2019 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 / ground state / spectral flow / K-theory |
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| Outline of Annual Research Achievements |
One-dimensional ground states of fermionic Hamiltonians were studied. A relationship was found between the topological phase of the ground state and the Z_2-valued spectral flow. These results were extended to infinite systems with higher order interactions using the split property and the study of states of the CAR C*-algebra. These results have been published in Reviews in Mathematical Physics.
In order to study wider symmetry classes, a KO-valued spectral flow was defined and its mathematical properties and relation to physics studied. This new spectral flow generalizes all previous spectral flow constructions. These results have been submitted and are currently under review.
A graded Cayley transform was also studied that maps between the different presentations of K-theory that emerge in the study of topological phases. The Cayley transform makes these maps explicit and computable in the examples of interest. These results have been submitted and are 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
We have defined a torsion-valued index of order two to pure states of the CAR C*-algebra that satisfy the split property. Using results of Matsui, such states include the unique ground state of one-dimensional Hamiltonians with a spectral gap. In particular, such Hamiltonians may have higher-order interactions.
These results put us slightly ahead of our initially planned schedule and give us more time to consider further extensions and applications.
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| Strategy for Future Research Activity |
We have defined topological phases for gapped fermionic ground states in infinite systems without any additional input data. Our next task is to consider the case of ground states with an on-site group symmetry and its connection to cohomology theory. We plan to undertake this research once again using the split property and an operator algebraic framework.
We also plan to further extend our studies of spectral flow and K-theory. Previous K-theory studies are usually applicable to free-fermionic Hamiltonians or quasi-free ground states. Our aim is to expand our K-theoretic framework to better accommodate systems with higher-order interactions.
We also plan to begin our studies on higher-dimensional lattice systems.
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