Higher-order topological corner states induced by gain and loss

TL;DR

This paper demonstrates that higher-order topological corner states can be induced in trivial phases by adding staggered gain and loss, establishing a bulk-corner correspondence in non-Hermitian systems with potential experimental realizations.

Contribution

It introduces a biorthogonal nested-Wilson-loop and edge-polarization theory to characterize non-Hermitian higher-order topological phases with gain and loss.

Findings

Higher-order topological corner states emerge from gain/loss in trivial phases.

A general bulk-corner correspondence is established for non-Hermitian systems.

Topological invariants for non-Hermitian multipole moments are defined.

Abstract

The concept of topological phases has been generalized to higher-order topological insulators and superconductors with novel boundary states on corners or hinges. Meanwhile, recent experimental advances in controlling dissipation (such as gain and loss) open new possibilities in studying non-Hermitian topological phases. Here, we show that higher-order topological corner states can emerge by simply introducing staggered on-site gain/loss to a Hermitian system in trivial phases. For such a non-Hermitian system, we establish a general bulk-corner correspondence by developing a biorthogonal nested-Wilson-loop and edge-polarization theory, which can be applied to a wide class of non-Hermitian systems with higher-order topological orders. The theory gives rise to topological invariants characterizing the non-Hermitian topological multipole moments (i.e., corner states) that are protected by reflection or chiral symmetry. Such gain/loss induced higher-order topological corner states can be experimentally realized using photons in coupled cavities or cold atoms in optical lattices.

Equations

$$H(\mathbf{k})$$

$$W_{h,\mathbf{k}}=\mathcal{P}\exp [i{\int }_{k_x}^{k_x+2\pi }{A}_h(k_x^{\prime },k_y)d{k}_x^{\prime }]\text{,}$$

$$p_v^{{\epsilon }_h}=-\frac{i}{4{\pi }^2}\int d{k}_x\log \det [{\tilde{W}}_{h,\mathbf{k}}]\text{.}$$

$$C_{\tau }=\frac{1}{2}\sum _{s\ne \bar{s}}\left[{\tau }_s^{+}(\frac{1-{\tau }_{\bar{s}}^z}{2})+{\tau }_s^{-}(\frac{1+{\tau }_{\bar{s}}^z}{2})\right],$$

$$E=\pm \sqrt{2J^2+{\lambda }_h^2+{\lambda }_v^2-{\gamma }^2\pm 2\sqrt{(J^2-{\gamma }^2)({\lambda }_h^2+{\lambda }_v^2)\pm 2J^2{\lambda }_h{\lambda }_v}}.$$

$$|w_{h,j,\mathbf{k}}^{R(L)}⟩=\sum _{m=1}^{N_{\text{occ}}}|u_{m,\mathbf{k}}^{R(L)}⟩[|{\epsilon }_{h,j,\mathbf{k}}^{R(L)}⟩{]}_m\text{,}$$

$${\tilde{W}}_{h,\mathbf{k}}=\mathcal{P}\exp [i{\int }_{k_y}^{k_y+2\pi }{\tilde{A}}_v^{{\epsilon }_h}(k_x,k_y^{\prime })d{k}_y^{\prime }]\text{,}$$

$$p_h^{i_v}=\frac{1}{2\pi }\int d{k}_x\sum _{j,n,m,l}{\epsilon }_{h,j}\cdot {\left[⟨u_{n,k_x}^L|\right]}_{i_v,l}[⟨{\epsilon }_{h,j}^L|{]}_n\cdot {\left[|u_{m,k_x}^R⟩\right]}_{i_v,l}[|{\epsilon }_{h,j}^R⟩{]}_m,$$

$$W=-\frac{i}{\pi }\text{Tr}\left[\mathcal{P}\oint ⟨u_{\pm ,m}^L|{\partial }_{k_y}|u_{\pm ,n}^R⟩dk\right],$$

$$H(\mathbf{k})$$

$$E(\mathbf{k})=\pm \sqrt{{\lambda }_h^2+{\lambda }_v^2+2J^2-2{\gamma }^2+2J{\lambda }_h\cos (k_x)+2J{\lambda }_v\cos (k_y)+2iJ\gamma [\sin (k_x)+\sin (k_y)]},$$

$$H(\mathbf{k})$$

Full text

Introduction.— Topological states of matter xiao2010berry ; hasan2010colloquium ; qi2011topological ; RevModPhys.88.035005 ; Sato2017Topological have been widely studied in various systems ranging from solid-state PhysRevLett.95.146802 ; Bernevig2006quantum ; Konig2007quantum , over cold atomic PhysRevLett.111.185301 ; PhysRevLett.111.185302 ; Goldman2014light ; Aidelsburger2014measuring ; Jotzu2014Experimental ; Li2016Bloch ; Flaschner2016Experimental ; Goldman2016topological ; Cooper2018topological to photonic haldane2008possible ; hafezi2011robust ; fang2012realizing ; lu2014topological ; kraus2012topological ; Hafezi2013imaging ; Ozawa2018Topological and acoustic He2018Topological ; Xiao2015Synthetic ; PhysRevLett.120.116802 ; PhysRevLett.114.114301 ; He2016Acoustic ; Ma2019Topological systems. The states are indexed by the bulk topological invariants that determine the boundary physics with lower dimensions. Recently, the concept has been generalized to higher-order topological insulators or superconductors with novel boundary states on corners or hinges PhysRevLett.110.046404 ; Benalcazar2017Quantized ; PhysRevLett.119.246401 ; PhysRevLett.119.246402 ; PhysRevB.96.245115 ; PhysRevB.98.241103 ; Schindler2018Higher ; Imhof2018Topolectrical-circuit ; Hassan2018Corner ; Serra-Garcia2018observation ; arXiv1812.09304 ; Peterson2018a ; PhysRevLett.120.026801 ; PhysRevB.97.241405 ; PhysRevB.97.205135 ; PhysRevLett.121.186801 ; PhysRevLett.121.096803 ; PhysRevX.9.011012 ; PhysRevB.98.205147 ; PhysRevLett.121.196801 ; arXiv1901.07579 . Different from conventional first-order topological states, the $d$-dimensional $n$-th order topological states can host $(d-n)$-dimensional gapless boundary states. The experimental realizations of such interesting higher-order topological states in photonic Hassan2018Corner ; Serra-Garcia2018observation ; PhysRevB.98.205147 ; arXiv1812.09304 and electrical circuit Imhof2018Topolectrical-circuit ; Peterson2018a systems further enlighten the research of these novel topological matters.

