Non-Hermitian non-Abelian topological insulators with $PT$ symmetry

We start with a Hermitian system capable to describe a non-Abelian topological insulator based of the one-dimensional lattice in Fig.1(a). We consider generators of $\mathfrak{so}\left(N\right)$ rotation $L_{\alpha\beta}$ indexed by $\alpha$ and $\beta$, whose $ab$ components are defined by

$$\left(L_{\alpha\beta}\right)_{ab}=\delta_{\alpha b}\delta_{\beta a}-\delta_{\alpha a}\delta_{\beta b}.$$

(1)

We consider a $PT$-invariant Hamiltonian in the momentum space given byWuScience

$$H_{\alpha\beta}\left(k\right)=R_{\alpha\beta}\left(\frac{k}{2}\right)\text{diag.}\left(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{N}\right)R_{\alpha\beta}\left(\frac{k}{2}\right)^{\text{t}},$$

(2)

where $0\leq k<2\pi$, $1\leq\alpha,\beta\leq N$, $\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{N}$ are real, and

$$R_{\alpha\beta}\left(\frac{k}{2}\right)=e^{\frac{k}{2}L_{\alpha\beta}}$$

(3)

is a rotation matrix given by

	$\displaystyle\left(R_{\alpha\beta}\left(\frac{k}{2}\right)\right)_{ab}$	$\displaystyle=\delta_{ab}+\left(\delta_{a\alpha}\delta_{b\alpha}+\delta_{a\beta}\delta_{b\beta}\right)\cos\frac{k}{2}$
		$\displaystyle+\left(\delta_{a\beta}\delta_{b\alpha}-\delta_{a\alpha}\delta_{b\beta}\right)\sin\frac{k}{2}.$		(4)

The Hamiltonian (2) is explicitly written as

	$\displaystyle H_{\alpha\beta}\left(k\right)$	$\displaystyle=\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}}{2}$
		$\displaystyle+\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2}\left(\delta_{a\alpha}\delta_{b\alpha}-\delta_{a\beta}\delta_{b\beta}\right)\cos k$
		$\displaystyle+\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2}\left(\delta_{a\beta}\delta_{b\alpha}+\delta_{a\alpha}\delta_{b\beta}\right)\sin k.$		(5)

It is decomposed into two parts,

$$H_{\alpha\beta}\left(k\right)=\bigoplus_{j\neq\alpha,\beta}H_{j}\oplus H_{\alpha\beta}^{\prime}\left(k\right),$$

(6)

where

	$\displaystyle H_{j}$	$\displaystyle=\varepsilon_{j}\mathbb{I}_{1},$		(7)
	$\displaystyle H_{\alpha\beta}^{\prime}\left(\theta\right)$	$\displaystyle=\left[\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2}\left(\begin{array}[]{cc}\cos k&\sin k\\ \sin k&-\cos k\end{array}\right)+\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}}{2}\mathbb{I}_{2}\right].$		(10)

The Hamiltonian is nontrivial only for the $\alpha$ and $\beta$ bands, with eigenvalues $\varepsilon_{\alpha}$ and $\varepsilon_{\beta}$. All other bands are spectators with respect to the $\alpha$ and $\beta$ bands. See Fig.1.

The energy spectrum of the bulk Hamiltonian does not change by the rotation (3) and is given by

$$E\left(k\right)=\varepsilon_{1},\quad\varepsilon_{2},\quad\cdots,\quad\varepsilon_{N}.$$

(11)

The eigenfunctions for the $2\times 2$ matrix $H_{\alpha\beta}^{\prime}\left(k\right)$ are

	$\displaystyle\psi_{a}^{+}$	$\displaystyle=\delta_{a\alpha}\sin\frac{k}{2}+\delta_{a\beta}\cos\frac{k}{2},$		(12)
	$\displaystyle\psi_{a}^{-}$	$\displaystyle=-\delta_{a\alpha}\cos\frac{k}{2}+\delta_{a\beta}\sin\frac{k}{2},$		(13)

while those for $H_{j}$ are $\psi_{a}=\delta_{aj}$.

The $\alpha$ and $\beta$ bands are perfectly flat. They are ($\ell$–$1$) fold degenerate in a finite chain, where $\ell$ is the number of sites in the chain. See Fig.2(a1).

