Hidden zero modes and topology of non-Hermitian quantum systems

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 lead 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 present a proper bulk-boundary correspondence for non-Hermitian systems using index theorems in Toeplitz theory which connect bulk invariants with the singular value spectrum.