Meanwhile, the search for topological states of matter has also turned to open quantum systems characterized by non-Hermitian Hamiltonians Rept.Prog.Phys.70.947 , which exhibit a rich variety of unique properties without Hermitian counterparts Eur.Phys.J.Spec.Top227 . States modeled by non-Hermitian Hamiltonians appear in systems such as photonic structures with loss or gain PhysRevLett.100.103904 ; PhysRevLett.115.200402 ; PhysRevLett.115.040402 ; NatMater16433 ; Photon.Rev.6.A51 ; Nature488.167 ; PhysRevLett.113.053604 ; Science346Loss ; Zhao2018topological ; PhysRevLett.120.113901 ; Bandres2018Topological ; St-Jean2017Lasing , and cold atomic systems or solid-state materials with finite (quasi-)particle lifetime Ashida2017Parity ; PhysRevLett.121.026403 ; arXiv1802.00443 ; PhysRevB.98.035141 ; PhysRevLett.118.045701 ; arXiv1608.05061 ; Muller2012Engineered ; arXiv1811.06046 . The eigenvalues are generally complex, and the right and left eigenstates, satisfying biorthonormolity constrains, are no longer equivalent to each other (neither of them form an orthogonal basis). Moreover, more than one right eigenstates can coalesce at exceptional points PhysRevLett.118.045701 . Such unique properties lead to a rich variety of interesting topological phenomena (e.g., the non-Hermitian skin effects, exceptional rings, bulk fermi arcs, etc.), with bulk-boundary correspondence very different from the Hermitian systems PhysRevB.84.205128 ; PhysRevLett.116.133903 ; PhysRevLett.121.136802 ; PhysRevLett.121.213902 ; PhysRevLett.121.086803 ; PhysRevX.8.031079 ; PhysRevLett.121.026808 ; arXiv1808.09541 ; PhysRevB.97.045106 ; PhysRevLett.118.040401 ; PhysRevLett.120.146402 ; PhysRevA.93.062101 ; Xiong2018why ; Science359Observation ; arXiv1804.04676 ; PhysRevB.99.081103 ; arXiv1812.02011 ; arXiv1809.02125 ; arXiv1902.07217 .

The effects of non-Hermiticity on higher-order topological physics have been considered recently in a few works  arXiv1810.04067 ; arXiv1811.12059 ; arXiv1810.04527 ; arXiv1810.11824 ; arXiv1812.09060 , where the non-Hermiticity is induced by asymmetric tunnelings, leading to the observation of interesting phenomena such as higher-order skin effect arXiv1810.04067 and biorthogonal bulk polarization arXiv1812.09060 . Nevertheless, a general bulk-corner correspondence of the non-Hermitian higher-order topological states is still elusive. In addition, compared to asymmetric tunnelings, a simpler and more tunable way for introducing non-Hermiticity in photonic and atomic experiments is to control the on-site particle dissipations directly. Therefore two natural questions arise: i) Can higher-order topological states be induced by simply controlling the on-site gain or loss? ii) Is there a general bulk-corner correspondence for the non-Hermitian higher-order topological states?

In this Letter, we address these two important questions by considering a 2-dimensional (2D) lattice model with staggered on-site particle gain/loss. Our main results are:

i) The non-Hermitian particle gain and loss can drive the system from a trivial phase to a second-order topological phase with the emergence of four degenerate corner states.

ii) We develop the biorthogonal nested-Wilson-loop and edge-polarization approach which gives rise to bulk topological invariants responsible for the gapless corner states. The topological invariants are protected by reflection or chiral symmetries. In the presence of additional $C_{4}$ rotation symmetry, the topology can also be characterized by a quantized biorthogonal winding number.

iii) Although we focus on 2D reflection-symmetric case, our model and the bulk-corner correspondence can be generalized to study $d$-dimensional $d$-th order non-Hermitian topological states with either reflection or chiral symmetries.

iv) Simple experimental schemes based on photons in coupled cavities and cold atoms in optical lattices are proposed. Our system only relies on the manipulation of on-site particle gain/loss, and is ready for experimental exploration.

The model.— We consider a 2D lattice model with staggered tunnelings along both horizontal and vertical directions, as shown in Fig. 1(a). There is an effective magnetic flux $\phi=\pi$ for each plaquette, which appears as the tunneling phases on the dashed lines. The non-Hermiticity is introduced by the particle loss (gain) on all blue (red) lattice sites. We choose 16 orbitals in Fig. 1(a) as our unit cell with horizontal and vertical primitive-lattice vectors. The Hamiltonian reads

[TABLE]

where $J_{h,v}>0$ ($\lambda_{h,v}>0$) are the nearest-neighbour tunneling amplitudes between red and blue (circle and square) sites, $\mathbf{\sigma}_{h,v}$ ($\mathbf{\tau}_{h,v}$) are the Pauli matrices for the degrees of freedom spanned by red and blue (circle and square) sites, and $h,v$ represent the horizontal and vertical directions, respectively. $\sigma_{h,v}^{\phi}=\sigma_{h,v}^{z}$ for $\phi=\pi$. The gain/loss rate $\gamma$ in Eq. Higher-order topological corner states induced by gain and loss is positive since the blue sites are lossy. Alternatively, we may consider a different gain/loss configuration with gain (loss) on blue (red) sites, which simply changes $\gamma$ to negative. In experiments, the Hamiltonian can be realized using cold atoms in optical lattices or photons in coupled cavities SM . Fig. 1 (b) is an example based on arrays of coupled micro-ring cavities, where the coupling amplitude and phase between neighbour cavities, and the photon gain/loss for each cavity can be controlled independently arXiv1812.09304 ; Zhao2018topological .

Corner states.— For simplicity, we assume $J_{h}=J_{v}\equiv J$ throughout this paper, and the physics for $J_{h}\neq J_{v}$ is similar. The system has 16 bands SM , which appear in pairs $E(\mathbf{k})=-E^{\ast}(\mathbf{k})$ due to the pseudo-anti-Hermiticity $\eta H\eta=-H^{\dagger}$ with $\eta=\sigma_{h}^{z}\sigma_{v}^{z}\tau_{h}^{z}\tau_{v}^{z}$. We are interested in the half-filling gap around $\text{Re}[E]=0$. We focus on the region $\lambda_{h(v)}\leq J$ (the system stays in the trivial insulating/metal phase at the Hermitian limit $\gamma=0$ Benalcazar2017Quantized ; PhysRevB.96.245115 ), and show that the second-order topological corner states can be induced solely by non-Hermitian gain and loss.

