Sensitivity of quantum walks to boundary of two-dimensional lattices: approaches from the CGMV method and topological phases

TL;DR

This paper analyzes how two-dimensional quantum walks with boundaries exhibit edge states linked to topological phases, using spectral analysis of CMV matrices and introducing a new topological invariant for gapless spectra.

Contribution

It connects quantum walks to CMV matrices, derives spectra of edge states, and introduces a new topological number applicable when bulk spectra are gapless.

Findings

Edge states are consistent with bulk-edge correspondence predictions.

A new topological number is defined for gapless bulk spectra.

Edge states influence the asymptotic behavior of quantum walks.

Abstract

In this paper, we treat quantum walks in a two-dimensional lattice with cutting edges along a straight boundary introduced by Asboth and Edge (2015 Phys.Rev. A 91 022324) in order to study one-dimensional edge states originating from topological phases of matter and to obtain collateral evidence of how a quantum walker reacts to the boundary. Firstly, we connect this model to the CMV matrix, which provides a 5-term recursion relation of the Laurent polynomial associated with spectral measure on the unit circle. Secondly,we explicitly derive the spectra of bulk and edge states of the quantum walk with the boundary using spectral analysis of the CMV matrix. Thirdly, while topological numbers of the model studied so far are well-defined only when gaps in the bulk spectrum exist, we find a new topological number defined only when there are no gaps in the bulk spectrum. We confirm that the existence of the spectrum for edge states derived from the CMV matrix is consistent with the prediction from a bulk-edge correspondence using topological numbers calculated in the cases where gaps in the bulk spectrum do or do not exist. Finally, we show how the edge states contribute to the asymptotic behavior of the quantum walk through limit theorems of the finding probability. Conversely, we also propose a differential equation using this limit distribution whose solution is the underlying edge state.