A unified diagrammatic approach to topological fixed point models
We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework established is reminiscent to that of state-sum models and lattice topological quantum field theories, but is formalized and unified in terms...
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| Format: | Article |
| Language: | English |
| Published: |
SciPost
2022-07-01
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| Series: | SciPost Physics Core |
| Online Access: | https://scipost.org/SciPostPhysCore.5.3.038 |
| _version_ | 1818188995423633408 |
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| author | A. Bauer, J. Eisert, C. Wille |
| author_facet | A. Bauer, J. Eisert, C. Wille |
| author_sort | A. Bauer, J. Eisert, C. Wille |
| collection | DOAJ |
| description | We introduce a systematic mathematical language for describing fixed point
models and apply it to the study to topological phases of matter. The framework
established is reminiscent to that of state-sum models and lattice topological
quantum field theories, but is formalized and unified in terms of tensor
networks. In contrast to existing tensor network ansatzes for the study of
ground states of topologically ordered phases, the tensor networks in our
formalism directly represent discrete path integrals in Euclidean space-time.
This language is more immediately related to the Hamiltonian defining the model
than other approaches, via a Trotterization of the respective imaginary time
evolution. We illustrate our formalism at hand of simple examples, and
demonstrate its full power by expressing known families of models in 2+1
dimensions in their most general form, namely string-net models and Kitaev
quantum doubles based on weak Hopf algebras. To elucidate the versatility of
our formalism, we also show how fermionic phases of matter can be described and
provide a framework for topological fixed point models in 3+1 dimensions. |
| first_indexed | 2024-12-11T23:35:47Z |
| format | Article |
| id | doaj.art-45de4e06d0b44bdebde8999743acc7ce |
| institution | Directory Open Access Journal |
| issn | 2666-9366 |
| language | English |
| last_indexed | 2024-12-11T23:35:47Z |
| publishDate | 2022-07-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics Core |
| spelling | doaj.art-45de4e06d0b44bdebde8999743acc7ce2022-12-22T00:45:52ZengSciPostSciPost Physics Core2666-93662022-07-015303810.21468/SciPostPhysCore.5.3.038A unified diagrammatic approach to topological fixed point modelsA. Bauer, J. Eisert, C. WilleWe introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework established is reminiscent to that of state-sum models and lattice topological quantum field theories, but is formalized and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism directly represent discrete path integrals in Euclidean space-time. This language is more immediately related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We illustrate our formalism at hand of simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.https://scipost.org/SciPostPhysCore.5.3.038 |
| spellingShingle | A. Bauer, J. Eisert, C. Wille A unified diagrammatic approach to topological fixed point models SciPost Physics Core |
| title | A unified diagrammatic approach to topological fixed point models |
| title_full | A unified diagrammatic approach to topological fixed point models |
| title_fullStr | A unified diagrammatic approach to topological fixed point models |
| title_full_unstemmed | A unified diagrammatic approach to topological fixed point models |
| title_short | A unified diagrammatic approach to topological fixed point models |
| title_sort | unified diagrammatic approach to topological fixed point models |
| url | https://scipost.org/SciPostPhysCore.5.3.038 |
| work_keys_str_mv | AT abauerjeisertcwille aunifieddiagrammaticapproachtotopologicalfixedpointmodels AT abauerjeisertcwille unifieddiagrammaticapproachtotopologicalfixedpointmodels |