In Figs. 2(a) and (b), we plot the energy spectrum as a function of $\gamma$, with open boundaries along both directions. Effectively, the particle loss reduces the tunnelings between gain and loss sites, while the tunnelings between two loss (gain) sites are not affected. We see that as $|\gamma|$ increases, the bulk gap closes and reopens (the small derivation is the finite size effect) at a critical point $\gamma_{c}$, leading to a topological phase transition with the emergency of four in-gap states. The typical density distributions of these in-gap states are shown in Fig. 2(c), which are well localized at four corners. We emphasize that our system does not suffer from the non-Hermitian skin effects due to the trivial eigenenergy vorticity PhysRevX.8.031079 $\oint\partial_{\mathbf{k}}\text{Arg}[E(\mathbf{k})]d\mathbf{k}=0$ for any loop in the momentum space, therefore it does not matter whether the right or/and left eigenstates are used to calculate the density distribution. As a result, the bulk states of $H(\mathbf{k})$ do distribute in the bulk [see Fig. 2(d)], and the open-boundary bulk spectrum is the same as that for periodic boundaries. We set $\lambda_{h}=\lambda_{v}$ in Fig. 2, therefore the system undergoes a bulk gap closing across the topological phase transition due to the $C_{4}$ symmetry PhysRevB.97.205135 . In general, the second-order topology can be altered by the gap closing in either the bulk or edge spectrum, and the emergency of corner states does not require bulk energy gap closing for $\lambda_{h}\neq\lambda_{v}$ PhysRevB.96.245115 ; PhysRevB.97.205135 , which will be further illustrated.

Topological invariants.— For Hermitian systems, it was shown that the topology of the nested Wilson loop and edge polarization are responsible for the corner states Benalcazar2017Quantized ; PhysRevB.96.245115 . Here we develop their non-Hermitian counterparts and show that the non-Hermitian corner states are originated from the topology of the generalized biorthogonal nested Wilson loops and edge polarizations. We consider a general Hamiltonian $H(\mathbf{k})$ on a torus with periodic boundaries and define the biorthogonal Wilson loop operator as

[TABLE]

where $A_{h}(\mathbf{k})=-i\langle u_{n,\mathbf{k}}^{L}|\partial_{k_{x}}|u_{m,\mathbf{k}}^{R}\rangle$ is the biorthogonal non-Abelian Berry connection in the horizontal direction, $|u_{m,\mathbf{k}}^{R,L}\rangle$ are the $m$-th occupied right and left Bloch eigenstates satisfying $H(\mathbf{k})|u_{m,\mathbf{k}}^{R}\rangle=E_{m}(\mathbf{k})|u_{m,\mathbf{k}}^{R}\rangle$, $H^{\dagger}(\mathbf{k})|u_{m,\mathbf{k}}^{L}\rangle=E_{m}^{\ast}(\mathbf{k})|u_{m,\mathbf{k}}^{L}\rangle$ and $\langle u_{n,\mathbf{k}}^{L}|u_{m,\mathbf{k}}^{R}\rangle=\delta_{n,m}$, and $\mathcal{P}$ is the path-ordering operator. Different from the Hermitian case Benalcazar2017Quantized , $W_{h,\mathbf{k}}$ may no longer be a unitary operator, and leads to a non-Hermitian Wannier Hamiltonian $H_{W_{h}}(\mathbf{k})=-\frac{i}{2\pi}\log W_{h,\mathbf{k}}$, which also has different left and right eigenstates, that is, $H_{W_{h}}(\mathbf{k})|\varepsilon_{h,j,\mathbf{k}}^{R}\rangle=\varepsilon_{h,j,k_{y}}|\varepsilon_{h,j,\mathbf{k}}^{R}\rangle$, $H_{W_{h}}^{\dagger}(\mathbf{k})|\varepsilon_{h,j,\mathbf{k}}^{L}\rangle=\varepsilon_{h,j,k_{y}}^{\ast}|\varepsilon_{h,j,\mathbf{k}}^{L}\rangle$ with $\langle\varepsilon_{h,j,\mathbf{k}}^{L}|\varepsilon_{h,j^{\prime},\mathbf{k}}^{R}\rangle=\delta_{j,j^{\prime}}$ and $j$ the Wannier band index. The non-Hermitian Wannier bands (independent from $k_{x}$), which obey the identification $\text{Re}[\varepsilon_{h,j,k_{y}}]\equiv\text{Re}[\varepsilon_{h,j,k_{y}}]\text{ mod }1$, can carry topological invariants if they are gapped.

The biorthogonal vertical polarization for the Wannier band sector $\varepsilon_{h}$ can be defined as

[TABLE]

Here $\tilde{W}_{h,\mathbf{k}}$ is the biorthogonal nested Wilson loop along the vertical direction, which is defined on the Wannier sector $\varepsilon_{h}$ with non-Hermitian Wannier-band basis $|w_{h,j,\mathbf{k}}^{R(L)}\rangle=\sum_{m=1}^{N_{\text{occ}}}|u_{m,\mathbf{k}}^{R(L)}\rangle[|\varepsilon_{h,j,\mathbf{k}}^{R(L)}\rangle]_{m}$ ($N_{\text{occ}}$ is the number of occupied energy bands and $\langle w_{h,j,\mathbf{k}}^{L}|w_{h,j^{\prime},\mathbf{k}}^{R}\rangle=\delta_{j,j^{\prime}}$) SM . Similarly, we can obtain the biorthogonal nested Wilson loop along the horizontal direction and the corresponding polarization $p_{h}^{\varepsilon_{v}}$. There would be corner states when $p_{h,v}^{\varepsilon_{v,h}}$ are non-trivial.

On the other hand, even for trivial $p_{h,v}^{\varepsilon_{v,h}}$, one may still have corner states if the edge polarization is non-trivial PhysRevB.96.245115 . For non-Hermitian systems, we should use the biorthogonal edge polarization, which are obtained by considering a cylindrical geometry and calculating the pseudo-one-dimensional biorthogonal Wannier values ($\varepsilon_{h,j}$ or $\varepsilon_{v,j}$) and polarization ($p_{h}^{i_{v}}$ or $p_{v}^{i_{h}}$ with $i_{v}$ or $i_{h}$ the unit-cell index along the open direction) along the periodic direction (horizontal or vertical) SM . The second-order corner modes are characterized by the vanishing bulk polarization (i.e., $i_{v,h}$ away from 1 and $N_{v,h}$), but quantized non-zero edge-localized polarization $p_{h}^{\text{edge}}$ and/or $p_{v}^{\text{edge}}$ (i.e., $i_{v,h}$ near $1$ or $N_{v,h}$) SM .

