Edge states and topological phases in non-Hermitian systems

Topological stability of the edge states is investigated for non-Hermitian systems. We examine two classes of non-Hermitian Hamiltonians supporting real bulk eigenenergies in weak non-Hermiticity: SU$(1,1)$ and SO$(3,2)$ Hamiltonians. As an SU$(1,1)$ Hamiltonian, the tight-binding model on the honeycomb lattice with imaginary onsite potentials is examined. Edge states with Re$E=0$ and their topological stability are discussed by the winding number and the index theorem based on the pseudo-anti-Hermiticity of the system. As a higher-symmetric generalization of SU$(1,1)$ Hamiltonians, we also consider SO$(3,2)$ models. We investigate non-Hermitian generalization of the Luttinger Hamiltonian on the square lattice and that of the Kane-Mele model on the honeycomb lattice, respectively. Using the generalized Kramers theorem for the time-reversal operator $\Theta$ with ${\Theta }^2=+1$ [M. Sato et al., e-print arXiv:1106.1806], we introduce a time-reversal-invariant Chern number from which topological stability of gapless edge modes is argued.

Figure 1

The honeycomb lattice.

Figure 2

The honeycomb lattice with zigzag edges at $m=1$ and 8 ($L_y=8$) along the $x$ direction.

Figure 3

The energy bands of the SU$(1,1)$ model (16) with zigzag edges along the $x$ direction for $t=1.0$ and ${\lambda }_V=0$ (Hermitian case). Here, $a$ is the lattice constant and $k_x$ the momentum in the $x$ direction. A zero-energy edge state with flat band appears for $2\pi /3<a{k}_x<4\pi /3$.

Figure 4

The real part of the energy bands of the SU$(1,1)$ model (16) with zigzag edges along the $x$ direction. We show the results for $t=1.0$ and various values of non-Hermitian parameter ${\lambda }_V$. Here, $a$ is the lattice constant and $k_x$ the momentum in the $x$ direction. For weak non-Hermiticity [(a) and (b)], the edge state with $\mathrm{Re}E=0$ survives around $a{k}_x\sim \pi$. For large ${\lambda }_V$ [(c)–(f)], no edge state with $\mathrm{Re}E=0$ appears.

Figure 5

The imaginary part of the energy bands of the SU$(1,1)$ model (16) with zigzag edges along the $x$ direction. We plot the results for $t=1.0$ and various values of the non-Hermitian parameter ${\lambda }_V$. Here, $a$ is the lattice constant and $k_x$ the momentum in the $x$ direction. For weak non-Hermiticity [(a) and (b)], there is no imaginary part in the bulk except near the gap-closing points at $a{k}_x=2\pi /3$ and $a{k}_x=4\pi /3$. The edge states with $\mathrm{Re}E=0$ [Figs. 4a and 4b] support the imaginary part in their energy. For large ${\lambda }_V$ [(c)–(f)], the bulk states support the imaginary part.

Figure 6

(a) Topologically protected state with Re$E=0$. (b) Topologically trivial states with Re$E=0$. Here, the closed (open) circles at Re$E=0$ represent states satisfying ${\sigma }_3|u\rangle \rangle =+|u\rangle$ [Eq. (27)] (${\sigma }_3|u\rangle \rangle =-|u\rangle$ [Eq. (28)]). In the latter case [(b)], the state can be nonzero mode since $n_{+}^0-n_{-}^0=0$.

Figure 7

The energy bands of the SO$(3,2)$ Luttinger Hamiltonian (67) with open boundary condition in the $x$ direction for $t=1.0$, $V=4.0$, $e_s=0.5$, and $d_1=d_2=0$ (Hermitian case). Here, $k_y$ is the momentum in the $y$ direction. Two gapless helical edge states appear on each edge.

Figure 8

The real part of the energy bands of the SO$(3,2)$ Luttinger Hamiltonian (67) with the open boundary condition in the $x$ direction. We plot the results for $t=1.0$, $V=4.0$, $e_s=0.5$, and various values of the non-Hermitian parameters $(d_1,d_2)$. Here, $k_y$ is the momentum in the $y$ direction. For weak non-Hermiticity [(a) and (b)], two gapless helical edge modes survive. When the bulk gap in the real part of the energy closes [(c) and (d)], the topological number $C_{\mathrm{TRI}}$ is not well defined. In this parameter region, only remnants of the edge states are observed.

