Anomalous helical edge states in a non-Hermitian Chern insulator

A non-Hermitian extension of a Chern insulator and its bulk-boundary correspondence are investigated. It is shown that in addition to the robust chiral edge states that reflect the nontrivial topology of the bulk (nonzero Chern number), anomalous helical edge states localized only at one edge can appear, which are unique to the non-Hermitian Chern insulator.

Figure 1

Phase diagram of the non-Hermitian Chern insulator with periodic boundaries given by Eq. (8). The white regions represent the gapped phases that are characterized by the Chern number $C$. The orange (${\nu }_0\ne 0$), blue (${\nu }_{\pi }\ne 0$), and green (${\nu }_0\ne 0,{\nu }_{\pi }\ne 0$) regions represent the gapless phases where pairs of exceptional points appear on $k_x=0,k_x=\pm \pi$, and both $k_x=0$ and $k_x=\pm \pi$, respectively.

Figure 2

Complex-band structures of the non-Hermitian Chern insulator $E_{\pm }=E_{\pm }\left(k_x,k_y\right)$ given by Eq. (9). The orange (blue) band represents $E_{+}$ ($E_{-}$). [(a) and (b)] Real and imaginary parts of the gapped and topologically trivial bands with zero Chern number $C=0$ ($t=1.0,m=-3.0,\gamma =0.5$). [(c) and (d)] Real and imaginary parts of the gapped and topologically nontrivial bands with nonzero Chern number $C=-1$ ($t=1.0,m=-1.0,\gamma =0.5$). [(e) and (f)] Real and imaginary parts of the gapless bands with a pair of exceptional points on $k_x=0$ ($t=1.0,m=-2.0,\gamma =1.0$). [(g) and (h)] Real and imaginary parts of the gapless bands with two pairs of exceptional points on both $k_x=0$ and $k_x=\pm \pi$ ($t=1.0,m=-0.2,\gamma =1.0$).

Figure 3

Complex spectrum of the non-Hermitian Chern insulator given by Eq. (16), which has periodic boundaries in the $x$ direction and open boundaries in the $y$ direction ($L_y=30$). [(a) and (b)] Real and imaginary parts of the complex spectrum in the gapped and topologically trivial phase ($t=1.0,m=-3.0,\gamma =0.5$). No gapless states appear between the gapped complex bands. [(c) and (d)] Real and imaginary parts of the complex spectrum in the gapped and topologically nontrivial phase ($t=1.0,m=-1.0,\gamma =0.5$). A pair of chiral edge states appears at both edges between the gapped complex bands whose dispersions are $E_{\mathrm{edge}}\left(k_x\right)=\pm \left(tsin{k}_x+\text{i}\gamma \right)$. [(e) and (f)] Real and imaginary parts of the complex spectrum in the gapless phase with an exceptional point at $k_x=0$ ($t=1.0,m=-2.0,\gamma =1.0$). [(g) and (h)] Real and imaginary parts of the complex spectrum in the gapless phase with exceptional points at both $k_x=0$ and $k_x=\pm \pi$ ($t=1.0,m=-0.2,\gamma =1.0$). A pair of chiral edge states appears at both edges, despite the absence of the energy gap.

Figure 4

Wave functions of the non-Hermitian Chern insulator given by Eq. (16), which has periodic boundaries in the $x$ direction and open boundaries in the $y$ direction ($L_y=30$). The system is in the gapped and topologically nontrivial phase that corresponds to Figs. 3 and 3 ($t=1.0,m=-1.0,\gamma =0.5;k_x=0.1\pi$). Whereas the bulk state with $E=0.985+0.157\text{i}$ (green dots) is delocalized throughout the bulk, the chiral edge states with $E=\pm \left(0.309+0.5\text{i}\right)$ (red and blue dots) are localized at both edges.

Figure 5

Complex spectrum of the non-Hermitian Chern insulator given by Eq. (19), which has open boundaries in the $x$ direction ($L_x=30$) and periodic boundaries in the $y$ direction. [(a) and (b)] Real and imaginary parts of the complex spectrum in the gapped and topologically trivial phase ($t=1.0,m=-3.0,\gamma =0.5$). No gapless states appear between the gapped complex bands. [(c) and (d)] Real and imaginary parts of the complex spectrum in the gapped and topologically nontrivial phase ($t=1.0,m=-1.0,\gamma =0.5$). A pair of chiral edge states appears between the gapped complex bands. [(e) and (f)] Real and imaginary parts of the complex spectrum for $t=1.0,m=-2.0$, and $\gamma =1.0$. Although the complex bands are gapless for the corresponding periodic systems [Figs. 2 and 2], the gap is open and the helical edge states localized only at the right edge appear. [(g) and (h)] Real and imaginary parts of the complex spectrum for $t=1.0,m=-0.2$, and $\gamma =1.0$. Although the complex bands are gapless for the corresponding periodic systems [Figs. 2 and 2], the gap is open and two pairs of helical edge states appear, one localized at the right edge and the other localized at the left edge.

