Relative homotopy approach to topological phases in quantum walks

TL;DR

This paper introduces a new topological invariant for basic discrete-time quantum walks, derived from a homotopy approach, which helps identify edge states and topological phases more generally than previous models.

Contribution

It develops a relative homotopy framework and a novel topological invariant for basic DTQWs, extending topological phase classification beyond split-step models.

Findings

Topological properties inferred from Brillouin zone mapping.

Homotopy relative to special points characterizes phases.

Invariant predicts number of edge states at interfaces.

Abstract

Discrete-time quantum walks (DTQWs) provide a convenient platform for a realisation of many topological phases in noninteracting systems. They often offer more possibilities than systems with a static Hamiltonian. Nevertheless, researchers are still looking for DTQW symmetries protecting topological phases and for definitions of appropriate topological invariants. Although majority of DTQW studies on this topic focus on the so called split-step quantum walk, two distinct topological phases can be observed in more basic models. Here we infer topological properties of the basic DTQWs directly from the mapping of the Brillouin zone to the Bloch Hamiltonian. We show that for translation symmetric systems they can be characterized by a homotopy relative to special points. We also propose a new topological invariant corresponding to this concept. This invariant indicates the number of edge states at the interface between two distinct phases.