# Bulk-edge correspondence and stability of multiple edge states of a $\mathcal{PT}$ symmetric non-Hermitian system by using non-unitary quantum walks

## TL;DR

This paper investigates topological phases and multiple edge states in a $ ext{PT}$ symmetric non-Hermitian quantum system using non-unitary quantum walks, confirming bulk-edge correspondence and analyzing edge state stability under symmetry-breaking perturbations.

## Contribution

It introduces a non-unitary three-step quantum walk model with $ ext{PT}$ symmetry, demonstrating multiple edge states and their stability, revealing a novel breakdown of bulk-edge correspondence in non-Hermitian systems.

## Key findings

- Multiple edge states appear consistent with bulk-edge correspondence.
- Edge states remain stable unless coalescing at an exceptional point.
- Proposed method to determine edge states from probability evolution.

## Abstract

Topological phases and the associated multiple edge states are studied for parity and time-reversal $(\mathcal{PT})$ symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining $\mathcal{PT}$ symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the $\mathbb{Z}$ topological phase and numerically confirm that multiple edge states appear as expected from the bulk-edge correspondence. Therefore, the bulk-edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge states against a symmetry-breaking perturbation so that the topological phase is reduced to $\mathbb{Z}_2$ from $\mathbb{Z}$. In this case, we find that the number of edge states does not become one unless a pair of edge states coalesce at an exceptional point. Thereby, this is a new kind of breakdown of the bulk-edge correspondence in non-Hermitian systems. The mechanism of the prolongation of edge states against the symmetry-breaking perturbation is unique to non-Hermitian systems with multiple edge states and anti-linear symmetry. Toward experimental verifications, we propose a procedure to determine the number of multiple edge states from the time evolution of the probability distribution.