Phase Diagram of a Non-Hermitian Chern Insulator: Destabilization of Chiral Edge States and Bulk–Boundary Correspondence

Bulk–boundary correspondence is a characteristic feature that manifests itself in various topological systems. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] A topological number defined in bulk geometry under a periodic boundary condition (pbc) plays a central role in the original scenario of bulk–boundary correspondence proposed by Hatsugai [11] and Ryu and Hatsugai. [12] The topological number governs the presence or absence of topological boundary states in the boundary geometry under an open boundary condition (obc); if it is nontrivial (trivial), topological boundary states appear (disappear) in the boundary geometry. Note that the original scenario employs the bulk and boundary geometries.

Inspired by an extension of quantum mechanics to the non-Hermitian regime, [13, 14, 15] extensive studies have been conducted on non-Hermitian topological systems. [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77] Bulk–boundary correspondence is an important subject of these studies. It was demonstrated that a straightforward application of the original scenario fails to describe the topological nature of some non-Hermitian topological systems. [20, 24]

A reason for the breaking of the bulk–boundary correspondence is a non-Hermitian skin effect; [28, 78, 79, 80, 81, 82, 83, 84, 85, 86] eigenfunctions in a non-Hermitian system under the obc are localized near a boundary of the system. This effect is hidden in the system under the pbc. Indeed, all eigenfunctions under the pbc extend over the entire system. Therefore, a topological number defined in the bulk geometry under the pbc never reflects the non-Hermitian skin effect in the boundary geometry. Several scenarios without using the bulk geometry have been proposed. [28, 38, 27, 52] Although these scenarios can avoid difficulty concerning a non-Hermitian skin effect, they do not share the notable feature of the original scenario that a topological number defined in closed geometry governs the presence or absence of topological boundary states.

To restore the bulk–boundary correspondence in non-Hermitian topological systems, another scenario has been proposed in Refs. \citenimura1 and \citenimura2. This scenario employs boundary geometry under the obc and bulk geometry under a modified periodic boundary condition (mpbc). Let $|\Psi^{R}\rangle=\sum_{n=1}^{N}\varphi(n)|n\rangle$ be a right eigenstate of a one-dimensional lattice system consisting of $N$ sites. The mpbc is defined by $\varphi(N+n)=b^{N}\varphi(n)$, where $b$ is a positive real constant. [47] The wavefunction $\varphi(n)=\beta^{n}$ satisfies this boundary condition when $\beta=be^{ika}$, where $a$ is the lattice constant. The case of $b=1$ corresponds to the ordinary pbc. If $b>1$ ($b<1$), the mpbc selects wavefunctions that exponentially increase (decrease) with increasing $n$. This behavior of $\varphi(n)$ resembles that caused by a non-Hermitian skin effect. Therefore, by using the bulk geometry under the mpbc, we can take the non-Hermitian skin effect into account in calculating a topological number.

The scenario in Refs. \citenimura1 and \citenimura2 succeeds in describing the bulk–boundary correspondence in one-dimensional non-Hermitian topological insulators. The present author [71] gave a recipe for executing the scenario in two-dimensional systems and applied it to a Chern insulator with gain/loss-type non-Hermiticity, [29] obtaining a phase diagram in the boundary geometry. The phase diagram separates three regions: a topologically nontrivial Chern insulator phase, a topologically trivial insulator phase, and a gapless phase. In Ref. \citentakane1, the phase boundary between the nontrivial and trivial regions and that between trivial and gapless regions are shown to be consistent with spectra in the boundary geometry. However, the phase boundary between the nontrivial and gapless regions is not fully consistent with spectra.

In this paper, we examine the bulk–boundary correspondence in the Chern insulator with gain/loss-type non-Hermiticity by using a revised recipe. A key observation is that the inconsistency reported in Ref. \citentakane1 is caused by a peculiar gap closing in the boundary geometry: if chiral edge states linking conduction and valence bands are destabilized by non-Hermiticity, the two bands are combined into one band without an obvious gap closing. This is beyond the scope of the recipe in Ref. \citentakane1. We thus revise the recipe of the bulk–boundary correspondence such that it can take the peculiar gap closing into account and show that the revised recipe enables us to describe the topological features of the system without any inconsistency. Indeed, we obtain a phase diagram that is fully consistent with spectra in the boundary geometry.

In the next section, we present a tight-binding Hamiltonian for the non-Hermitian Chern insulator with gain/loss-type non-Hermiticity [29] and specify two symmetries of it. Complex spectra of this system are constrained by these symmetries. In Sect. 3, we introduce right and left eigenstates of the Hamiltonian as a preparation for theoretical analyses. In Sect. 4, we consider chiral edge states in the geometry with the obc in one direction and the mpbc in the other direction. This consideration reveals important characteristics of chiral edge states in the boundary geometry, which are used in the following two sections. Section 5 is the central part of this paper. We give a revised recipe for executing the scenario of the bulk–boundary correspondence and obtain a phase diagram in the boundary geometry by using it. In Sect. 6, we justify the resulting phase diagram by comparing it with spectra of the system in the boundary geometry. The last section is devoted to a summary and discussion.

