Index Theory of Chiral Unitaries and Split-Step Quantum Walks
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...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211976 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Index Theory of Chiral Unitaries and Split-Step Quantum Walks. Chris Bourne. SIGMA 19 (2023), 053, 39 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862743801565544448 |
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| author | Bourne, Chris |
| author_facet | Bourne, Chris |
| citation_txt | Index Theory of Chiral Unitaries and Split-Step Quantum Walks. Chris Bourne. SIGMA 19 (2023), 053, 39 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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 *-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 *-modules, extending a result by Matsuzawa.
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| first_indexed | 2026-04-17T18:35:21Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211976 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T18:35:21Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bourne, Chris 2026-01-20T16:12:56Z 2023 Index Theory of Chiral Unitaries and Split-Step Quantum Walks. Chris Bourne. SIGMA 19 (2023), 053, 39 pages 1815-0659 2020 Mathematics Subject Classification: 46L80; 47L90 arXiv:2211.10601 https://nasplib.isofts.kiev.ua/handle/123456789/211976 https://doi.org/10.3842/SIGMA.2023.053 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 *-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 *-modules, extending a result by Matsuzawa. The author thanks C. Cedzich, S. Richard, and Y. Tanaka for helpful discussions. We also thank the anonymous referees for their numerous suggestions that have helped improve the manuscript. This work is supported by a JSPS Grant-in-Aid for Early-Career Scientists (No. 19K14548). An earlier version of this paper was completed while the author was affiliated with the WPI-AIMR, Tohoku University. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Index Theory of Chiral Unitaries and Split-Step Quantum Walks Article published earlier |
| spellingShingle | Index Theory of Chiral Unitaries and Split-Step Quantum Walks Bourne, Chris |
| title | Index Theory of Chiral Unitaries and Split-Step Quantum Walks |
| title_full | Index Theory of Chiral Unitaries and Split-Step Quantum Walks |
| title_fullStr | Index Theory of Chiral Unitaries and Split-Step Quantum Walks |
| title_full_unstemmed | Index Theory of Chiral Unitaries and Split-Step Quantum Walks |
| title_short | Index Theory of Chiral Unitaries and Split-Step Quantum Walks |
| title_sort | index theory of chiral unitaries and split-step quantum walks |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211976 |
| work_keys_str_mv | AT bournechris indextheoryofchiralunitariesandsplitstepquantumwalks |