Topological Nature of Edge States for One-Dimensional Systems without Symmetry Protection

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 non-zero 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.