We adopt a tight-binding model for non-Hermitian Chern insulators on a square lattice with lattice constant $a$. The Hamiltonian is given by $H=H_{\rm d}+H_{x}+H_{y}$ with [29]

	$\displaystyle H_{\rm d}$	$\displaystyle=\sum_{m,n}|m,n\rangle\left[i\gamma\left(\sigma_{x}+\sigma_{y}\right)+M\sigma_{z}\right]\langle m,n|,$		(1)
	$\displaystyle H_{x}$	$\displaystyle=\sum_{m,n}|m+1,n\rangle\left[\frac{i}{2}v\sigma_{x}-\frac{1}{2}\sigma_{z}\right]\langle m,n|+{\rm h.c.},$		(2)
	$\displaystyle H_{y}$	$\displaystyle=\sum_{m,n}|m,n+1\rangle\left[\frac{i}{2}v\sigma_{y}-\frac{1}{2}\sigma_{z}\right]\langle m,n|+{\rm h.c.},$		(3)

where $M$, $\gamma$, and $v$ are real parameters, $m$ and $n$ specify the location of each site in the $x$- and $y$-directions, and $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$ are the $x$-, $y$-, and $z$-components of Pauli matrices, respectively. $|m,n\rangle$ and $\langle m,n|$ consist of spin-up and -down components:

	$\displaystyle|m,n\rangle$	$\displaystyle=\left[\begin{array}[]{cc}|m,n\rangle_{\uparrow}&|m,n\rangle_{\downarrow}\end{array}\right],$		(5)
	$\displaystyle\langle m,n|$	$\displaystyle=\left[\begin{array}[]{c}{}_{\uparrow}\langle m,n|\\ {}_{\downarrow}\langle m,n|\end{array}\right],$		(8)

where $|m,n\rangle_{\sigma}$ and ${}_{\sigma}\langle m,n|$ with $\sigma=\uparrow$, $\downarrow$ are, respectively. column and row vectors. If the number of sites in the system is $N_{\rm site}$, the column and row vectors consist of $2N_{\rm site}$ components, where the factor $2$ reflects the spin degrees of freedom. In $|m,n\rangle_{\sigma}$ and ${}_{\sigma}\langle m,n|$, only one component corresponding to the $(m,n)$th site with $\sigma$ is $1$ and other components are $0$. They satisfy ${}_{\sigma}\langle m,n|={}^{t}|m,n\rangle_{\sigma}$ and

$\displaystyle{}_{\sigma}\langle m,n|m^{\prime},n^{\prime}\rangle_{\sigma^{\prime}}=\delta_{m,m^{\prime}}\delta_{n,n^{\prime}}\delta_{\sigma,\sigma^{\prime}}.$

(9)

As a consequence, $H$ is a $2N_{\rm site}\times 2N_{\rm site}$ matrix. The Hamiltonian $H$ is normalized such that the coefficients of $\sigma_{z}$ in $H_{x}$ and $H_{y}$ become $-\frac{1}{2}$ and therefore $H$ is dimensionless. The non-Hermiticity of the system is characterized by $\gamma$. For simplicity, we focus on the case of $0\leq M$ and $0\leq\gamma$ with $v=1$ throughout this paper. The model becomes topologically nontrivial when $M<2$ in the Hermitian limit of $\gamma=0$.

In the remainder of this section, we point out two symmetries that $H$ possesses. The first symmetry is

$\displaystyle\sigma_{x}H^{*}\sigma_{x}=-H,$

(10)

where $\sigma_{x}$ acts on $2\times 2$ spin space. From this symmetry, we can show that, if $|\Psi^{R}\rangle$ satisfies $H|\Psi^{R}\rangle=E|\Psi^{R}\rangle$, then $\sigma_{x}|\Psi^{R}\rangle^{*}$ satisfies $H\sigma_{x}|\Psi^{R}\rangle^{*}=-E^{*}\sigma_{x}|\Psi^{R}\rangle^{*}$. That is, an eigenstate of $H$ with an eigenvalue of $E$ is paired with its counterpart with the eigenvalue of $-E^{*}$ if ${\rm Re}\{E\}\neq 0$. The second symmetry is

$\displaystyle{}^{t}(UP)({}^{t}H)UP=-H,$

(11)

where $U=i\sigma_{y}$ acts on $2\times 2$ spin space and $P$ represents the space inversion operator. If the system is square shaped with $N\times N$ sites (i.e., $1\leq m,n\leq N$), then $P$ is expressed as

$\displaystyle P=\sum_{m,n}|N+1-m,N+1-n\rangle\langle m,n|.$

(12)

From this symmetry, we can show that, if $|\Psi^{R}\rangle$ satisfies $H|\Psi^{R}\rangle=E|\Psi^{R}\rangle$, then ${}^{t}(UP|\Psi^{R}\rangle)$ satisfies ${}^{t}(UP|\Psi^{R}\rangle)H={}^{t}(UP|\Psi^{R}\rangle)(-E)$. That is, an eigenstate of $H$ with an eigenvalue of $E$ is paired with its counterpart with the eigenvalue of $-E$ if $|E|\neq 0$.

These results tell us that if an eigenstate with an eigenvalue of $E$ with ${\rm Re}\{E\}\neq 0$ and ${\rm Im}\{E\}\neq 0$ is present, this eigenstate and its three counterparts form a quartet: four eigenstates with $E$, $E^{*}$, $-E^{*}$, and $-E$ are related by the two symmetries. An important exception is the case of ${\rm Re}\{E\}\neq 0$ and ${\rm Im}\{E\}=0$, where an eigenstate and its counterpart form a pair; eigenstates with $E$ and $-E$ are related by the symmetries.