In general, higher-order topological phases are protected by symmetries Benalcazar2017Quantized ; PhysRevB.96.245115 . We consider a Hamiltonian that respects either reflection symmetries $M_{h}H(k_{x},k_{y})M_{h}^{-1}=H(-k_{x},k_{y})$ and $M_{v}H(k_{x},k_{y})M_{v}^{-1}=H(k_{x},-k_{y})$, or chiral (sublattice) symmetry $\Xi H(k_{x},k_{y})\Xi^{-1}=-H(k_{x},k_{y})$, with symmetry operators given by $M_{h}$, $M_{v}$ or $\Xi$. Since the biorthogonal Wannier bands or values (on a torus or cylinder) change the signs under reflection operation, they are either flat bands locked at [math] or $\frac{1}{2}$, or appear in $\pm\varepsilon$ pairs for reflection-symmetric systems. The reflection symmetries also ensure the quantization of ($p_{h}^{\varepsilon_{v}}$, $p_{v}^{\varepsilon_{h}}$) and ($p_{h}^{\text{edge}}$, $p_{v}^{\text{edge}}$) with value [math] or $\frac{1}{2}$. Similar properties hold for the chiral-symmetric systems with non-Hermiticity induced by asymmetric tunneling SM .

The Wannier bands correspond to the position of the particle density cloud Benalcazar2017Quantized ; PhysRevB.96.245115 . We focus on the Wannier sectors $\in(0,\frac{1}{2})$ [or $\in(\frac{1}{2},1)$] which are responsible for the edge topology and corner states. Based on the Wannier-sector and edge polarizations, we define two topological invariants: $Q_{1}=4p_{v}^{\varepsilon_{h}}p_{h}^{\varepsilon_{v}}$ mod 2 [with $\varepsilon_{v,h}$ the Wannier sector $\in(0,\frac{1}{2})$] and $Q_{2}=2(p_{h}^{\text{edge}}+p_{v}^{\text{edge}})$ mod 2. For the topological phase, we have either $Q_{1}=1$ or $Q_{2}=1$; while for the trivial phase, we have both $Q_{1}=0$ and $Q_{2}=0$. The above bulk-corner correspondence can apply to any non-Hermitian systems with reflection or chiral symmetries, and are reduced to the normal nested-Wilson-loop and edge-polarization theory Benalcazar2017Quantized in the Hermitian limit.

Phase diagram.— As an example, we study the phase diagram of the model in Fig. 1 based on the biorthogonal topological invariants. The corresponding Hamiltonian satisfies reflection symmetries with $M_{h}=\sigma_{v}^{\phi}\sigma_{h}^{x}\tau_{h}^{x}$ and $M_{v}=\sigma_{v}^{x}\tau_{v}^{x}$. It also possesses the rotational symmetry $C_{4}H(k_{x},k_{y})C_{4}^{-1}=H(k_{y},-k_{x})$ if $\lambda_{h}=\lambda_{v}$, where $C_{4}=C_{\tau}\otimes C_{\sigma}$ with

[TABLE]

and $C_{\sigma}$ has a similar expression with $s,\bar{s}=\{h,v\}$. In Fig. 3(a), we show the phase diagram in the $\gamma$-$\lambda_{v}$ plane with $J/\lambda_{h}=\sqrt{2}$. The phase diagram is symmetric with respect to $\gamma=0$, so we focus on $\gamma\geq 0$. The right and left parts of the phase diagram belong to topological and trivial phases, with their boundary given by the solid line. The trivial phase enlarges with the phase boundary shifting rightward as we increase $J$. There are two topological phases: T-I with $(Q_{1},Q_{2})=(1,0)$ and T-II with $(Q_{1},Q_{2})=(0,1)$.

We first consider the $C_{4}$ symmetric case for $\lambda_{v}=\lambda_{h}$, with the open-boundary spectra shown in Fig. 2(a). The typical Wannier bands for the Hamiltonian Eq. Higher-order topological corner states induced by gain and loss with periodic boundaries are shown in Fig. 3(b). There are 8 Wannier bands, with four located around $\varepsilon=0$, two at $0<\text{Re}[\varepsilon]<\frac{1}{2}$ and two at $0>\text{Re}[\varepsilon]>-\frac{1}{2}$, forming three Wannier sectors labeled by $0,\pm$, as shown in Fig. 3(b). Only the ‘$\pm$’-Wannier sectors are responsible for the edge topology and corner states. In fact, the ‘0’-Wannier sector is trivial in the whole parameter space and the ‘$\pm$’-Wannier sectors always have the same topology. Due to the $C_{4}$ symmetry, we have $p_{h}^{\varepsilon_{v}=\pm}=p_{v}^{\varepsilon_{h}=\pm}$ and $p_{h}^{\text{edge}}=p_{v}^{\text{edge}}$, all of which jump from [math] to $\frac{1}{2}$ across the phase transition as $\gamma$ increases [see Fig. 3(c)]. We would like to point out that, with $C_{4}$ symmetry, the topology can also be characterized by the biorthogonal winding number $W$ along the high-symmetry line $k_{x}=k_{y}$ in the reflection-rotation ($C_{4}M_{h}$) subspace SM .

For $\lambda_{v}\neq\lambda_{h}$, the bulk energy gap persists [see Fig. 3(d)], and the phase transitions are driven by gap close/reopen in the edge spectra and the Wannier bands, which lead to polarization jumps. In the following, we focus on $\lambda_{v}<\lambda_{h}$ without loss of generality, and show how the topological invariants and phases change as we increase $\gamma$, as shown in Figs. 4(a) and (b). (i) First, the vertical Wannier bands $\varepsilon_{v,j,\mathbf{k}}$ close the gap between ‘0’ and ‘$\pm$’ sectors in the patterned region in Fig. 3(a) SM . Further increasing $\gamma$ reopens the gap and leads to the the jump of $p_{h}^{\varepsilon_{v}=\pm}$ from [math] to $\frac{1}{2}$. (ii) Then, the gap for the edge spectra closes and reopens on the red solid line, where $p_{v}^{\text{edge}}$ jumps from [math] to $\frac{1}{2}$ and the system enters the T-II phase with the emergence of corner states. Shown in Figs. 4(c) and (d) are the Wannier values ($\varepsilon_{h,j}$ and $\varepsilon_{v,j}$) and edge-polarization distribution ($p_{h}^{i_{v}}$ and $p_{v}^{i_{h}}$) for the phase T-II on a cylinder. (iii) Finally, $\varepsilon_{h,j,\mathbf{k}}$ close the gap between ‘+’ and ‘$-$’ sectors on the black solid line, where both $p_{v}^{\varepsilon_{h}=\pm}$ and $p_{h}^{\text{edge}}$ jump from [math] to $\frac{1}{2}$, and we reach the T-I phase. Both T-I and T-II phases support corner states, and they are distinguished by the edge topology SM . The T-II phase region shrinks to zero as $\lambda_{v}$ approaches $\lambda_{h}$, where all edge and Wannier-sector polarizations jump at the same $\gamma$ due to the $C_{4}$ symmetry. These phenomena are very different from the Hermitian case. Especially, one can only have the topological phase T-I for the Hermitian limit, where all edge polarizations must vanish as long as $Q_{1}=0$ PhysRevB.96.245115 . The appearance of phase T-II is a result of the interplay between the non-Hermiticity and the $C_{4}$ symmetry breaking SM .

