A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics
We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We star...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2022 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2022
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211542 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics. Eric J. Pap, Daniël Boer and Holger Waalkens. SIGMA 18 (2022), 003, 42 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862604747627823104 |
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| author | Pap, Eric J. Boer, Daniël Waalkens, Holger |
| author_facet | Pap, Eric J. Boer, Daniël Waalkens, Holger |
| citation_txt | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics. Eric J. Pap, Daniël Boer and Holger Waalkens. SIGMA 18 (2022), 003, 42 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we demonstrate that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection that yields the (generalized) geometric phase. This bundle also provides a natural generalization of the quantum geometric tensor and derived tensors, and we demonstrate how it can incorporate the non-geometric dynamical phase as well. We conclude by demonstrating how the bundle can be recast as a principal bundle, allowing both the geometric phases and the permutations of eigenstates to be expressed simultaneously using standard holonomy theory.
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| first_indexed | 2026-03-14T04:12:43Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211542 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T04:12:43Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Pap, Eric J. Boer, Daniël Waalkens, Holger 2026-01-05T12:30:38Z 2022 A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics. Eric J. Pap, Daniël Boer and Holger Waalkens. SIGMA 18 (2022), 003, 42 pages 1815-0659 2020 Mathematics Subject Classification: 81Q70; 81Q12; 55R99 arXiv:2107.02497 https://nasplib.isofts.kiev.ua/handle/123456789/211542 https://doi.org/10.3842/SIGMA.2022.003 We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we demonstrate that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection that yields the (generalized) geometric phase. This bundle also provides a natural generalization of the quantum geometric tensor and derived tensors, and we demonstrate how it can incorporate the non-geometric dynamical phase as well. We conclude by demonstrating how the bundle can be recast as a principal bundle, allowing both the geometric phases and the permutations of eigenstates to be expressed simultaneously using standard holonomy theory. The authors thank the anonymous referees whose careful remarks contributed to the quality of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics Article published earlier |
| spellingShingle | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics Pap, Eric J. Boer, Daniël Waalkens, Holger |
| title | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics |
| title_full | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics |
| title_fullStr | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics |
| title_full_unstemmed | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics |
| title_short | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics |
| title_sort | unified view on geometric phases and exceptional points in adiabatic quantum mechanics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211542 |
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