2022 Fiscal Year Final Research Report
A rigorous analysis of the stability of magnetism in the Hubbard model: An approach using operator inequalities.
| Project/Area Number |
18K03315
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | Hokkaido University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | 金属強磁性 / 基底状態 / 作用素環論 / 作用素不等式 / 普遍性 |
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| Outline of Final Research Achievements |
The mathematical understanding of the origin of metallic ferromagnetism remains insufficient at present. In this study, we constructed a mathematical framework using operator algebra and operator inequalities that can uniformly describe the fundamental theorems in the theory of metallic ferromagnetism, including Marshall-Lieb-Mattis' theorem, Nagaoka-Thouless' theorem, and Lieb's theorem. On the other hand, we proved that the magnetic properties of the ground state remain stable even when many-electron systems described by the Kondo lattice model or Anderson model interact with phonons. These results were analyzed from an overview perspective using the aforementioned unified framework and the relationship with Marshall-Lieb-Mattis' theorem was clarified.
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| Free Research Field |
数理物理学
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| Academic Significance and Societal Importance of the Research Achievements |
これまでの金属強磁性に関する理論研究では，個々の模型を解析することが中心で，普遍的な構造を数学的に解明する試みはほとんどなかった．本研究では，多電子系を記述する様々な模型の基底状態が持つ共通の数学的構造を解明することで，磁気秩序の安定性に関する新しい知見を得た．その数学的構造の記述には，作用素環論，特にstandard formの理論が有効であることが判明した．このことは，関連がほとんど意識されていない分野へ作用素環論が応用可能であることを意味し，今後の発展が期待できる．
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