The above statement is useful in considering a complex spectrum of our system. Let us denote a complex eigenvalue of energy $E=E_{\rm R}+iE_{\rm I}$ as $E=(E_{\rm R},E_{\rm I})$. All eigenvalues are real as $E=(\varepsilon,0)$ in the Hermitian limit of $\gamma=0$. With increasing $\gamma$, eigenvalues gradually become complex as $E=(\varepsilon,\delta$). We focus on a pair of eigenstates with eigenvalues of $(\varepsilon_{1},0)$ and $(-\varepsilon_{1},0)$ at $\gamma=0$. Let us suppose that $(\varepsilon_{1},0)$ deviates from the real axis as $(\varepsilon_{1},\delta)$ with increasing $\gamma$. This statement requires that this eigenstate forms a quartet together with three other eigenstates with eigenvalues of $(\varepsilon_{1},-\delta)$, $(-\varepsilon_{1},\delta)$, and $(-\varepsilon_{1},-\delta)$. Such a quartet should be formed by a combination of two pairs of eigenstates. That is, an eigenvalue can become complex only when a pair of eigenstates with $(\varepsilon_{1},0)$ and $(-\varepsilon_{1},0)$ and another pair of eigenstates with $(\varepsilon_{2},0)$ and $(-\varepsilon_{2},0)$ are degenerate in energy under the condition of $\varepsilon_{1}=\varepsilon_{2}$.

We consider the system of $N\times N$ sites in the boundary and bulk geometries. Right and left eigenstates of $H$ are expressed as

	$\displaystyle|\Psi^{R}\rangle=\sum_{m,n}|m,n\rangle\cdot|\psi^{R}(m,n)\rangle,$		(13)
	$\displaystyle\langle\Psi^{L}|=\sum_{m,n}\langle\psi^{L}(m,n)|\cdot\langle m,n|,$		(14)

where $|\psi^{R}(m,n)\rangle$ and $\langle\psi^{L}(m,n)|$ are, respectively, two component column and row vectors:

	$\displaystyle|\psi^{R}(m,n)\rangle$	$\displaystyle=\left[\begin{array}[]{c}\psi_{\uparrow}^{R}(m,n)\\ \psi_{\downarrow}^{R}(m,n)\end{array}\right],$		(17)
	$\displaystyle\langle\psi^{L}(m,n)|$	$\displaystyle=\left[\begin{array}[]{cc}\psi_{\uparrow}^{L}(m,n)&\psi_{\downarrow}^{L}(m,n)\end{array}\right].$		(19)

In the boundary geometry, the following obc is imposed on $|\psi^{R}(m,n)\rangle$ and $\langle\psi^{L}(m,n)|$:

		$\displaystyle|\psi^{R}(0,n)\rangle=|\psi^{R}(N+1,n)\rangle$
		$\displaystyle=|\psi^{R}(m,0)\rangle=|\psi^{R}(m,N+1)\rangle=\left[\begin{array}[]{c}0\\ 0\end{array}\right],$		(22)
		$\displaystyle\langle\psi^{L}(0,n)|=\langle\psi^{L}(N+1,n)|$
		$\displaystyle=\langle\psi^{L}(m,0)|=\langle\psi^{L}(m,N+1)|=\left[\begin{array}[]{cc}0&0\end{array}\right].$		(24)

In the bulk geometry, we introduce plane-wave-like right and left eigenstates to define a topological number. To do this, we set

	$\displaystyle|\psi^{R}(m,n)\rangle$	$\displaystyle=\varphi^{R}(m,n)|\psi^{R}\rangle,$		(25)
	$\displaystyle\langle\psi^{L}(m,n)|$	$\displaystyle=\langle\psi^{L}|\varphi^{L}(m,n),$		(26)

where $|\psi\rangle^{R}$ and $\langle\psi^{L}|$ are, respectively, two component column and row vectors independent of $m$ and $n$, and

	$\displaystyle\varphi^{R}(m,n)$	$\displaystyle=\beta_{x}^{m}\beta_{y}^{n},$		(27)
	$\displaystyle\varphi^{L}(m,n)$	$\displaystyle={\beta^{\prime}}_{\!\!x}^{m}{\beta^{\prime}}_{\!\!y}^{n}.$		(28)

The eigenvalue equation $H|\Psi^{R}\rangle=E|\Psi^{R}\rangle$ is reduced to

$\displaystyle H_{\rm rd}^{R}|\psi^{R}\rangle=E|\psi^{R}\rangle,$

(29)

where $H_{\rm rd}^{R}=\sigma_{x}\eta_{x}+\sigma_{y}\eta_{y}+\sigma_{z}\eta_{z}$ with

		$\displaystyle\eta_{x}=\frac{1}{2i}\left(\beta_{x}-\beta_{x}^{-1}\right)+i\gamma,$		(30)
		$\displaystyle\eta_{y}=\frac{1}{2i}\left(\beta_{y}-\beta_{y}^{-1}\right)+i\gamma,$		(31)
		$\displaystyle\eta_{z}=M-\frac{1}{2}\left(\beta_{x}+\beta_{x}^{-1}\right)-\frac{1}{2}\left(\beta_{y}+\beta_{y}^{-1}\right).$		(32)