We generalize the Hermitian non-Abelian system (2) to a non-Hermitian non-Abelian system, keeping $PT$ symmetry. We consider the Hamiltonian

$$H_{\alpha\beta}^{\prime}\left(k;\gamma,\xi\right)=H_{\alpha\beta}^{\prime}\left(k\right)+i\gamma\sigma_{y}+\xi\sigma_{x}\sin k,$$

(14)

whose eigenenergies are

$$E_{\alpha\beta}^{\prime}\left(k;\gamma,\xi\right)=\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}\pm\sqrt{g(k;\gamma,\xi)}}{2},$$

(15)

with

$$g(k;\gamma,\xi)=\left(\varepsilon_{\alpha}-\varepsilon_{\beta}\right)^{2}-\gamma^{2}+4\xi\left(\varepsilon_{\alpha}-\varepsilon_{\beta}+\xi\right)\sin^{2}k.$$

(16)

We explain the meanings of the $\gamma$ term and the $\xi$ term. The Hamiltonian (14) is Hermitian when $\gamma=0$. When $\xi=0$ in addition, the band structure is highly degenerate as in Fig.2(a1). This degeneracy is resolved by introducing the $\xi$ term as shown in Fig.2(a2). We show the band structure with $\gamma=0$ as a function of $\xi$ in Fig.2(b1). The perfect flat bands at $\varepsilon_{\alpha}$ and $\varepsilon_{\beta}$ become bended and have dispersions. We also show the band structure with $\gamma=0.25$ as a function of $\xi$ in Fig.2(b2).

We show Re[$E_{\alpha\beta}^{\prime}\left(k;\gamma,\xi=0\right)$] in Fig.2(c1) and Im[$E_{\alpha\beta}^{\prime}\left(k;\gamma,\xi=0\right)$] in Fig.2(d1) as a function of $\gamma$. They are real for $|\gamma|\leq\gamma_{0}$ with

$$\gamma_{0}=\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2},$$

(17)

where $PT$ symmetry is preserved. On the other hand, they are complex for $|\gamma|>\gamma_{0}$, and hence $PT$ symmetry is spontaneously broken there. Namely, although the Hamiltonian is $PT$-symmetric, eigenvalues and eigenfunctions are no longer real in the spontaneous symmetry broken phase. We show the real and imaginary parts of the energy as a function of $\gamma$ in Fig.2(c2) and (c2), where the bulk band has a finite width. We also show the real and imaginary parts of the energy as a function of the momentum $k$ in Fig.3, where the bands have dispersions.

Especially, we have

$$E_{\alpha\beta}^{\prime}\left(\frac{\pi}{2};\gamma,\xi\right)=\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}\pm\sqrt{h(\gamma,\xi)}}{2},$$

(18)

with

$$h(\gamma,\xi)=\left(\varepsilon_{\alpha}-\varepsilon_{\beta}+2\xi-2\gamma\right)\left(\varepsilon_{\alpha}-\varepsilon_{\beta}+2\xi+2\gamma\right).$$

(19)

By solving the condition that $E_{\alpha\beta}^{\prime}\left(\frac{\pi}{2};\gamma,\xi\right)$ is complex, or $h(\gamma,\xi)<0$, we find a phase transition point $\gamma_{1}$ in addition to the phase transition point $\gamma_{0}$ as

$$\gamma_{1}=\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2}-\xi=\gamma_{0}-\xi.$$

(20)

When $\xi>0$, the bulk energy is real for $\left|\gamma\right|\leq\gamma_{1}$, complex for $\left|\gamma\right|>\gamma_{1}$. On the other hand, when $\xi<0$, the bulk energy is real for $\left|\gamma\right|\leq\gamma_{0}$, complex for $\left|\gamma\right|>\gamma_{0}$.

The tight-binding Hamiltonian (10) is written in the coordinate space as

$$H_{\alpha\beta}^{\prime}=H_{0}+H_{\gamma}+H_{\xi},$$

(21)

