Hidden Zero Modes and Topology of Multiband Non-Hermitian 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 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 Hover ˜, which is in general distinct from the adjoint H^†, to properly relate the number of protected zeros to the winding number of H.