Figure 6

Wave functions of the non-Hermitian Chern insulator given by Eq. (19), which has open boundaries in the $x$ direction ($L_x=30$) and periodic boundaries in the $y$ direction. (a) The system is in the gapped and topologically nontrivial phase that corresponds to Figs. 5 and 5 ($t=1.0,m=-1.0,\gamma =0.5;k_x=0.1\pi$). The chiral edge states with $E=\pm 0.309$ (red and blue dots) at both edges are more localized than the bulk state with $E=1.04+0.48\text{i}$ (green dots). (b) The system is in the gapped phase that corresponds to Figs. 5 and 5 ($t=1.0,m=-2.0,\gamma =1.0;k_x=0.1\pi$). The helical edge states with $E=\pm 0.309$ (red and blue dots) are localized only at the right edge, one of which is more localized than the bulk state with $E=0.968+0.204\text{i}$ (green dots) and the other of which is not.

Figure 7

Phase diagram of the non-Hermitian Chern insulator given by Eq. (19), which has open boundaries in the $x$ direction and periodic boundaries in the $y$ direction. The gapped phases are distinguished by the presence (red regions) or absence (gray regions) of the edge states, and $N_L$ ($N_R$) is the number of them localized at the left (right) edge. Across the blue lines, the number of edge states localized at either left or right edge changes, whereas the total number of edge states does not. The four green dots represent the set of parameters used in Fig. 5. In stark contrast to the Hermitian counterparts, the phase diagram is drastically different from that under the periodic boundary condition (Fig. 1).

Figure 8

Complex spectrum of the non-Hermitian Chern insulator given by Eq. (22), which has open boundaries in both $x$ and $y$ directions ($L_x=L_y=30$). Real and imaginary parts of the complex eigenenergies are shown for (a) $t=1.0,m=-3.0,\gamma =0.5$, (b) $t=1.0,m=-1.0,\gamma =0.5$, (c) $t=1.0,m=-2.0,\gamma =1.0$, and (d) $t=1.0,m=-0.2,\gamma =1.0$. The complex gap is open in all the cases, and the gapless edge states appear for (b)–(d).

Figure 9

Wave functions of the non-Hermitian Chern insulator given by Eq. (22), which has open boundaries in both $x$ and $y$ directions ($L_x=L_y=30$). The system is in the gapped phase that corresponds to Fig. 8 ($t=1.0,m=-1.0,\gamma =0.5$) for (a) and (b) and Fig. 8 ($t=1.0,m=-2.0,\gamma =1.0$) for (c) and (d). The gapless states [(b) and (d)] are more localized at the right edge than the bulk states [(a) and (c)].

Figure 10

Stability of the chiral and helical edge states. [(a)–(d)] Complex spectrum of the disordered non-Hermitian Chern insulator given by Eq. (B1) as a function of the wave number $k_x$. The periodic boundary condition is imposed in the $x$ direction, while the open boundary condition is imposed in the $y$ direction ($L_y=30$). The chiral edge states are robust against disorder for both [(a) and (b)] $t=1.0,m_y=-1.0+0.5{\epsilon }_y$, and ${\gamma }_y=0.5+0.5{\epsilon }_y^{\prime }$, and [(c) and (d)] $t=1.0,m_y=-0.2+0.2{\epsilon }_y$, and ${\gamma }_y=1.0+0.2{\epsilon }_y^{\prime }$, where ${\epsilon }_y$ and ${\epsilon }_y^{\prime }$ are random variables uniformly distributed over $\left[-0.5,0.5\right]$. [(e) and (f)] Complex spectrum of the disordered non-Hermitian Chern insulator given by Eq. (B2) as a function of the wave number $k_y$. The open boundary condition is imposed in the $x$ direction ($L_x=30$), while the periodic boundary condition is imposed in the $y$ direction. The helical edge states are robust against disorder for $t=1.0,m_x=-2.0+0.5{\epsilon }_x$, and ${\gamma }_x=1.0+0.5{\epsilon }_x^{\prime }$.

Figure 11

Complex spectrum of the non-Hermitian Chern insulator given by Eq. (19), which has open boundaries in the $x$ direction ($L_x=200$) and periodic boundaries in the $y$ direction. The same sets of parameters are used as in Fig. 5. The band gap is open and the chiral or helical edge states survive even in this larger system.