with

	$\displaystyle H_{0}$	$\displaystyle=$	$\displaystyle\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2}\sum_{j=1}^{\ell-1}(\left|\alpha_{j}\right\rangle\left\langle\alpha_{j+1}\right|+\left|\beta_{j}\right\rangle\left\langle\beta_{j+1}\right|,$		(22)
			$\displaystyle+i\left|\alpha_{j}\right\rangle\left\langle\beta_{j+1}\right|-i\left|\beta_{j}\right\rangle\left\langle\alpha_{j+1}\right|)+\text{h.c.},$
	$\displaystyle H_{\gamma}$	$\displaystyle=$	$\displaystyle\gamma\sum_{j=1}^{\ell}(\left|\alpha_{j}\right\rangle\left\langle\beta_{j}\right|-\left|\beta_{j}\right\rangle\left\langle\alpha_{j}\right|),$		(23)
	$\displaystyle H_{\xi}$	$\displaystyle=$	$\displaystyle i\xi\sum_{j=1}^{\ell-1}(\left|\alpha_{j}\right\rangle\left\langle\beta_{j+1}\right|-\left|\beta_{j}\right\rangle\left\langle\alpha_{j+1}\right|)+\text{h.c.},$		(24)

where the first two terms in $H_{0}$ represent normal hoppings, while the last two terms represent spin-orbit-like imaginary hoppings. The $\xi$ term modifies the spin-orbit-like imaginary hoppings. The $\gamma$ term represents nonreciprocal hoppings, which make the system non-Hermitian.

The tight-binding Hamiltonians for the spectator bands are simply given by

$$H_{j\neq\alpha,\beta}=\sum_{j=1}^{\ell}\varepsilon_{j}\left|j\right\rangle\left\langle j\right|+\sum_{j=1}^{\ell-1}t_{j}\left|j\right\rangle\left\langle j+1\right|+\text{h.c.},$$

(25)

where $\varepsilon_{j}$ is the on-site energy and $t_{j}$ is the hopping parameter.

In this sense, it is enough to consider only the $\alpha$ and $\beta$ bands for an arbitrary $N$ band system. We illustrate the tight-binding model in Fig.1.

We illustrate the tight-binding model (21) in Fig.1(b). In a finite chain, two localized states emerge at the edges with the energy

$$E(\xi,\gamma)=\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}}{2}\pm i\gamma$$

(26)

in the presence of the $\gamma$ term and the $\xi$ term. They are degenerate only in the Hermitian limit ($\gamma=0$). In contrasted to the bulk band, the eigenenergy (26) is complex once $\gamma$ is introduced even for the $PT$ symmetric phase. We show Eq.(26) as a function of $\gamma$ in Fig.2. In contrast to the bulk band, the eigenenergy (26) has no $\xi$ dependence: See Figs.2(d1) and (d2).

When $\xi=0$, the eigenfunctions for the edge states $\psi_{\alpha}\left(j\right)$ and $\psi_{\beta}\left(j\right)$ at the $j$ site are perfectly localized at the edges and given by

$$\psi_{\alpha}\left(j\right)=\frac{1}{\sqrt{2}}\delta_{1,j},\qquad\psi_{\beta}\left(j\right)=\frac{-i}{\sqrt{2}}\delta_{1,j}$$

(27)

for the left edge, and

$$\psi_{\alpha}\left(j\right)=\frac{1}{\sqrt{2}}\delta_{\ell,j},\qquad\psi_{\beta}\left(j\right)=\frac{i}{\sqrt{2}}\delta_{\ell,j}$$

(28)

for the right edge. Here, $1$ in $\delta_{1,j}$ represent the left edge, while $\ell$ in $\delta_{\ell,j}$ represents the right edge. The perfectly localized edge states for $\xi=0$ are transformed to edge states with finite penetration depth for $\xi\neq 0$.

We define a non-Hermitian non-Abelian Berry connection or a non-Hermitian Berry-Wilczek-Zee (BWZ) connection byWZ

$$A_{\alpha\beta}^{\text{RL}}\left(\theta\right)=\left\langle\psi_{\alpha}^{\text{R}}\right|\partial_{\theta}\left|\psi_{\beta}^{\text{L}}\right\rangle,$$

(29)

where

$$H\left|\psi_{\alpha}^{\text{L}}\right\rangle=\varepsilon_{\alpha}\left|\psi_{\alpha}^{\text{L}}\right\rangle$$

(30)

is the left eigenfunction, and

$$H^{\dagger}\left|\psi_{\alpha}^{\text{R}}\right\rangle=\varepsilon_{\alpha}\left|\psi_{\alpha}^{\text{R}}\right\rangle$$

(31)

is the right eigenfunction.

