bottom and top two bands at $\alpha=1/4$ respectively, there would be a gap opening, providing the possibility of supporting a nontrivial Chern insulator phase. We demonstrate these two points in Fig. 4(b), where we do observe one chiral surface wave (for a given propagating direction) residing within each gap. The zero modes naturally persist but are shifted with the topological phase transition points. The characterization of the topological properties are similar and a case study of a mix phase is presented in Appendix D.

Appendix C $\mathcal{PT}$-symmetric phase by gain/loss is fragile

In main text, we state that the usual $\mathcal{PT}$-symmetric phase by gain/loss is fragile, therefore the spectrum cannot be purely real in the separable regime YuceC2014 . To illustrate the fragile $\mathcal{PT}$-symmetric phase, we consider a simple model described by the following tight-binding Hamiltonian

[TABLE]

where $\gamma>0$ is on-site gain/loss rate. The Hamiltonian is defective at the exceptional point and the system enters the partially broken regime (part of the spectrum becomes complex) at the smallest $\gamma_{c}$ satisfying $\text{Det}(H_{PT}(\gamma_{c}))=0$. The determinant can be solved through the recursive equation $D_{n}=(-1)^{n}\gamma D_{n-1}-t^{2}D_{n-2}$ with boundary conditions $D_{1}=-i\gamma$ and $D_{2}=\gamma^{2}-t^{2}$. The critical values $\gamma_{c}$ for different chain lengths are shown in Fig. 5, where we see a fast drop of $\gamma_{c}$ when the chain length starts to increase and ultimately approaches zero. In the insets (a) and (b), we show the spectrum for varying $\gamma$ at different chain lengths 2 and 10. This argument also applies to more general cases such as larger unit cell, quasiperiodic potential, or higher dimensions. This explains the complex edge states observed in many models with on-site gain and loss YuceC2014 ; TakataK2018 .

Such a fragility does not happen when the non-Hermiticity is introduced by asymmetric hopping as illustrated in Fig. 3(a) and (b), thus our model enjoys real edge modes and is more accessible to experimental observation.

Appendix D Topological characterization of the coexisting phase

Although we have studied two topological phases individually, the coexisting phase can be characterized in a similar way if the band degeneracy is treated carefully. We still consider a small perturbation $\delta\phi_{L}$ when compute the Berry phase and the results are shown in Fig. 6(a). Due to the gapless phase, the Berry phase does not have the periodicity any more. However, it is still quantized at inversion-symmetric point because a small $\delta\phi_{L}$ only perturbs the system slightly. In Fig. 3(e), the chiral surface wave still crosses at the high-symmetry points, at which the Berry phases are quantized to non-trivial values [Fig. 6(a)].

We notice that the Chern number is not well-defined due to the topological phase transition point and degeneracies between top/bottom four bands. While $\delta\phi_{L}$ term allows computing the Chern number, it breaks the 2$\mathbb{Z}$ constraint, and the Chern numbers are odd for the top/bottom two bands (see also the charge pumping in Fig. 6(a)). When $\delta\phi_{L}$ is gradually tuned to 0, the band Chern number should not change because there is no topological phase transition. When $\delta\phi_{L}=0$, two bands become degenerate. If we take both bands as a single band, the Chern number is even and the chiral surface wave appears in pair in each gap.

In comparison, the zero modes can be directly characterized by the $\mathbb{Z}_{2}$ invariant, which is shown in Fig. 6(b), and it is consistent with the zero modes observed in Fig. 3(e).