# Bulk-boundary correspondence for non-Hermitian Hamiltonians via Green functions

## TL;DR

This paper investigates the bulk-boundary correspondence in non-Hermitian topological phases, revealing that the non-Hermitian winding number indicates a bulk phase transition and does not always predict boundary states.

## Contribution

It demonstrates that in non-Hermitian systems, the traditional bulk-boundary correspondence breaks down, and the non-Hermitian winding number signals bulk phase transitions through Green function growth.

## Key findings

- No general correspondence between topological invariants and boundary states in non-Hermitian systems.
- Non-Hermitian winding number indicates a bulk phase transition via Green function growth.
- Boundary eigenstates are not always predicted by periodic boundary condition invariants.

## Abstract

Genuinely non-Hermitian topological phases can be realized in open systems with sufficiently strong gain and loss; in such phases, the Hamiltonian cannot be deformed into a gapped Hermitian Hamiltonian without energy bands touching each other. Comparing Green functions for periodic and open boundary conditions we find that, in general, there is no correspondence between topological invariants computed for periodic boundary conditions, and boundary eigenstates observed for open boundary conditions. Instead, we find that the non-Hermitian winding number in one dimension signals a topological phase transition in the bulk: It implies spatial growth of the bulk Green function.