The eigenvalue equation $\langle\Psi^{L}|H=\langle\Psi^{L}|E$ is reduced to

$\displaystyle\langle\psi^{L}|H_{\rm rd}^{L}=\langle\psi^{L}|E,$

(33)

where $H_{\rm rd}^{L}=\sigma_{x}{\eta^{\prime}}_{\!\!x}+\sigma_{y}{\eta^{\prime}}_{\!\!y}+\sigma_{z}{\eta^{\prime}}_{\!\!z}$ with

	$\displaystyle{\eta^{\prime}}_{\!\!x}=-\frac{1}{2i}\left({\beta^{\prime}}_{\!\!x}-{\beta^{\prime}}_{\!\!x}^{-1}\right)+i\gamma,$		(34)
	$\displaystyle{\eta^{\prime}}_{\!\!y}=-\frac{1}{2i}\left({\beta^{\prime}}_{\!\!y}-{\beta^{\prime}}_{\!\!y}^{-1}\right)+i\gamma,$		(35)
	$\displaystyle{\eta^{\prime}}_{\!\!z}=M-\frac{1}{2}\left({\beta^{\prime}}_{\!\!x}+{\beta^{\prime}}_{\!\!x}^{-1}\right)-\frac{1}{2}\left({\beta^{\prime}}_{\!\!y}+{\beta^{\prime}}_{\!\!y}^{-1}\right).$		(36)

We impose the following mpbc on $\varphi^{R}(m,n)$:

$\displaystyle\varphi^{R}(m+N,n)=\varphi^{R}(m,n+N)=b^{N}\varphi^{R}(m,n),$

(37)

which results in

$\displaystyle\beta_{x}=be^{ik_{x}a},\hskip 8.53581pt\beta_{y}=be^{ik_{y}a},$

(38)

where $k_{i}=2n_{i}\pi/N$ and $n_{i}=0,1,2,\dots,N-1$ ($i=x,y$). To present a biorthogonal set of right and left basis functions, [15] we determine ${\beta^{\prime}}_{\!\!x}$ and ${\beta^{\prime}}_{\!\!y}$ as

$\displaystyle{\beta^{\prime}}_{\!\!x}=\beta_{x}^{-1},\hskip 28.45274pt{\beta^{\prime}}_{\!\!y}=\beta_{y}^{-1},$

(39)

which results in

$\displaystyle{\beta^{\prime}}_{\!\!x}=(be^{ik_{x}a})^{-1},\hskip 8.53581pt{\beta^{\prime}}_{\!\!y}=(be^{ik_{y}a})^{-1}.$

(40)

This indicates that $\varphi^{L}(m,n)$ obeys the following mpbc:

$\displaystyle\varphi^{L}(m+N,n)=\varphi^{L}(m,n+N)=b^{-N}\varphi^{L}(m,n),$

(41)

which is different from Eq. (37).

We find $H_{\rm rd}^{L}=H_{\rm rd}^{R}$ from Eq. (39) and thus denote them as $H_{\rm rd}$ hereafter. The reduced Hamiltonian $H_{\rm rd}$ is expressed as

$\displaystyle H_{\rm rd}=\sigma_{x}\eta_{x}+\sigma_{y}\eta_{y}+\sigma_{z}\eta_{z},$

(42)

with

		$\displaystyle\eta_{x}=b_{+}\sin(k_{x}a)+i\left[\gamma-b_{-}\cos(k_{x}a)\right],$		(43)
		$\displaystyle\eta_{y}=b_{+}\sin(k_{y}a)+i\left[\gamma-b_{-}\cos(k_{y}a)\right],$		(44)
		$\displaystyle\eta_{z}=M-b_{+}\left[\cos(k_{x}a)+\cos(k_{y}a)\right]$
		$\displaystyle\hskip 56.9055pt-ib_{-}\left[\sin(k_{x}a)+\sin(k_{x}a)\right].$		(45)

where

$\displaystyle b_{\pm}=\frac{1}{2}\left(b\pm b^{-1}\right).$

(46)

By solving the reduced eigenvalue equation, we find that the energy of an eigenstate characterized by $\mib{k}=(k_{x},k_{y})$ and $b$ is given by $E=\pm\epsilon$ with

$\displaystyle\epsilon=\sqrt{\eta_{x}^{2}+\eta_{y}^{2}+\eta_{z}^{2}},$

(47)

where ${\rm Re}\{\epsilon\}\geq 0$. The right and left eigenstates of $H$ with the eigenvalue of $\pm\epsilon$ are expressed as

	$\displaystyle|\Psi_{\mib{k}}^{R\pm}\rangle$	$\displaystyle=\frac{1}{N}\sum_{m,n}(be^{ik_{x}a})^{m}(be^{ik_{y}a})^{n}|m,n\rangle\cdot|\psi_{\mib{k}}^{R\pm}\rangle,$		(48)
	$\displaystyle\langle\Psi_{\mib{k}}^{L\pm}|$	$\displaystyle=\frac{1}{N}\sum_{m,n}\langle\psi_{\mib{k}}^{L\pm}|\cdot\langle m,n|(be^{ik_{x}a})^{-m}(be^{ik_{y}a})^{-n},$		(49)

where

	$\displaystyle|\psi_{\mib{k}}^{R\pm}\rangle$	$\displaystyle=\frac{1}{\sqrt{A_{\pm}}}\left[\begin{array}[]{c}\eta_{z}\pm\epsilon\\ \eta_{x}+i\eta_{y}\end{array}\right],$		(52)
	$\displaystyle\langle\psi_{\mib{k}}^{L\pm}|$	$\displaystyle=\frac{1}{\sqrt{A_{\pm}}}^{t}\!\left[\begin{array}[]{c}\eta_{z}\pm\epsilon\\ \eta_{x}-i\eta_{y}\end{array}\right],$		(55)

with $A_{\pm}=\eta_{x}^{2}+\eta_{y}^{2}+(\eta_{z}\pm\epsilon)^{2}$. We easily find