Discussions.— It is possible to generalize our study by considering different flux configurations. As a simple example, one may consider $\phi=0$ and set $\sigma_{h,v}^{\phi}=\sigma_{h,v}^{0}$ in the Hamiltonian Eq. Higher-order topological corner states induced by gain and loss. For such a zero flux model, the Hermitian part is a gapless metal when $|\lambda_{h}-\lambda_{v}|\leq 2J$. The gain/loss term effectively reduces the tunneling $J$ and can open a topological gap with in-gap corner states SM . Moreover, it is straightforward to generalize our non-Hermitian model and bulk-corner correspondence to higher-dimensional systems (e.g., $3$D system supporting third-order topological phases with quantized octupole moment) SM . Finally, we consider a general asymmetric-tunneling model (without on-site gain/loss) as an example of chiral symmetric systems, and confirm the bulk-corner correspondence numerically SM . The asymmetric tunnelings break both the Hermiticity and reflection symmetries (other symmetries like $C_{4}$ rotation or reflection-rotation $C_{4}M_{h}$ are also broken). As we increase the strength of the non-Hermiticity (i.e., asymmetry), the system can transform from the trivial phase to the second-order topological phase with zero-energy modes at four corners, which are characterized by the non-trivial topology of the biorthogonal nested Wilson loops SM .

Conclusion.— In summary, we propose a scheme to realize non-Hermitian higher-order topological insulators by simply controlling the on-site gain or loss, and show that the non-Hermitian corner states are characterized by the bulk topology in the form of biorthogonal nested Wilson loops or edge polarizations. The generalized bulk-corner correspondence may work for a wide class of non-Hermitian $d$-dimensional $d$-order topological systems with reflection or chiral symmetries. The proposed model can be realized easily in experiments. Our work offers a tunable method for manipulating corner states through dissipation control, and paves the way for the study of various non-Hermiticity induced higher-order topological states of matter and the classifications of them.

Supplementary Materials

.1 Energy band structures

The Hamiltonian Eq. 1 in the main text has 16 energy bands, and the typical band structure is shown in Fig. S1. The bands are two-fold degenerate in both real and imaginary parts. There is an exceptional loop (or exceptional ring) for the occupied or unoccupied bands, which is trivial in the sense that a loop surrounding it has zero vorticity $\oint\partial_{\mathbf{k}}\text{Arg}[E(\mathbf{k})]d\mathbf{k}=0$ PhysRevLett.120.146402S . In the calculation of the Wannier bands, the exceptional loop is excluded which does not affect the integral because they are only two points in the horizontal or vertical direction. The numerical band structures show that the band gap is minimized at $\mathbf{k}=0$, which should be the band touching point if there are any gap closing. The energy bands have analytic expressions at $\mathbf{k}=0$, which are given by (all bands are two-fold degenerate)

[TABLE]

For the gap closing at zero energy $E=0$, we have the only solution $\lambda_{h}=\lambda_{v}$ and $\gamma=\sqrt{2(J^{2}-\lambda_{h}^{2})}$. As a result, the bulk band gap persists across the phase transition for $\lambda_{h}\neq\lambda_{v}$, while the gap closes at the phase transition point $\gamma=\gamma_{c}\equiv\sqrt{2(J^{2}-\lambda_{h}^{2})}$ for $\lambda_{h}=\lambda_{v}$.

Although our system has no parity-time (PT) symmetry for $\lambda_{h},\lambda_{v}\neq 0$, we find that when $\gamma$ is small, half of the eigenenergies are real inside the exceptional ring. As we increase $\gamma$, the exceptional ring shrinks to the point $\mathbf{k}=(0,0)$ and disappears when $\gamma>{J(\lambda_{h}+\lambda_{v})}/\sqrt{\lambda_{h}^{2}+\lambda_{v}^{2}}$, where Im$[E(\mathbf{k})]$ opens a gap. For the parameters in Figs. 2(a)(b) of the main text, Im$[E(\mathbf{k})]$ opens a gap at $\gamma\simeq 1.94$. Since the exceptional ring corresponds to degeneracy within the occupied (or unoccupied) bands Re$[E(\mathbf{k})]<0$ (or Re$[E(\mathbf{k})]>0$), it does not affect the topology for the gap at Re$[E(\mathbf{k})]=0$. The exceptional ring can be understood by noticing the pseudo-Hermiticity of the Hamiltonian in Eq. 1 of the main text, that is, $\chi_{h}H(\mathbf{k})\chi_{h}^{-1}=H^{{\dagger}}(\mathbf{k})$ and $\chi_{v}H(\mathbf{k})\chi_{v}^{-1}=H^{{\dagger}}(\mathbf{k})$ with $\chi_{h}=\exp(ik_{x}{\tau_{h}^{z}}/{2})\tau_{h}^{x}$ and $\chi_{v}=\exp(ik_{x}{\tau_{v}^{z}}/{2})\tau_{v}^{x}$. The pseudo-Hermiticity guarantees that for $H|u_{m}\rangle=E_{m}|u_{m}\rangle$, one has $E_{m}^{\ast}\langle u_{m}|\chi_{h}|u_{m}\rangle=E_{m}\langle u_{m}|\chi_{h}|u_{m}\rangle$ and $E_{m}^{\ast}\langle u_{m}|\chi_{v}|u_{m}\rangle=E_{m}\langle u_{m}|\chi_{v}|u_{m}\rangle$, which means that the spectrum $E_{m}=E_{m}^{\ast}$ is real when either $\langle u_{m}|\chi_{h}|u_{m}\rangle\neq 0$ or $\langle u_{m}|\chi_{v}|u_{m}\rangle\neq 0$. Outside the exceptional ring, the pseudo-Hermiticity is fully broken, i.e., $\langle u_{m}|\chi_{h}|u_{m}\rangle=\langle u_{m}|\chi_{h}|u_{m}\rangle=0$ for all states $m$. While inside the exceptional ring, the pseudo-Hermiticity is partially broken, namely, $\langle u_{m}|\chi_{h}|u_{m}\rangle=\langle u_{m}|\chi_{h}|u_{m}\rangle=0$ for one half of the states $m$, and the other half satisfy either $\langle u_{m}|\chi_{h}|u_{m}\rangle\neq 0$ or $\langle u_{m}|\chi_{v}|u_{m}\rangle\neq 0$. For strong enough gain/loss rate, the exceptional ring disappears with gap opening in Im$[E(\mathbf{k})]$, where the pseudo-Hermiticity is fully broken in the whole Brillouin zone.

