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Symmetry, Integrability and Geometry: Methods and Applications, 2023, Volume 19, 053, 39 pp.
DOI: https://doi.org/10.3842/SIGMA.2023.053
(Mi sigma1948)
 

Index Theory of Chiral Unitaries and Split-Step Quantum Walks

Chris Bourneab

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
References:
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.
Keywords: index theory, $K$-theory, quantum walk, operator algebras.
Funding agency Grant number
Japan Society for the Promotion of Science 19K14548
This work is supported by a JSPS Grant-in-Aid for Early-Career Scientists (No. 19K14548).
Received: December 4, 2022; in final form July 24, 2023; Published online July 28, 2023
Document Type: Article
MSC: 46L80, 47L90
Language: English
Citation: Chris Bourne, “Index Theory of Chiral Unitaries and Split-Step Quantum Walks”, SIGMA, 19 (2023), 053, 39 pp.
Citation in format AMSBIB
\Bibitem{Bou23}
\by Chris~Bourne
\paper Index Theory of Chiral Unitaries and Split-Step Quantum Walks
\jour SIGMA
\yr 2023
\vol 19
\papernumber 053
\totalpages 39
\mathnet{http://mi.mathnet.ru/sigma1948}
\crossref{https://doi.org/10.3842/SIGMA.2023.053}
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