$\displaystyle\langle\Psi_{\mib{k}}^{L\pm}|\Psi_{\mib{k^{\prime}}}^{R\pm}\rangle=\delta_{\mib{k},\mib{k^{\prime}}},\hskip 17.07164pt\langle\Psi_{\mib{k}}^{L\pm}|\Psi_{\mib{k^{\prime}}}^{R\mp}\rangle=0.$

(56)

That is, $\{|\Psi_{\mib{k}}^{R\pm}\rangle\}$ and $\{\langle\Psi_{\mib{k}}^{L\pm}|\}$ constitute a biorthogonal set of basis functions. [15] The eigenvectors satisfy

$\displaystyle\langle\psi_{\mib{k}}^{L\pm}|\psi_{\mib{k}}^{R\pm}\rangle=1,\hskip 17.07164pt\langle\psi_{\mib{k}}^{L\pm}|\psi_{\mib{k}}^{R\mp}\rangle=0.$

(57)

We determine the spectrum of chiral edge states in the $N\times N$ system when an obc is imposed in one direction and a mpbc is imposed in the other direction. Considering the resulting spectrum, we find important characteristics of chiral edge states in the boundary geometry.

First, we consider the system [see Fig. 1(a)] under the obc in the $y$-direction,

$\displaystyle\varphi^{R}(m,0)$

$\displaystyle=\varphi^{R}(m,N+1)=0,$

(58)

and the mpbc in the $x$-direction,

$\displaystyle\varphi^{R}(m,n)$

$\displaystyle=b^{N}\varphi^{R}(m+N,n).$

(59)

To obtain fundamental solutions, we set

$\displaystyle\varphi^{R}(m,n)$

$\displaystyle=\beta_{x}^{m}\rho^{n},$

(60)

where $\rho$ characterizes the penetration of chiral edge states in the $y$-direction and $\beta_{x}=be^{ik_{x}a}$. We decompose the reduced Hamiltonian as $H_{\rm rd}=H_{\rm rd}^{(0)}+H_{\rm rd}^{(1)}$ with

	$\displaystyle H_{\rm rd}^{(0)}$	$\displaystyle=i\left[\gamma-\frac{\rho-\rho^{-1}}{2}\right]\sigma_{y}+\left[\tilde{M}-\frac{\rho+\rho^{-1}}{2}\right]\sigma_{z},$		(61)
	$\displaystyle H_{\rm rd}^{(1)}$	$\displaystyle=\eta_{x}\sigma_{x},$		(62)

where $\eta_{x}$ is given in Eq. (43) and

$\displaystyle\tilde{M}=M-\frac{1}{2}\left(\beta_{x}+\beta_{x}^{-1}\right).$

(63)

A simple method for obtaining edge localized solutions is described in Appendix of Ref. \citentakane2. By adapting the method to $H_{\rm rd}^{(0)}|\psi^{R}\rangle=E_{\bot}|\psi^{R}\rangle$, we find two eigenstates at $E_{\bot}=0$:

	$\displaystyle|\Psi_{1}^{R}\rangle$	$\displaystyle=c\sum_{m}(be^{ik_{x}a})^{m}\sum_{n=0}^{N+1}\left(\rho_{1}^{n}-\tilde{\rho}_{1}^{n}\right)$
		$\displaystyle\hskip 5.69054pt\times|m,n\rangle\cdot|\psi_{1}^{R}\rangle,$		(64)
	$\displaystyle|\Psi_{2}^{R}\rangle$	$\displaystyle=c\sum_{m}(be^{ik_{x}a})^{m}\sum_{n=0}^{N+1}\left(\frac{\rho_{2}^{n}}{\rho_{2}^{N+1}}-\frac{\tilde{\rho}_{2}^{n}}{\tilde{\rho}_{2}^{N+1}}\right)$
		$\displaystyle\hskip 5.69054pt\times|m,n\rangle\cdot|\psi_{2}^{R}\rangle,$		(65)

where $c$ is a normalization constant,

$\displaystyle|\psi_{1}^{R}\rangle=\frac{1}{\sqrt{2}}\left[\begin{array}[]{c}1\\ 1\end{array}\right],\hskip 14.22636pt|\psi_{2}^{R}\rangle=\frac{1}{\sqrt{2}}\left[\begin{array}[]{c}1\\ -1\end{array}\right],$

(70)

and

$\displaystyle\rho_{1}=\tilde{M}+\gamma,\hskip 14.22636pt\rho_{2}=\frac{1}{\tilde{M}-\gamma}.$

(71)

The limit of $\tilde{\rho}_{1}\to 0$ and $\tilde{\rho}_{2}\to\infty$ is assumed in the above equations, so that

