Pole and zero edge state invariant for 1D non-Hermitian sublattice symmetry

TL;DR

This paper proves the equivalence of two topological criteria for edge states in 1D non-Hermitian sublattice models and extends the pole-zero approach to more general Hamiltonians, supported by numerical examples.

Contribution

It provides an explicit proof of the equivalence between the GBZ and pole-zero invariants and extends the pole-zero method to non-off-diagonal Hamiltonians.

Findings

Both criteria correctly predict edge states.

The pole-zero approach is extended to more general models.

Numerical examples validate the invariants.

Abstract

There have been several criteria for the existence of topological edge states in 1D non-Hermitian two-band sublattice-symmetric tight-binding Hamiltonians. The generalized Brillouin zone (GBZ) approach uses the integration of the Berry connection over the GBZ contour in the complex wavevector space. An alternate `pole-zero' approach uses algebraic properties of the off-diagonal matrix elements of the sublattice-symmetric Hamiltonian in off-diagonal form. Both correctly predict the presence or absence of edge states, but there has not been an explicit proof of their equivalence. Here we provide such an explicit proof and moreover we extend the pole-zero approach so that it also applies for sublattice-symmetric models when the Hamiltonian is not in off-diagonal form. We give numerical examples for these invariants.