Interacting topological phases and operator algebras
| 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 | Tohoku University |
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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 |
Granted (Fiscal Year 2022)
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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 | Index theory / Quantum walk / Topological phases / 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 Annual Research Achievements |
The project has been quite successful in adapting techniques from operator algebras, index theory and noncommutative geometry to certain classes of quantum mechanical Hamiltonians and gapped ground states.
Building from our established framework, we began a systematic study of topological and index theoretic properties of quantum walks, which can be regarded as quantum (discrete) analogues of random walks. In particular, quantum walks are expected to be useful in the implementation of algorithms in quantum computing. Determining robust properties of such quantum walks via a concrete link to topological invariants is therefore of significant value.
We were able to adapt our mathematical techniques in the study of Hamiltonians and gapped ground states to quantum walk systems and define several topologically robust indices. This work has been submitted for publication and is currently under review.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The key expected outcomes and goals of the submitted proposal have largely been accomplished. Robust topological indices have been defined for a large class of Hamiltonians that support a unique gapped ground state. However, due to the COVID pandemic, most of the proposed travel plans have had to be delayed or cancelled. As such, we have extended the project for one more year to further expand upon the project's original goals as well as reschedule delayed travel plans.
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| Strategy for Future Research Activity |
Our new results on topological phases of quantum walks provide a strong starting point to further explore new and novel connections between index theory and operator algebras with quantum information theory.
As a first step in this direction, our aim is to characterise quantum walks with additional anti-linear symmetries such as charge-conjugation symmetry. We also aim to study quantum walks that arise from periodically driven systems and their connection with secondary invariants.
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Report
(4 results)
Research Products
(38 results)
