| Project/Area Number |
19K14548
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | Nagoya University (2023)
Tohoku University (2019-2022) |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2022: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | Topological phase / Operator algebras / Index theory / 作用素環論 / トポロジカル相 / 指数理論 / Quantum walks / Topological phases / Quantum walk / Operator Algebras / Gapped ground state / SPT phase / operator algebras / gapped ground state / ground state / spectral flow / K-theory |
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| Outline of Research at the Start |
We will use tools from operator algebras to prove precise statements that characterise gapped ground states of quantum spin chains. Cohomology theories can be incorporated into this framework which will clarify the manifestly topological nature of such ground states.
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| Outline of Final Research Achievements |
The project was an exploration of topological properties of gapped ground states. That is, properties of low-energy quantum mechanical systems which are stable under small perturbations and deformations. Our primary method for studying such problems was to use methods from operator algebras and non-commutative index theory.
Homology and cohomology are mathematical tools that give a simple algebraic description of a potential complicated setting (for example, how many holes in a shape). By using homology and cohomology theories for operator algebras, which describe quantum mechanical systems, we mathematically characterised stable properties of a wide variety of gapped ground states.
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| Academic Significance and Societal Importance of the Research Achievements |
Ground states give the most basic information about quantum mechanical system. By understanding the topological properties of ground states, we can understand which systems can be loosely connected and which are manifestly distinct. This aids our conceptual understanding of materials.
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