$\displaystyle\sum_{n=0}^{N+1}\tilde{\rho}_{1}^{n}|m,n\rangle=|m,0\rangle,\hskip 5.69054pt\sum_{n=0}^{N+1}\frac{\tilde{\rho}_{2}^{n}}{\tilde{\rho}_{2}^{N+1}}|m,n\rangle=|m,N+1\rangle.$

(72)

We observe that $|\Psi_{1}^{R}\rangle$ satisfies the obc at $n=0$. When $|\rho_{1}|<1$, it exponentially decreases with increasing $n$ and satisfies the obc at $n=N+1$ if $N$ is sufficiently large. That is, $|\Psi_{1}^{R}\rangle$ is localized near the bottom edge at $n=1$ when $|\rho_{1}|<1$. In a manner similar to this, we can show that $|\Psi_{2}^{R}\rangle$ is localized near the top edge at $n=N$ when $1<|\rho_{2}|$. Since $H_{\rm rd}^{(1)}|\psi_{1}^{R}\rangle=\eta_{x}|\psi_{1}^{R}\rangle$ and $H_{\rm rd}^{(1)}|\psi_{2}^{R}\rangle=-\eta_{x}|\psi_{2}^{R}\rangle$, we can conclude that the energy dispersion relations of the edge states localized near the bottom edge and top edge respectively are

		$\displaystyle E_{\rm bottom}(k_{x})=b_{+}\sin(k_{x}a)+i\left[\gamma-b_{-}\cos(k_{x}a)\right],$		(73)
		$\displaystyle E_{\rm top}({k^{\prime}}_{\!\!x})=-b_{+}\sin({k^{\prime}}_{\!\!x}a)-i\left[\gamma-b_{-}\cos({k^{\prime}}_{\!\!x}a)\right],$		(74)

where $k_{x}$ in $E_{\rm top}$ is replaced with ${k^{\prime}}_{\!\!x}$ for later convenience.

We next consider the system [see Fig. 1(b)] under the obc in the $x$-direction,

$\displaystyle\varphi^{R}(0,n)$

$\displaystyle=\varphi^{R}(N+1,n)=0,$

(75)

and the mpbc in the $y$-direction,

$\displaystyle\varphi^{R}(m,n)$

$\displaystyle=b^{N}\varphi^{R}(m,n+N),$

(76)

and find that the energy dispersion relations of the edge states localized near the left edge and right edge respectively are

		$\displaystyle E_{\rm left}({k^{\prime}}_{\!\!y})=-b_{+}\sin({k^{\prime}}_{\!\!y}a)-i\left[\gamma-b_{-}\cos({k^{\prime}}_{\!\!y}a)\right],$		(77)
		$\displaystyle E_{\rm right}(k_{y})=b_{+}\sin(k_{y}a)+i\left[\gamma-b_{-}\cos(k_{y}a)\right],$		(78)

where $k_{y}$ in $E_{\rm left}$ is replaced with ${k^{\prime}}_{\!\!y}$ for later convenience.

In the above argument, we assume $|\rho_{1}|<1<|\rho_{2}|$ such that two solutions are spatially separated: one is localized near the bottom (right) edge whereas the other is localized near the top (left) edge. Although this assumption is satisfied when $\gamma$ is sufficiently small, it eventually breaks down with increasing $\gamma$. However, as briefly described in Appendix, even though the assumption breaks down, the dispersion relations in Eqs. (73), (74), (77), and (78) are valid as long as $N$ is sufficiently large and $|\rho_{1}|<|\rho_{2}|$. Here, $|\rho_{1}|<|\rho_{2}|$ is satisfied under the condition of $M<2\sqrt{1+\gamma^{2}}$ if $\gamma<1$. In Sect. 5, we show that this condition describes one part of phase boundaries.

Let us consider chiral edge states in the boundary geometry shown in Fig. 1(c). A chiral edge state circulates along the square loop consisting of the bottom, top, left, and right edges. Thus, we expect that such a chiral edge state can be allowed only when the four dispersion relations in Eqs. (73), (74), (77), and (78) become identical for appropriately chosen $k_{x}$, ${k^{\prime}}_{\!\!x}$, ${k^{\prime}}_{\!\!y}$, and $k_{y}$:

$\displaystyle E_{\rm bottom}(k_{x})=E_{\rm top}({k^{\prime}}_{\!\!x})=E_{\rm left}({k^{\prime}}_{\!\!y})=E_{\rm right}(k_{y}).$

(79)

This is satisfied when $k_{x}=-{k^{\prime}}_{\!\!x}=-{k^{\prime}}_{\!\!y}=k_{y}\equiv k$ with

$\displaystyle b_{-}\cos(ka)=\gamma.$

(80)

Note that the mpbc is essential in obtaining this result.

Equation (80) requires that the energy $E$ of a chiral edge state must be real. This result is consistent with the argument based on the symmetries of $H$. In Sect. 2, we show that an eigenvalue of energy can become complex only when a pair of eigenstates with $(\varepsilon_{1},0)$ and $(-\varepsilon_{1},0)$ and another pair of eigenstates with $(\varepsilon_{2},0)$ and $(-\varepsilon_{2},0)$ are degenerate in energy under the condition of $\varepsilon_{1}=\varepsilon_{2}$. Let us apply this to the chiral edge states in the Hermitian limit of $\gamma=0$, where its spectrum is almost equally distributed on the real axis of $E$ in a pairwise manner. Even when $\gamma$ is increased from $0$, two pairs of chiral edge states cannot be degenerate when the chiral edge states are topologically protected and the spectrum is almost equally distributed on the real axis. Therefore, the spectrum of the chiral edge states cannot deviate from the real axis as long as the states are stable. Conversely, we can say that a chiral edge state is destabilized if its energy becomes complex as $E=(\varepsilon,\delta)$.

