Exponentially growing bulk Green functions as signature of nontrivial non-Hermitian winding number in one dimension

TL;DR

This paper demonstrates that in non-Hermitian topological systems, the bulk Green function exhibits exponential growth related to the non-Hermitian winding number, revealing a breakdown of bulk-boundary correspondence due to boundary sensitivity.

Contribution

It introduces a bulk Green function that captures the response growth in non-Hermitian systems and links this to the non-Hermitian winding number, explaining boundary condition effects.

Findings

Bulk Green function grows exponentially with distance when the winding number is nonzero.

Exponential growth explains boundary sensitivity and breakdown of bulk-boundary correspondence.

Non-Hermitian topological invariants do not reliably predict boundary states under open boundary conditions.

Abstract

A nonzero non-Hermitian winding number indicates that a gapped system is in a nontrivial topological class due to the non-Hermiticity of its Hamiltonian. While for Hermitian systems nontrivial topological quantum numbers are reflected by the existence of edge states, a nonzero non-Hermitian winding number impacts a system's bulk response. To establish this relation, we introduce the bulk Green function, which describes the response of a gapped system to an external perturbation on timescales where the induced excitations have not propagated to the boundary yet, and show that it will grow in space if the non-Hermitian winding number is nonzero. Such spatial growth explains why the response of non-Hermitian systems on longer timescales, where excitations have been reflected at the boundary repeatedly, may be highly sensitive to boundary conditions. This exponential sensitivity to boundary conditions explains the breakdown of the bulk-boundary correspondence in non-Hermitian systems: topological invariants computed for periodic boundary conditions no longer predict the presence or absence of boundary states for open boundary conditions.