2023 Fiscal Year Annual Research Report
Interacting topological phases and operator algebras
| Project/Area Number |
19K14548
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| Research Institution | Nagoya University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | Index theory / Quantum walks / Topological phases |
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| Outline of Annual Research Achievements |
The goal of the project is to construct and understand topological indices for a gapped Hamiltonians and ground states in a variety of settings. Methods from quantum information theory have also been quite effective in understanding further properties of ground states. In FY2022 and FY2023 we also began to investigate topological properties of a few simple descriptions of time evolutions and operations on quantum states.
Quantum walks provide a flexible model for a discrete time step of a Hamiltonian. We were able to define topological indices for quantum walks with an additional chiral symmetry in the very general setting of Hilbert C*-modules. This result also has implications for edge/boundary properties of Hamiltonians on half-space systems and properties. This work has been published in Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), an open-access journal.
Furthermore, we developed a general method to study the essential spectrum of a wide variety of quantum walk unitaries. This method provides the tools to give a more systematic study of higher-dimensional quantum walks. These results have been disseminated in a variety of conferences and seminars.
We also produced a chapter concerning the K-theoretic classification of topological insulators and superconductors for the upcoming revised version of the Encyclopedia of Mathematical Physics. This submission is currently under review.
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