Complete homotopy invariants for translation invariant symmetric quantum walks on a chain

TL;DR

This paper classifies one-dimensional translation invariant symmetric quantum walks using topological invariants, providing explicit formulas and analyzing the effects of locality, symmetry, and translation invariance on their topological phases.

Contribution

It offers a complete classification of translation invariant quantum walks with explicit winding number formulas, refining previous results by including translation invariance considerations.

Findings

Classification matches previous non-translation invariant results with finer distinctions.

Provides explicit winding number formulas for all symmetry types.

Shows exponential decay in stationary Schrödinger equations for large bulk pieces.

Abstract

We provide a classification of translation invariant one-dimensional quantum walks with respect to continuous deformations preserving unitarity, locality, translation invariance, a gap condition, and some symmetry of the tenfold way. The classification largely matches the one recently obtained (arXiv:1611.04439) for a similar setting leaving out translation invariance. However, the translation invariant case has some finer distinctions, because some walks may be connected only by breaking translation invariance along the way, retaining only invariance by an even number of sites. Similarly, if walks are considered equivalent when they differ only by adding a trivial walk, i.e., one that allows no jumps between cells, then the classification collapses also to the general one. The indices of the general classification can be computed in practice only for walks closely related to some translation invariant ones. We prove a completed collection of simple formulas in terms of winding numbers of band structures covering all symmetry types. Furthermore, we determine the strength of the locality conditions, and show that the continuity of the band structure, which is a minimal requirement for topological classifications in terms of winding numbers to make sense, implies the compactness of the commutator of the walk with a half-space projection, a condition which was also the basis of the general theory. In order to apply the theory to the joining of large but finite bulk pieces, one needs to determine the asymptotic behaviour of a stationary Schr\"odinger equation. We show exponential behaviour, and give a practical method for computing the decay constants.