.2 Topological invariants and phases

Biorthogonal nested Wilson loop. In the main text, we define the biorthogonal nested Wilson loop, based on which we obtain the topological invariants. In particular, the biorthogonal nested Wilson loop is defined on the non-Hermitian Wannier band basis

[TABLE]

with $\langle w_{h,j,\mathbf{k}}^{L}|w_{h,j^{\prime},\mathbf{k}}^{R}\rangle=\delta_{j,j^{\prime}}$ and $N_{\text{occ}}$ the number of occupied bands. The biorthogonal nested Berry connection in the vertical direction can be defined as $\tilde{A}_{v}^{\varepsilon_{h}}(\mathbf{k})=-i\langle w_{j,h,\mathbf{k}}^{L}|\partial_{k_{y}}|w_{j^{\prime},h,\mathbf{k}}^{R}\rangle$ with $j$ and $j^{\prime}$ running over the Wannier band sector $\varepsilon_{h}$. The corresponding biorthogonal nested Wilson loop is

[TABLE]

Edge polarizations. For Hermitian systems with trivial Wannier bands, it was shown that the topological invariant is characterized by the edge polarizations PhysRevB.96.245115S . We find that this can also be applied to our non-Hermitian system by calculating edge polarizations in the biorthogonal basis. In particular, we consider a cylindrical geometry with open boundary along vertical direction, and treat the system as a pseudo-one-dimensional system along the horizontal direction with $N_{\text{occ}}\times N_{v}$ occupied bands ($N_{v}$ is number of unit cells along the vertical direction). Similar to the torus case with a fixed $k_{y}$ (as described in the main text), we can obtain the biorthogonal Wilson loop $\left[W_{h,k_{x}}\right]_{m,n}$, and the Wannier bands $\varepsilon_{h,j}$ with $m,n,j\in 1,2,...,N_{\text{occ}}\times N_{v}$. We define the horizontal polarization as a function of vertical site index $i_{v}$ as

[TABLE]

where $l$ is the orbital index in the unit cell $i_{v}$, $m,n$ are the occupied band indexes, and $j$ is the Wannier band index. Similarly, we may consider open boundary along the horizontal direction and obtain the vertical polarization $p_{v}^{i_{h}}$ as a function of horizontal unit-cell index $i_{h}$. The edge polarization $p_{h(v)}^{\text{edge}}$ is given by the summation of $p_{h}^{i_{v}}$ ($p_{v}^{i_{h}}$) near one edge PhysRevB.96.245115S .

Winding number. In the presence of additional $C_{4}$ symmetry, the Hermitian part of the Hamiltonian in the main text can be characterized by the winding number of its projection $H_{\pm}(\mathbf{k})$ onto the reflection-rotation ($C_{4}M_{h}$) subspace along the high-symmetry line $k_{x}=k_{y}=k$ [$H_{\pm}(k,k)$ acts on the subspace satisfying $C_{4}M_{h}=\pm 1$] Imhof2018Topolectrical-circuitS . For the non-Hermitian system, the winding number is evaluated on the biorthogonal basis, leading to the following biorthogonal winding number

[TABLE]

where $(m,n)$ runs over the occupied bands of $H_{\pm}(k,k)$ and $|u_{\pm,n}^{R,L}\rangle$ are the corresponding right and left eigenstates.

Topological phases. For $\lambda_{v}\neq\lambda_{h}$, we have $p_{h}^{\varepsilon_{v}=\pm}\neq p_{v}^{\varepsilon_{h}=\pm}$ and $p_{h}^{\text{edge}}\neq p_{v}^{\text{edge}}$, and these polarizations jump at different loss rate $\gamma$ due to the lack of $C_{4}$ symmetry. We focus on $\lambda_{v}<\lambda_{h}$ in the following (for $\lambda_{v}>\lambda_{h}$, the physics is similar except that the horizontal and vertical directions exchange their roles). In the patterned region in Fig. 3(a) of the main text, the vertical Wannier bands $\varepsilon_{v,j,\mathbf{k}}$ close the gap between ‘0’ and ‘$\pm$’ sectors, as shown in Fig. S2(a). The horizontal Wannier bands $\varepsilon_{h,j,\mathbf{k}}$ close and reopen the gap between ‘+’ and ‘$-$’ sectors at the phase boundary between topological phases T-I ($Q_{1}=1$, $Q_{2}=0$) and T-II ($Q_{1}=0$, $Q_{2}=1$), as shown in Fig. S2(b). For other $\gamma$, the Wannier bands are gapped. As an example, we plot the typical Wannier bands for the T-I phase in Figs. S2(c) and (d). The phase boundary between the trivial phase ($Q_{1}=0$, $Q_{2}=0$) and the topological phase T-II is characterized by the polarization jump only for $Q_{2}$ (i.e., $p_{v}^{\text{edge}}$), which is induced by the gap close/reopen in the edge spectra. Both $Q_{1}$ and $Q_{2}$ jump at the phase boundary between T-II and T-I, and both $p_{h,v}^{\text{edge}}$ and $p_{h,v}^{\varepsilon_{v,h}=\pm}$ become non-trivial (i.e., equals to $\frac{1}{2}$) in the phase T-I. Near the phase boundary, $p_{h}^{i_{v}}$ exponentially penetrates into the bulk, therefore we need to consider a large system to obtain the quantized edge polarization. The global polarization [i.e., the summation of $p_{h}^{i_{v}}$ ($p_{v}^{i_{h}}$) over all unit cell $i_{v}$ ($i_{h}$)] is always zero (mod 1).

These features are quite different from the Hermitian case, where all edge polarizations must be zero as long as $Q_{1}=0$. The topological phase T-II is a result of the interplay between the non-Hermiticity and the $C_{4}$ symmetry breaking. Even though the Wannier-sector polarization is trivial along vertical direction in phase T-II, the non-Hermitian particle loss can induce non-trivial vertical polarization on the horizontal edge (i.e., $i_{h}=1$ and $i_{h}=N_{h}$). In particular, the gapped edge states first appear on the horizontal boundaries due to the stronger horizontal coupling (i.e., $\lambda_{h}>\lambda_{v}$), then the particle loss $\gamma$ drives the phase transition of these edge states (from trivial phase $p_{v}^{\text{edge}}=0$ to T-II phase $p_{v}^{\text{edge}}=\frac{1}{2}$) prior to the jump of $Q_{1}$. Upon the transition from trivial phase to T-II phase, the edge-state gaps close and reopen.