We define a non-Hermitian BWZ phase by

$$\Gamma_{\alpha\beta}^{\text{RL}}=\frac{1}{2\pi}\int_{0}^{2\pi}\text{Re}\left[A_{\alpha\beta}^{\text{RL}}\left(\theta\right)\right]d\theta,$$

(32)

which we use as a non-Hermitian non-Abelian topological charge. The eigenfunctions are analytically solved as

	$\displaystyle\left|\psi_{\pm}^{\text{L}}\right\rangle$	$\displaystyle=\frac{1}{\sqrt{\left(\psi_{1}^{\text{L}}\right)^{2}+\left(\psi_{2}^{\text{L}}\right)^{2}}}\left\{\psi_{1}^{\text{L}},\psi_{2}^{\text{L}}\right\},$		(33)
	$\displaystyle\psi_{1}^{\text{L}}$	$\displaystyle=-\gamma_{0}\cos k\pm\sqrt{\gamma_{0}^{2}-\gamma^{2}+\xi\left(2\gamma+\xi\right)\sin^{2}k},$		(34)
	$\displaystyle\psi_{2}^{\text{L}}$	$\displaystyle=\gamma-\left(\gamma_{0}+\xi\right)\sin k,$		(35)

and

	$\displaystyle\left|\psi_{\pm}^{\text{R}}\right\rangle$	$\displaystyle=\frac{1}{\sqrt{\left(\psi_{1}^{\text{R}}\right)^{2}+\left(\psi_{2}^{\text{R}}\right)^{2}}}\left\{\psi_{1}^{\text{R}},\psi_{2}^{\text{R}}\right\},$		(36)
	$\displaystyle\psi_{1}^{\text{R}}$	$\displaystyle=\gamma_{0}\cos k\pm\sqrt{\gamma_{0}^{2}-\gamma^{2}+\xi\left(2\gamma+\xi\right)\sin^{2}k},$		(37)
	$\displaystyle\psi_{2}^{\text{R}}$	$\displaystyle=\gamma+\left(\gamma_{0}+\xi\right)\sin k.$		(38)

When $\gamma=0$ and $\xi=0$, using (33) and (36), we calculate the non-Hermitian BWZ connection (29) numerically and find that

$$A_{\alpha\beta}^{\text{RL}}=\frac{1}{2}\left(\begin{array}[]{cc}0&-1\\ 1&0\end{array}\right),$$

(39)

which leads to the non-Hermitian BWZ phase (32) as

$$\Gamma_{\alpha\beta}^{\text{RL}}=\frac{1}{2}\left(\begin{array}[]{cc}0&-1\\ 1&0\end{array}\right),$$

(40)

where we have explicitly written only the nontrivial $2\times 2$ submatrix within the $N\times N$ matrix. Hence, the topological charges are given by

$$\Gamma_{\alpha\beta}^{\text{RL}}=-\frac{1}{2}L_{\alpha\beta}$$

(41)

with Eq.(1). The topological charges $\Gamma_{\alpha\beta}^{\text{RL}}$ obey essentially the same non-Abelian algebra as $L_{\alpha\beta}$.

When $\gamma\neq 0$ and $\xi=0$, we calculate the non-Hermitian BWZ connection (29) to find that it is no longer a constant. However, the non-Hermitian BWZ phase (32) is calculated as in Eq.(40), and hence the topological charge is quantized as in Eq.(41) for any $\gamma$. Nevertheless, the eigenfunctions as well as the eigenvalues are real (i.e., real-line-gap topological insulator phase) only for $\gamma^{2}\leq\gamma_{0}^{2}$, while the eigenvalues and the eigenfunctions are complex (i.e. imaginary-line-gap topological insulator phase) for $\gamma^{2}>\gamma_{0}^{2}$. Hence, PT symmetry is preserved only for $\gamma^{2}\leq\gamma_{0}^{2}$, and it is spontaneously broken for $\gamma^{2}>\gamma_{0}^{2}$. The system undergoes a phase transition at $\gamma=\pm\gamma_{0}$.

When $\gamma\neq 0$ and $\xi\neq 0$, we have numerically calculated the topological charge (32) with the use of (33) and (36). We have shown the $\left(2,1\right)$ component of the $2\times 2$ matrix $\Gamma_{\alpha\beta}^{\text{RL}}$ for various values of $\xi$ in Fig.4. It is quantized to be 1/2 for $\gamma^{2}\leq\gamma_{1}^{2}$ and $\gamma^{2}>\gamma_{0}^{2}$ when $\xi>0$, while $\gamma^{2}\leq\gamma_{0}^{2}$ and $\gamma^{2}>\gamma_{1}^{2}$ when $\xi<0$, where $\gamma_{1}=\gamma_{0}-\xi$ as in Eq.(20). On the other hand, it is not quantized for the metallic phase that emerges between $\gamma_{0}$ and $\gamma_{1}$, as in Fig.4. It is concluded that the topological charges are quantized and given by Eq.(41) in the insulator phases.

