Topological nature of edge states for one-dimensional systems without symmetry protection

TL;DR

This paper introduces a new winding number invariant for one-dimensional two-band models that predicts edge states without relying on symmetry protection, broadening the understanding of topological phases.

Contribution

It analytically and numerically establishes a topological invariant based on complex wave-vector continuation, applicable to models with complex couplings and open boundaries, independent of symmetry constraints.

Findings

Winding number predicts edge states in complex, non-Hermitian models.

Invariant remains valid under unitary and similarity transformations.

Transition points do not necessarily involve gap closing.

Abstract

We numerically verify and analytically prove a winding number invariant that correctly predicts the number of edge states in one-dimensional, nearest-neighbor (between unit cells), two-band models with any complex couplings and open boundaries. Our winding number uses analytical continuation of the wave-vector into the complex plane and involves two special points on the full Riemann surface band structure that correspond to bulk eigenvector degeneracies. Our winding number is invariant under unitary or similarity transforms. We emphasize that the topological criteria we propose here differ from what is traditionally defined as a topological or trivial phase in symmetry-protected classification studies. It is a broader invariant for our model that supports nonzero energy edge states and its transition does not coincide with the gap closing condition. When the relevant symmetries are applied, our invariant reduces to well-known Hermitian and non-Hermitian symmetry-protected topological invariants.