As $\gamma$ increases further, the system enters the T-I phase with the appearance of fully separated edge states on both the horizontal and vertical boundaries (There are no fully separated edge states on the vertical edges in the trivial and T-II phases). In particular, we consider periodic (open) boundary condition in horizontal (vertical) direction. The edge spectrum exists for all momentum $k_{x}$ and is fully separated from the bulk spectrum in phase T-I; while in phase T-II, the edge spectrum merges into the bulk at certain momentum $k_{x}$ and disappears for a finite momentum interval (see Fig. S3). The change of edge spectrum leads to the change of edge polarization at the phase boundary between T-I and T-II phases.

Symmetry breaking perturbations. In general, higher-order topological phases are protected by symmetries PhysRevB.96.245115S . As a result, when both reflection and chiral symmetries are broken, both the multipole moments and the biorthogonal topological invariants are no longer quantized. In particular, if we introduce perturbations (e.g., $\delta\sigma_{h}^{z}\sigma_{v}^{z}$) such that the system breaks both reflection and chiral symmetries, the four corner states break their degeneracy and shift toward the bulk as the perturbation strength increases PhysRevB.96.245115S . Moreover, the biorthogonal nested-Wilson-loop and edge-polarization theory does not require additional symmetries such as the $C_{4}$ rotational or reflection-rotation symmetries.

.3 Zero flux $\phi=0$ case

For the zero flux case, the Hamiltonian becomes

[TABLE]

Here we consider $\lambda_{v}>\lambda_{h}\geq J$. When $\gamma=0$ and $|\lambda_{h}-\lambda_{v}|\leq 2J$, the Hamiltonian is in the gapless metal phase PhysRevB.96.245115S . As $\gamma$ increases (the tunneling $J$ is effectively reduced), a topological gap opens with the emergency of in-gap corner states [see Figs. S4 (a) and (b)]. $\lambda_{h}\neq\lambda_{v}$ is required for the appearance of the gap. Our numerical results show that the gap opens at momentum $\mathbf{k}=0$, where we have analytic solutions for the eigenenergies: $E(\mathbf{k}=0)=\pm\lambda_{v}\pm\lambda_{h}\pm i\gamma$ and $E(\mathbf{k}=0)=\pm\lambda_{v}\pm\lambda_{h}\pm\sqrt{4J^{2}-\gamma^{2}}$. The gap opens when $\lambda_{v}-\lambda_{h}-\sqrt{4J^{2}-\gamma^{2}}=0$, leading to the critical loss rate $\gamma_{c}=\sqrt{4J^{2}-(\lambda_{v}-\lambda_{h})^{2}}$.

In the metal phase, these is no well defined topological invariant. Even if there exist zero-energy corner states, they are embedded in the bulk spectra (no gap protection) and cannot be distinguished. Unfortunately, the topological invariant of such a model cannot be extracted from the biorthogonal nested Wilson loop because all the Wannier bands are locked at [math] or $\frac{1}{2}$ with trivial Wannier-sector polarizations. For Hermitian systems with trivial flat Wannier bands, the second-order topology is characterized by the biorthogonal edge polarizations PhysRevB.96.245115S . In Figs. S4 (c)-(f), we show the Wannier bands $\varepsilon_{h,j}$ and $\varepsilon_{v,j}$, as well as the polarization $p_{h}^{i_{v}}$ and $p_{v}^{i_{h}}$ in the topological insulator phase at a large $\gamma$. We see that the polarization is well localized at the edges with vanishing bulk distributions, leading to the non-trivial topological invariant $Q_{2}=1$.

.4 Chiral symmetric model

As we discussed in the main text, the biorthogonal topological invariant not only applies to reflection symmetric systems, but also to chiral symmetric systems with non-Hermiticity induced by asymmetric tunnelings. Because of the chiral symmetry, the occupied and unoccupied energy bands have the same Wannier bands (values), and their total summation should be flat and locked at [math] mod $1$. Thus, the Wannier bands (values) for occupied energy bands should be either flat bands locked at [math] or $\frac{1}{2}$, or appear in $\pm\varepsilon$ pairs. Interestingly, we find that the biorthogonal Wannier-sector or edge polarization is also quantized to [math] or $\frac{1}{2}$ mod $1$. This might be understood by considering the projection (both occupied and unoccupied bands) onto the Wannier basis as a smooth mapping to an effective one-dimensional model. The biorthogonal polarization should be quantized due to the chiral symmetry and the unoccupied bands give the chiral partner of the occupied bands.

As an example, we consider a similar asymmetric-tunneling model as that in Ref. arXiv1810.04067S . The chiral symmetric Hamiltonian under periodic boundary is

[TABLE]

where $\mathbf{\sigma}_{h,v}$ are the Pauli matrices for the degrees of freedom spanned by circle and square sites, and the inter-cell tunneling is asymmetric with $\lambda_{v,\pm}=\lambda_{v}\pm\gamma$, $\lambda_{h,\pm}=\lambda_{h}\pm\gamma$ and $\gamma>0$, as shown in Fig. S5 (a). The chiral symmetry is given by $\Xi=\sigma_{h}^{z}\sigma_{v}^{z}$, which flips the sign of all square sites, leading to $\Xi H(\mathbf{k})\Xi^{-1}=-H(\mathbf{k})$. The (two-fold degenerate) eigenenergies are

[TABLE]

which is gapless in the region $\sqrt{\frac{(J-\lambda_{h})^{2}+(J-\lambda_{v})^{2}}{2}}<\gamma<\sqrt{\frac{(\lambda_{h}+J)^{2}+(\lambda_{v}+J)^{2}}{2}}$ and gapped otherwise.

In the Hermitian limit, the system stays in the topological trivial phase in the region $J>\lambda_{h,v}$ (we assume $J>\lambda_{v}\geq\lambda_{h}$ without loss of generality). We find that a non-zero $\gamma$ not only breaks the Hermiticity and the reflection symmetry, but also drives the system to a second-order topological phase with corner states. The non-trivial topology is characterized by the biorthogonal nested Wilson loops (we also calculate the biorthogonal edge polarization which either leads to a trivial topological invariant or becomes ill defined due to the non-Hermitian skin effect). Three phases are identified for different $\gamma$: phase (I) for $\gamma<J-\lambda_{v}$; phase (II) for $J-\lambda_{v}<\gamma<J+\lambda_{v}$; phase (III) for $\gamma>J+\lambda_{v}$. In phases (I) and (III), both $E(\mathbf{k})$ and the Wannier bands $\varepsilon_{h(v),j,\mathbf{k}}$ [see Figs. S5 (b) and (c)] are gapped. We have $Q_{1}=1$ in phase (III) and $Q_{1}=0$ in phase (I). Therefore phase (III) is topological non-trivial with four zero-energy corner modes under open boundaries, and they are located at the four corners, respectively, leading to quantized quadruple moment [see Figs. S5 (d)]. In phase (II), either $E(\mathbf{k})$ or $\varepsilon_{h(v),j,\mathbf{k}}$ is gapless, and we do not have well defined $Q_{1}$. In all phases, the bulk states are located at one corner [see Fig. S5 (e)] when open boundaries are considered.

