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

## TL;DR

This paper establishes a topological correspondence between Hermitian and non-Hermitian systems, showing that anomalous boundary states in Hermitian systems have non-Hermitian counterparts characterized by matching topological invariants, revealing new insights into non-Hermitian topological phases.

## Contribution

It demonstrates a systematic construction linking Hermitian anomalous boundary modes to non-Hermitian systems via point-gap topological invariants, expanding the understanding of topological classifications.

## Key findings

- Non-Hermitian counterparts of Hermitian boundary modes exist.
- Point-gap invariants match between Hermitian and non-Hermitian systems.
- Established a topological classification correspondence across dimensions.

## Abstract

The hallmark of symmetry-protected topological (SPT) 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.