Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 053, 39 pages      arXiv:2211.10601      https://doi.org/10.3842/SIGMA.2023.053

Index Theory of Chiral Unitaries and Split-Step Quantum Walks

Chris Bourne ab
a) Institute for Liberal Arts and Sciences and Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
b) RIKEN iTHEMS, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

Received December 04, 2022, in final form July 24, 2023; Published online July 28, 2023

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.

Key words: index theory; $K$-theory; quantum walk; operator algebras.

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