Edge States and Topological Invariants of Non-Hermitian Systems

The bulk-boundary correspondence is among the central issues of non-Hermitian topological states. We show that a previously overlooked “non-Hermitian skin effect” necessitates redefinition of topological invariants in a generalized Brillouin zone. The resultant phase diagrams dramatically differ from the usual Bloch theory. Specifically, we obtain the phase diagram of the non-Hermitian Su-Schrieffer-Heeger model, whose topological zero modes are determined by the non-Bloch winding number instead of the Bloch-Hamiltonian-based topological number. Our work settles the issue of the breakdown of conventional bulk-boundary correspondence and introduces the non-Bloch bulk-boundary correspondence.

Figure 1

Non-Hermitian SSH model. The dotted box indicates the unit cell.

Figure 2

Numerical spectra of an open chain with length $L=40$ (unit cell). $t_2=1$, $\gamma =4/3$; $t_1$ varies in $[-3,3]$. (a) $|E|$ as functions of $t_1$. The zero-mode line is shown in red (twofold degenerate, ignoring an indiscernible split). The true transition point [$\sqrt{t_2^2+(\gamma /2{)}^2}\approx 1.20$] and the $H(k)$-gap-closing points ($t_2\pm \gamma /2$) are indicated by arrows. (b),(c) The real and imaginary parts of $E$. (d) The same as (a) except that the value of $t_1$ at the leftmost bond is replaced by $t_1-0.8$, which generates additional nonzero modes, but the zero modes are unaffected.

Figure 3

(a) $|{\beta }_j|-E$ curves from Eq. (7). $t_1=1$ (dark color) and $\sqrt{t_2^2+(\gamma /2{)}^2}\approx 1.20$ (light color). (b) Complex-valued ${\beta }_j$’s form a closed loop $C_{\beta }$, which is a circle for the present model [by Eq. (13)]. The shown one is for $t_1=1$. $C_{\beta }$ can be viewed as a deformed Brillouin zone that generalizes the usual one. In Hermitian cases, $C_{\beta }$ is a unit circle (dashed line). (c) Profile of a zero mode (main figure) and eight randomly chosen bulk eigenstates (inset), illustrating the non-Hermitian skin effect found in the analytic solution, namely, all the bulk eigenstates are localized near the boundary. $t_1=1$. Common parameters: $t_2=1,\gamma =4/3$.

Figure 4

Numerical result of topological invariant. $N_{\beta }$ is the number of grid points on $C_{\beta }$. $t_2=1,\gamma =4/3$.

Figure 5

The nonzero $t_3$ case. (a) Upper panel: Spectrum of an open chain; $t_2=1,\gamma =4/3,t_3=1/5$; $L=100$. Lower panel: topological invariant calculated using 200 grid points on $C_{\beta }$. The transition points are $t_1\approx \pm 1.56$. (b) $C_{\beta }$ for $t_1=1.1$.