Figure 9

The imaginary part of the energy bands of the SO$(3,2)$ Luttinger Hamiltonian (67) with the open boundary condition in the $x$ direction. We plot the results for $t=1.0$, $V=4.0$, $e_s=0.5$, and various values of the non-Hermitian parameters $(d_1,d_2)$. Here, $k_y$ is the momentum in the $y$ direction. For weak non-Hermiticity [(a) and (b)], there is no imaginary part in the bulk bands. Only the edge states with $\mathrm{Re}E=0$ [Figs. 8a and 8b] support the imaginary part in their energy. When the bulk gap in the real part of the energy closes [Figs. 8c and 8d], the bulk states also support the imaginary part as well.

Figure 10

The honeycomb lattice.

Figure 11

The sign of ${\nu }_{ij}$.

Figure 12

$\mathit{d}$, ${\mathit{d}}_1$, and ${\mathit{d}}_2$.

Figure 13

The nearest-neighbor hopping integrals of imaginary Rashba terms. The hopping integrals of each bond are asymmetric. [$U_1=\frac{1}{2}{\sigma }_x-\frac{\sqrt{3}}{2}{\sigma }_y$ and $U_2=\frac{1}{2}{\sigma }_x+\frac{\sqrt{3}}{2}{\sigma }_y$.]

Figure 14

The energy bands of the Kane-Mele model (108) with zigzag edges along the $x$ direction for $t=1.0$, ${\lambda }_{\mathrm{SO}}=0.06$, ${\lambda }_R=0$ (Hermitian case), and (a) ${\lambda }_V=0.1$, (b) ${\lambda }_V=0.4$. Here, $a$ is the lattice constant and $k_x$ the momentum in the $x$ direction. In (a), a gapless helical edge state appears on each edge, while in (b), no gapless edge state is obtained.

Figure 15

The real part of the energy bands of the non-Hermitian Kane-Mele model (108) with zigzag edge along the $x$ direction. We plot the results for $t=1.0$, ${\lambda }_{\mathrm{SO}}=0.06$, ${\lambda }_V=0.1$, and various values of non-Hermitian parameter ${\lambda }_R$. Here, $a$ is the lattice constant and $k_x$ is the momentum in the $x$ direction. In (a) and (b), we have gapless edge states. In (c), the bulk gap closes, and no gapless edge state exists.

Figure 16

The real vs imaginary part of the energy bands of non-Hermitian Kane-Mele model (108) with zigzag edges along the $x$ direction. We plot the results for $t=1.0$, ${\lambda }_{\mathrm{SO}}=0.06$, ${\lambda }_V=0.1$, and various values of the non-Hermitian parameter ${\lambda }_R$. The data (a) and (b) indicate that the energy of the gapless edge states [Figs. 15a and 15b] is real. On the other hand, the bulk state supports the imaginary part of the energy.

Figure 17

The real part of the energy bands of the non-Hermitian Kane-Mele model (108) with zigzag edge along the $x$ direction. We plot the results for $t=1.0$, ${\lambda }_{\mathrm{SO}}=0.06$, ${\lambda }_V=0.4$, and various values of non-Hermitian parameter ${\lambda }_R$. Here, $a$ is the lattice constant and $k_x$ is the momentum in the $x$ direction. In (a), no gapless edge state exists, while in (b), topologically protected gapless edge states appear. In (c), the bulk gap closes and the topologically protected gapless edge states disappear.

Figure 18

The real vs imaginary part of the energy bands of non-Hermitian Kane-Mele model (108) with zigzag edges along the $x$ direction. We plot the results for $t=1.0$, ${\lambda }_{\mathrm{SO}}=0.06$, ${\lambda }_V=0.4$, and various values of the non-Hermitian parameter ${\lambda }_R$. The data (b) indicate that the energy of the gapless edge states [Fig. 17b] is real. On the other hand, the bulk state supports the imaginary part of the energy.

Figure 19

Tight-binding model on the square lattice with imaginary SU$\left(2\right)$ gauge fields.