In non-Hermitian systems, there are line-gap insulators and point-gap insulatorsUedaPRX ; Kawabata in general. In the point-gap insulator, there is a gap in $|E|$. On the other hand, there are two types of line-gap insulators. A real-line gap topological insulator has a gap in Re$\left[E\right]$, while an imaginary-line-gap topological insulator has a gap in Im$\left[E\right]$. We first consider the case $\xi>0$. For $\left|\gamma\right|<\gamma_{1}$, the system is a non-Hermitian line-gap topological insulator along the Re$\left[E\right]$. The systems is metallic for $\gamma_{1}\leq\left|\gamma\right|\leq\gamma_{0}$. For $\left|\gamma\right|>\gamma_{0}$, the system is a non-Hermitian line-gap topological insulator along the Im$\left[E\right]$. If $\xi<0$, the system is a real-line-gap topological insulator for $\left|\gamma\right|<\gamma_{0}$, it is a metal for $\gamma_{0}\leq\left|\gamma\right|\leq\gamma_{1}$ and it is an imaginary-line-gap topological insulator for $\left|\gamma\right|>\gamma_{1}$. We show the topological phase diagram in Fig.5(b). It is consistent with the real and imaginary parts of the energy in the $\gamma$-$\xi$ plane as shown in Fig.5(a).

The band structure as well as edge states are well observed by impedance resonance, which is definedTECNature ; ComPhys ; Hel by

$$Z_{ab}=V_{a}/I_{b}=G_{ab},$$

(50)

where $G=J^{-1}$ is the Green function. Taking the nodes $a=b$ at an edge, we show the real and imaginary parts of the impedance for a finite chain as a function of $\omega$ in Fig.7, which are marked in red. For comparison, we also show the impedance for a periodic boundary condition in cyan, where the edge states are absent.

We first study the Hermitian case ($\gamma=0$) with $\xi=0$, where the impedance is shown in Fig.7(a1) and (a2). The edge impedance resonance is clear by comparing the periodic boundary condition and the open boundary condition. There are only two bulk peaks in cyan at Re[$E_{\alpha\beta}^{\prime}\left(k;\gamma,\xi\right)$]. On the other hand, there is an additional peak in red due to the edge states between two bulk peaks, as corresponds to Fig.2(a1).

Next, we show the impedance for various nonreciprocity $\gamma$ with $\xi=0$ in Fig.7(a1)$\sim$(f1) and (a2)$\sim$(f2). The edge impedance resonance rapidly decreases as the increase of $\gamma$, as shown in Fig.7(b1). This is due to the imaginary contribution in Eq.(26). Then, the distance between two bulk peaks becomes narrower, which is consistent with Re[$E_{\alpha\beta}^{\prime}\left(k;\gamma,\xi=0\right)$] as shown in Fig.2(c1). The two bulk peaks merge into one peak at the spontaneous $PT$ symmetry breaking point $\gamma_{0}$, as shown in Fig.7(e1). The bulk impedance resonance is very strong due to the gap closing of the bulk band. We also observe the edge impedance resonance in the imaginary-line gap topological insulating phase, where the impedance resonance is weak comparing to Fig.7(a1) as shown in Fig.7(f1). This is also the imaginary contribution in Eq.(26).

We also show the impedance for finite $\xi$ in Fig.7(a3)$\sim$(f3) and (a4)$\sim$(f4), as corresponds to Fig.2(c2). The bulk impedance peaks become broad, which reflects the broadening of the bulk bands. As a result, the edge impedance peak becomes clearer as in Fig.7(a3) in comparison to Fig.7(a1). There are strong cyan resonances at the phase transition point $\gamma_{1}$ point as shown in Fig.7(c3) and (c4). It is due to the gap closing of the bulk band. In Fig.7(d3) and (d4), the impedance structure is complicated, which reflects the metallic band structure. The effect of the $\xi$ term is negligible for the imaginary-line-gap topological phase as shown in Fig.7(f3) and (f4) since the peak of the impedance is broad even for $\xi=0$ in Fig.7(f1) and (f2). Here, note that $\xi$ appears only in the form of $(1+\xi)$ in Eq.(45).