2020 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 / SPT phase / gapped ground state |
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| Outline of Annual Research Achievements |
The research results submitted in FY2019 have been peer reviewed and published in mathematics and mathematical physics journals of good quality.
We have defined topological phase labels for unique gapped ground states of one-dimensional lattice fermions with an on-site finite group symmetry. These indices are defined using the split property and take value in a twisted sum of cohomology groups. This result gives a mathematically precise formulation of one-dimensional fermionic symmetry protected topological (SPT) phases in infinite systems. The work also further develops the theory of fermionic matrix product states via endomorphisms on graded Cuntz algebras. These results have been published in Forum of Mathematics, Sigma.
In separate work, we studied the question of Gabor frames and localized Wannier bases for Schroedinger operators with an aperiodic point potential supported around a Delone set. This result provides a connection between groupoids, operator K-theory and condensed matter physics. We have submitted this work for publication and it is currently under review.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
The project's desired aims for one-dimensional gapped ground states have now been completed. This gives us more time than initially expected to study higher-dimensional systems as well as systems with non-degenerate gapped ground states arising from long-range topological order.
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| Strategy for Future Research Activity |
The next task of the project is to investigate gapped ground states in two-dimensions and higher. A key challenge is to study such systems without using the split property, which is no longer applicable in many cases of interest.
We also aim to reconnect our study of SPT phases and ground state indices with (generalized) notions of spectral flow. We intend to study this question by further investigating flux insertions and charge pumping in gapped ground states.
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| Causes of Carryover |
Due to COVID-19, all planned business trips (domestic and international) for FY2020 were postponed or cancelled. This has changed our predicted budget, which included funds for domestic and international travel. The FY2020 funding was used on equipment (macbook pro and monitor, iPad, textbooks). The unspent funds will be used on future business trips when it becomes safe to do so.
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