Topological edge states in two-gap unitary systems: A transfer matrix approach

TL;DR

This paper introduces a transfer matrix approach to analyze topological edge states in two-gap unitary systems, revealing their winding behavior and associated topological invariants on Riemann surfaces.

Contribution

It develops a novel transfer matrix formalism for two-gap unitary systems and links edge states to topological invariants via Riemann surface constructions.

Findings

Edge states are localized and wind around non-contractible loops.

A topological invariant is associated with each energy gap.

The approach provides detailed insights into the band structure topology.

Abstract

We construct and investigate a family of two-band unitary systems living on a cylinder geometry and presenting localized edge states. Using the transfer matrix formalism, we solve and investigate in details such states in the thermodynamic limit. Analitycity considerations then suggest the construction of a family of Riemman surfaces associated to the band structure of the system. In this picture, the corresponding edge states naturally wind around non contractile loops, defining by the way a topological invariant associated to each gap of the system.