Relative homotopy approach to topological phases in quantum walks

• URL: http://arxiv.org/abs/2209.12820v1
• Date: Mon, 26 Sep 2022 16:27:56 GMT
• Title: Relative homotopy approach to topological phases in quantum walks
• Authors: Andrzej Grudka, Marcin Karczewski, Pawel Kurzynski, Jan Wojcik, and Antoni Wojcik
• Abstract summary: We show that for translation symmetric systems they can be characterized by a homotopy relative to special points. We propose a new topological invariant corresponding to this concept. This invariant indicates the number of edge states at the interface between two distinct phases.
• Score: 0.08896991256227595
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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