Index Theorems for One-dimensional Chirally Symmetric Quantum Walks with Asymptotically Periodic Parameters

Abstract

We focus on index theory for chirally symmetric discrete-time quantum walks on the one-dimensional integer lattice. Such a discrete-time quantum walk model can be characterised as a pair of a unitary self-adjoint operator $\Gamma$ and a unitary time-evolution operator $U,$ satisfying the chiral symmetry condition $U^{\ast }=\Gamma U\Gamma .$ The significance of this index theory lies in the fact that the index we assign to the pair $(\Gamma ,U)$ gives a lower bound for the number of symmetry protected edge-states associated with the time-evolution $U.$ The symmetry protection of edge-states is one of the important features of the bulk-edge correspondence. The purpose of the present paper is to revisit the well-known bulk-edge correspondence for the split-step quantum walk on the one-dimensional integer lattice. The existing mathematics literature makes use of a fundamental assumption, known as the $2$-phase condition, but we completely replace it by the so-called asymptotically periodic assumption in this article. This generalisation heavily relies on analysis of some topological invariants associated with Toeplitz operators.

References in corpus (5)

• Environment-Assisted Quantum Walks in Photosynthetic Energy Transfer
• Quantum walks of correlated particles
• Exploring Topological Phases With Quantum Walks
• A Constructive Approach to Topological Invariants for One-dimensional Strictly Local Operators
• The bulk-edge correspondence for the split-step quantum walk on the one-dimensional integer lattice

Cited by in corpus (1)

• Index Theory of Chiral Unitaries and Split-Step Quantum Walks