Topological Correspondence between Hermitian and Non-Hermitian Systems: Anomalous Dynamics

The hallmark of symmetry-protected topological phases is the existence of anomalous boundary states, which can only be realized with the corresponding bulk system. In this work, we show that for every Hermitian anomalous boundary mode of the ten Altland-Zirnbauer classes, a non-Hermitian counterpart can be constructed, whose long-time dynamics provides a realization of the anomalous boundary state. We prove that the non-Hermitian counterpart is characterized by a point-gap topological invariant, and furthermore, that the invariant exactly matches that of the corresponding Hermitian anomalous boundary mode. We thus establish a correspondence between the topological classifications of ($d+1$)-dimensional gapped Hermitian systems and $d$-dimensional point-gapped non-Hermitian systems. We illustrate this general result with a number of examples in different dimensions. This work provides a new perspective on point-gap topological invariants in non-Hermitian systems.

Figure 1

(a) Dispersion for the 1D chiral hopping model in the complex plane. The topological invariant $w$ is defined with respect to $E_B$ as in Eq. (3). Depending on the value of $E_B$ and other parameters, $w$ can differ. In this case, $w=1$. (b) Dispersion for the 2D model in Eq. (19) that resembles the surface of a 3D chiral topological insulator. Here, ${\gamma }_{\mathit{k}}=2\cos k_x+\cos k_y$, $b_{1,\mathit{k}}=\sin k_x$, $b_{2,\mathit{k}}=\sin k_y$, and $b_{3,\mathit{k}}=0$. Each white dot represents a Dirac cone with $\pm$ chirality. In this case, depending on $E_B$, the topological invariant can be 1,0,-1. (c) The 1D system characterized by $w\in \mathbb{Z}$ in the chiral hopping model corresponds to the edge of a 2D system characterized by an integer quantum Hall state with Chern number $n=w\in \mathbb{Z}$.

Figure 2

Diagrams defining (a) Hermitian classes $s$ and (b) non-Hermitian real classes $s^{†}$ (${\mathrm{AZ}}^{†}$). Hermitian AZ classes are defined by time reversal $\mathcal{T}$, particle-hole $\mathcal{P}$, and chiral $\mathcal{C}$ symmetries. Similarly, non-Hermitian ${\mathrm{AZ}}^{†}$ classes are defined by $K$-, $C$-, and $Q$-type symmetries. Real (complex) classes are given by blue (red) dots. There are two complex classes ($s=0$, 1) depending on the absence or presence of $\mathcal{C}$ or $q$.