Hidden zero modes and topology of multiband non-Hermitian systems

TL;DR

This paper explains the breakdown of bulk-boundary correspondence in finite non-Hermitian systems, revealing hidden zero modes through singular value analysis and emphasizing the importance of considering the reflected Hamiltonian for topological characterization.

Contribution

It demonstrates the existence of hidden zero modes in finite non-Hermitian systems and introduces the need to analyze the reflected Hamiltonian to correctly relate zero modes to topological invariants.

Findings

Hidden zero modes are revealed via singular value spectrum analysis.

The reflected Hamiltonian $ ilde H$ is essential for topological classification.

Breakdown of bulk-boundary correspondence explained in finite systems.

Abstract

In a finite one-dimensional non-Hermitian system, the number of zero modes does not necessarily reflect the topology of the system. This is known as the breakdown of the bulk-boundary correspondence and has led to misconceptions about the topological protection of edge modes in such systems. Here we show why this breakdown does occur and that it typically results in hidden zero modes, extremely long-lived zero energy excitations, which are only revealed when considering the singular value instead of the eigenvalue spectrum. We point out, furthermore, that in a finite multiband non-Hermitian system with Hamiltonian $H$, one needs to consider also the reflected Hamiltonian $\tilde H$, which is in general distinct from the adjoint $H^\dagger$, to properly relate the number of protected zeroes to the winding number of $H$.