Relative homotopy approach to topological phases in quantum walks

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.

6 pages

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Cited by in corpus (4)

• Edge-dependent anomalous topology in synthetic photonic lattices subject to discrete step walks
• Demonstration of Discrete-Time Quantum Walks and Observation of Topological Edge States in a Superconducting Qutrit Chain
• Entanglement and fidelity across quantum phase transitions in locally perturbed topological codes with open boundaries
• Quantum walk with coherent multiple translations induces fast quantum gate operations