In the boundary geometry, Eq. (80) also requires that the wavefunction amplitude of the chiral edge states varies in the $x$- and $y$-directions with the same rate of increase:

$\displaystyle b=\sqrt{\left(\frac{\gamma}{\cos(ka)}\right)^{2}+1}+\frac{\gamma}{\cos(ka)}.$

(81)

This result is used in the argument in Sect. 5.

In Sect. 3, it is shown that the plane-wave-like states under the mpbc of Eq. (37) consist of a conduction band of $E=\epsilon$ and a valence band of $E=-\epsilon$, where $\epsilon$ is given in Eq. (47). When the two bands are separated by a gap, that is, ${\rm Re}\{\epsilon\}\neq 0$ for an arbitrary $\mib{k}$, we can define the Chern number $\nu$ by using $|\psi_{\mib{k}}^{R-}\rangle$ and $\langle\psi_{\mib{k}}^{L-}|$ given in Eqs. (52) and (55). The result is [29]

$\displaystyle\nu=\frac{1}{2\pi i}\int d^{2}k\left(\frac{\partial\langle\psi_{\mib{k}}^{L-}|}{\partial k_{x}}\frac{\partial|\psi_{\mib{k}}^{R-}\rangle}{\partial k_{y}}-\frac{\partial\langle\psi_{\mib{k}}^{L-}|}{\partial k_{y}}\frac{\partial|\psi_{\mib{k}}^{R-}\rangle}{\partial k_{x}}\right),$

(82)

which is rewritten as [71]

$\displaystyle\nu=\int\frac{d^{2}k}{4\pi}\frac{1}{\epsilon^{3}}\left(\eta_{x}\frac{\partial\eta_{y}}{\partial k_{y}}\frac{\partial\eta_{z}}{\partial k_{x}}+\frac{\partial\eta_{x}}{\partial k_{x}}\eta_{y}\frac{\partial\eta_{z}}{\partial k_{y}}-\frac{\partial\eta_{x}}{\partial k_{x}}\frac{\partial\eta_{y}}{\partial k_{y}}\eta_{z}\right).$

(83)

Here, the limit of $N\to\infty$ is implicitly assumed. The Chern number $\nu$ takes an integer that depends on not only $M$ and $\gamma$ but also $b$. In the Hermitian limit of $\gamma=0$, Eq. (82) at $b=1$ becomes equivalent to an ordinary expression of the Chern number. [1, 2]

Our system takes three phases: a topologically trivial phase of $\nu=0$, a nontrivial phase of $\nu=1$, and a gapless phase, in which $\nu$ cannot be defined. We consider $\nu$ for a given $M$ in a parameter space of $\gamma$ and $b$, where these phases are separated by lines on which the gap closes. Let us find such gap closing lines. [71] A gap closing takes place in our system when ${\rm Re}\{\epsilon\}={\rm Im}\{\epsilon\}=0$. Equation (47) indicates that ${\rm Im}\{\epsilon^{2}\}=0$ for $\mib{k}=(0,0)$, $(0,\frac{\pi}{a})$, $(\frac{\pi}{a},0)$, and $(\frac{\pi}{a},\frac{\pi}{a})$, where $\mib{k}=(\frac{\pi}{a},\frac{\pi}{a})$ can be ignored because this point has a gap in relevant situations. For $\mib{k}=(0,0)$, we find that ${\rm Re}\{\epsilon^{2}\}=0$ when $(M-2b_{+})^{2}-2(-b_{-}+\gamma)^{2}=0$, which yields four solutions of $b$:

	$\displaystyle b_{\rm 1}=\frac{M-\sqrt{2}\gamma+\sqrt{(M-\sqrt{2}\gamma)^{2}-2}}{2-\sqrt{2}},$		(84)
	$\displaystyle b_{\rm 2}=\frac{M+\sqrt{2}\gamma+\sqrt{(M+\sqrt{2}\gamma)^{2}-2}}{2+\sqrt{2}},$		(85)
	$\displaystyle b_{\rm 3}=\frac{M-\sqrt{2}\gamma-\sqrt{(M-\sqrt{2}\gamma)^{2}-2}}{2-\sqrt{2}},$		(86)
	$\displaystyle b_{\rm 4}=\frac{M+\sqrt{2}\gamma-\sqrt{(M+\sqrt{2}\gamma)^{2}-2}}{2+\sqrt{2}}.$		(87)

For $\mib{k}=(0,\frac{\pi}{a})$ and $(\frac{\pi}{a},0)$, we find that ${\rm Re}\{\epsilon^{2}\}=0$ when $2b_{-}^{2}-M^{2}+2\gamma^{2}=0$, which yields two solutions of $b$:

	$\displaystyle b_{5}$	$\displaystyle=\sqrt{\frac{M^{2}}{2}-\gamma^{2}+1}+\sqrt{\frac{M^{2}}{2}-\gamma^{2}},$		(88)
	$\displaystyle b_{6}$	$\displaystyle=\sqrt{\frac{M^{2}}{2}-\gamma^{2}+1}-\sqrt{\frac{M^{2}}{2}-\gamma^{2}}.$		(89)

