The topological classification of one-dimensional symmetric quantum walks

TL;DR

This paper provides a comprehensive topological classification of one-dimensional symmetric quantum walks, identifying three indices that characterize their properties and stability, extending previous translation-invariant results to more general cases.

Contribution

It introduces a complete topological classification for 1D quantum walks with discrete symmetries, including non-translation-invariant cases, using three indices that capture asymptotic and perturbation-sensitive features.

Findings

Classification is parameterized by three indices in different groups.

Two indices are stable under compact perturbations, one is sensitive.

The classification extends known translation-invariant results and relates to bulk-boundary correspondence.

Abstract

We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition. No translation invariance is assumed. The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm continuous path along which all the mentioned properties remain valid. Of the three indices, two are related to the asymptotic behaviour far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. The results apply to the Hamiltonian case as well. In this case all compact perturbations can be contracted, so the third index is not defined. Our classification extends the one known in the translation invariant case, where the asymptotic right and left indices add up to zero, and the third one vanishes, leaving effectively only one independent index. When two translationally invariant bulks with distinct indices are joined, the left and right asymptotic indices of the joined walk are thereby fixed, and there must be eigenvalues at $1$ or $-1$ (bulk-boundary correspondence). Their location is governed by the third index. We also discuss how the theory applies to finite lattices, with suitable homogeneity assumptions.