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

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.