2021 Fiscal Year Research-status Report
Interacting topological phases and operator algebras
| Project/Area Number |
19K14548
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| Research Institution | Tohoku University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | Operator Algebras / Gapped ground state / SPT phase / Topological phases |
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| Outline of Annual Research Achievements |
The previously submitted work on localised Wannier bases for aperiodic Schroedinger operators has been accepted and published in the Journal of Fourier Analysis and Applications.
We studied the connection between quasifree states of the canonical anti-commutation relations (CAR) algebra and coarse index theory. We found that by imposing a local equivalence condition on pure quasifree states of the CAR algebra with respect to a metric space, we can construct a K-homology class for this metric space. The coarse assembly map then relates this K-homology class to the classification of free-fermionic gapped Hamiltonians via K-theory. This work was submitted to and recently published in Journal of Physics A: Mathematical and Theoretical.
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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
We have made good progress in understanding and characterizing topological invariants of gapped ground states in one dimension. We have also made new and novel connections of gapped ground states to index theory and spectral flow in higher dimensions provided these ground states are quasifree. Higher dimensional systems with quartic and higher order interactions continue to be a technical challenge, though by imposing additional physically reasonable assumptions, we hope to provide further index-theoretic connections. As such, we are confident that the project's central goals are largely on track for completion.
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| Strategy for Future Research Activity |
Our recent work demonstrated a link between gapped ground states and coarse geometry in a specialized setting. Our plan is to further study gapped ground states using a coarse geometric framework and to find new topological properties of gapped ground states under less restrictive assumptions. In doing so, we also provide a pathway to understand such ground states using a possibly generalized version of spectral flow as well as other analytic characterizations of topological invariants.
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| Causes of Carryover |
Due to the ongoing COVID-19 pandemic, all business trips for FY2021 were postponed or cancelled. We hope to use the remaining funds for travel expenses when it is safe to do so.
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