Symmetries, Topological Phases and Bound States in the One-Dimensional Quantum Walk

TL;DR

This paper explores the topological phases of a one-dimensional quantum walk, revealing that particle-hole symmetry alone can induce non-trivial topology and edge states, even without chiral symmetry.

Contribution

It introduces a simple method to calculate the Z2 × Z2 topological invariant for 1D periodically driven quantum systems based on particle-hole symmetry.

Findings

Topological phases arise from particle-hole symmetry without chiral symmetry.

Edge states can exist between bulks with identical Floquet operators.

The bulk Floquet operator alone does not determine the topological invariant.

Abstract

Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the Z2 \times Z2 topological invariant in a simple way that is consistent with a general definition for 1-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondance. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.