Each $b_{i}$ ($i=1,\dots,6$) is irrelevant if it becomes a complex number. In addition to $b_{i}$ ($i=1,\dots,6$), we need to consider $b_{0}$ given below. When $(M-\sqrt{2}\gamma)^{2}<2$, $b_{1}$ and $b_{3}$ become the complex numbers $b_{0}e^{\pm ik_{0}a}$, where

$\displaystyle b_{0}=1+\sqrt{2}$

(90)

and $k_{0}$ is defined by

$\displaystyle\tan\left(k_{0}a\right)=\frac{\sqrt{2-(M-\sqrt{2}\gamma)^{2}}}{M-\sqrt{2}\gamma}.$

(91)

We can show that ${\rm Re}\{\epsilon^{2}\}={\rm Im}\{\epsilon^{2}\}=0$ at $b=b_{0}$ if $\mib{k}=\pm(k_{0},k_{0})$. Therefore, $b_{0}$ is a gap closing line.

The phase diagrams in the bulk geometry for $M=1.0$, $2.0$, $2.5$, and $3.0$ are shown in Figs. 2(a)–2(d) in the $\gamma b$-plane, where the gap closing lines separate the topologically trivial region ($\nu=0$), nontrivial region ($\nu=1$), and gapless region in which $\nu$ cannot be defined. For example, the nontrivial region ($\nu=1$) in panel (a) is bounded by $b_{2}$, $b_{5}$, and $b_{6}$, and its outside is the gapless region.

Using the phase diagrams shown in Fig. 2, let us consider which of the three phases appears in the boundary geometry. In the Hermitian limit of $\gamma=0$, a phase realized in the boundary geometry is governed by $\nu$ at $b=1$, which corresponds to the ordinary pbc. If $\nu=1$ ($\nu=0$) at $b=1$, the nontrivial (trivial) phase is realized in the boundary geometry. To extend this bulk–boundary correspondence to the non-Hermitian regime of $\gamma>0$, we need to determine $b$ as a function of $\gamma$ such that $\nu(\gamma,b)$ is in one-to-one correspondence with a phase realized in the boundary geometry.

A recipe for determining $b(\gamma)$ is given in Ref. \citentakane1. We here give a revised version of it:

1. 1.

   $b(0)=1$ at the Hermitian limit of $\gamma=0$.

2. 2.

   With the exception described in (3), $b(\gamma)$ is allowed to cross gap closing lines only at a crossing point between the two.

3. 3.

   If a peculiar gap closing is caused by the destabilization of topological boundary states, $b(\gamma)$ in the nontrivial region is determined in accordance with it.

The third requirement is added in this revised version. Below, we explain the second and third requirements as well as the meaning of a peculiar gap closing.

We briefly describe the reasoning on which the second requirement is based. [71] If $b(\gamma)$ crosses a gap closing line, a zero-energy solution appears at the crossing point, giving rise to a gapless spectrum in the bulk geometry. Hence, to verify the bulk–boundary correspondence, the spectrum in the boundary geometry must also be gapless at this point. A single solution is insufficient to construct a general solution compatible with the obc. [28, 38] A crossing point between two $b_{i}$ yields two zero-energy solutions, which should enable us to construct a zero-energy solution in the boundary geometry under the obc. We thus expect that $b(\gamma)$ is allowed to cross $b_{i}$ only at such a crossing point. The second requirement does not uniquely determine $b(\gamma)$, except at a crossing point, so that $b(\gamma)$ is arbitrary in a region between two neighboring gap closing lines in which the two bands have a gap. Since $\nu(\gamma,b)$ is uniquely determined in such a gapped region, this arbitrariness does not affect the bulk–boundary correspondence.

The above reasoning implicitly assumes that a phase transition accompanies a gap closing (i.e., a touching of the conduction and valence bands) at zero energy, and that a zero-energy bulk state in the boundary geometry is captured by the method of a separation of variables. Here and hereafter, a bulk state represents an eigenstate in the two bands being distinguishable from chiral edge states. Although this assumption is plausible, there is an exception. Let us consider the Chern insulator phase in the boundary geometry, where the conduction and valence bands are linked by chiral edge states. If the chiral edge states are destabilized by non-Hermiticity, the two bands are combined into one band by a bridge of destabilized chiral edge states (see Fig. 7). Here, the destabilized states should be regarded as bulk states, and cannot be captured by the method of a separation of variables. This is referred to as a peculiar gap closing, which is intrinsic to two- and three-dimensional non-Hermitian topological systems. The third requirement is added to take this into account.

When a peculiar gap closing takes place in the boundary geometry, a chiral edge state at zero energy is transformed into a bulk state at a transition point to the gapless phase. Such a state is characterized by

$\displaystyle b=\sqrt{\gamma^{2}+1}+\gamma,$

(92)

which is identical to Eq. (81) at $k=0$. Thus, in the nontrivial region, we determine $b(\gamma)$ in the third requirement using Eq. (92). At a crossing point of $b(\gamma)$ and a gap closing line, the bulk geometry becomes gapless. To verify the bulk–boundary correspondence, this crossing must coincide with the destabilization of chiral edge states in the boundary geometry. This is justified at the end of this section.