In Figs. S5 (g)-(i), we plot the complex energies as functions of $\gamma$ with open boundaries. We notice that the open-boundary bulk spectra is gapped everywhere (the imaginary energy gap opens before the real energy gap closes), which are quite different with the periodic case. Under open boundaries, there might still exist anomalous zero modes in phase (II); however, the four in-gap states are located at one (or two) corner(s) [see Fig. S5 (f)]. Thus they do not correspond to quantized quadruple moment, which is why we do not have a well-defined $Q_{1}$ in phase (II). Nevertheless, the in-gap states in phase (II) might be characterized by, for example, the non-Bloch theory PhysRevLett.121.136802S ; PhysRevLett.121.086803S ; arXiv1810.04067S , and the anomalous in-gap states are believed to be a result of the interplay between skin effect and finite size effect. Developing a general bulk-corner correspondence for such anomalous zero modes is also very interesting and can be addressed in future work.

We would like to emphasize that the direct diagonalization of the Hamiltonian Eq. S7 with open boundaries may not give the correct corner states in phase (III). This is because there are couplings (exponentially weak) between four corner states for a finite system, which mix the states at four corners. For such a mixed state, the skin effects wash out the components in three corners. However, we can isolate the state at each corner by an infinitesimal on-site detuning $\delta\sigma_{h}^{z}\sigma_{v}^{z}$ PhysRevB.96.245115S or by considering open boundaries with broken unit cell PhysRevLett.121.026808S . The skin effects only affect the spatial profiles (decay rates) of the corner states without changing their localizing corner positions in phase (III) with $Q_{1}=1$. While in phase (II), the skin effects become strong enough and all corner states are shifted to a single (or two) corner(s).

.5 Third-order topological phases in 3D systems

In this section, we show how to generalize our non-Hermitian model and bulk-corner correspondence to higher-dimensional systems. As an example, we consider a $3$D system supporting third-order topological phases with quantized octupole moment, as shown in Fig. S6. There is an effective magnetic flux $\phi=\pi$ for each plaquette, which appears as the tunneling phases on the dashed lines. The non-Hermiticity is introduced by the particle loss (gain) on all blue (red) lattice sites with rate $\gamma$. For simplicity, we choose the tunneling strengths to be the same for all green (black) links and denote them as $\lambda$ ($J$). The Hamiltonian in momentum space reads

[TABLE]

$\mathbf{\sigma}_{a,h,v}$ ($\mathbf{\tau}_{a,h,v}$) are the Pauli matrices for the degrees of freedom spanned by red and blue (circle and square) sites, and $a,h,v$ represent the $z$, $x$ and $y$ directions, respectively.

For $J>\lambda$, the system is in a trivial phase, and as we increase the gain/loss rate $\gamma$, the system undergoes a phase transition to the third-order topological phase at $\gamma=\gamma_{c}\equiv\sqrt{3(J^{2}-\lambda^{2})}$. In general, we have $\gamma_{c}\equiv\sqrt{d(J^{2}-\lambda^{2})}$ for such $C_{4}$-rotational symmetric case, with $d$ the system dimension. To obtain the biorthogonal topological invariants, we can define the biorthogonal Wilson loop operator along $k_{x}$, just like the 2D case. From which we can obtain the 2D Wannier Hamiltonian $H_{W_{h}}(k_{y},k_{z})$ and Wannier bands $\varepsilon_{h}(k_{y},k_{z})$ in the $k_{y}$-$k_{z}$ plane. Then we calculate the 2D biorthogonal topological invariants (i.e., the quadrupole moment) of the non-Hermitian Hamiltonian $H_{W_{h}}(k_{y},k_{z})$ by taking the Wannier sector $\varepsilon_{h}(k_{y},k_{z})\in(0,\frac{1}{2})$ as the effective “occupied bands” PhysRevB.96.245115S . Similarly, the biorthogonal edge polarizations can be generalized to the biorthogonal corner polarizations. We may consider periodic boundary along the $x$ direction and open boundary along the other two directions, then we treat the system as a pseudo-one-dimensional system and calculate the biorthogonal Wannier values $\varepsilon_{h,j}$ along the periodic direction. Using Eq. S4, we can get the $x$-direction polarization as a function of $y$- and $z$-direction site index $i_{v},i_{a}$, which should be well localized at the corners in the $i_{v}$-$i_{a}$ plane for the third-order topological phases. Moreover, one may also calculate the edge polarization of the 2D Wannier Hamiltonian $H_{W_{h}}(k_{y},k_{z})$, which should be non-trivial for the third-order topological phases PhysRevB.96.245115S .

.6 Experimental implementation

In the main text, we have shown that the Hamiltonian Eq. (1) in the main text can be realized using coupled arrays of micro-ring cavities, as shown in Fig. 1 in the main text. Each site is represented by a main cavity, which is coupled to its neighbor cavities through the auxiliary coupling cavities with controllable coupling strength and phase arXiv1812.09304S . The loss/gain of each cavity can also be controlled independently Zhao2018topologicalS .

The Hamiltonian can also be realized using cold atoms in optical lattices, with lattice potential shown in Fig. S7 (a). The Hermitian part can be realized within current techniques as proposed in Ref. Benalcazar2017QuantizedS . To obtain the on-site loss, we introduce the resonance couplings between the ground state and the excited state with a strong loss rate $\kappa$ arXiv1608.05061S ; Ashida2017ParityS , where the excited state feels the same lattice potential as the ground state [see Fig. S7 (b)]. The coupling $\Omega(x,y)$ between the ground state and excited state gives rise to the effective loss for the ground state $2\gamma=\frac{\Omega^{2}(x,y)}{\kappa}$ PhysRevA.85.032111S ; PhysRevX.8.031079S , and the staggered loss can be controlled easily by $\Omega(x,y)$ [Fig. S7 (b)]. Notice that the staggered loss configuration (without gain) is equivalent to the staggered gain-loss configuration up to a constant.

We also would like to point out that both the coupled cavities and optical lattices are able to realize the chiral non-Hermitian model with asymmetric tunnelings. The optical-lattice scheme has been proposed in arXiv1810.04067S . For the coupled cavities, the asymmetric coupling can be realized by introducing gain and loss to the two arms of the coupling cavity, respectively Sci.Rep.5.13376S .