Index Theory of Chiral Unitaries and Split-Step Quantum Walks

TL;DR

This paper develops an index theory for chiral unitaries, linking topological invariants to quantum walk models and extending winding number formulas to broader mathematical settings.

Contribution

It introduces a new index framework for chiral unitaries using projections and Cayley transforms, applicable to Hilbert spaces and modules, and extends winding number results for split-step quantum walks.

Findings

Defined topological indices for chiral unitaries.

Related the index of chiral unitaries to Hamiltonian indices.

Extended winding number formulas to Hilbert C*-modules.

Abstract

Building from work by Cedzich et al. and Suzuki et al., we consider topological and index-theoretic properties of chiral unitaries, which are an abstraction of the time evolution of a chiral-symmetric self-adjoint operator. Split-step quantum walks provide a rich class of examples. We use the index of a pair of projections and the Cayley transform to define topological indices for chiral unitaries on both Hilbert spaces and Hilbert $C^*$-modules. In the case of the discrete time evolution of a Hamiltonian-like operator, we relate the index for chiral unitaries to the index of the Hamiltonian. We also prove a double-sided winding number formula for anisotropic split-step quantum walks on Hilbert $C^*$-modules, extending a result by Matsuzawa.