A mathematical theory of topological invariants of quantum lattice systems
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917363760234496 |
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| author | Artymowicz, Adam Kapustin, Anton Yang, Bowen |
| author_facet | Artymowicz, Adam Kapustin, Anton Yang, Bowen |
| contents | We show that Hall conductance and its non-abelian and higher-dimensional analogs are obstructions to promoting a symmetry of a state to a gauge symmetry. To do this, we define a local Lie algebra over a Grothendieck site as a pre-cosheaf of Lie algebras with additional properties and propose that a gauge symmetry should be described by such an object. We show that infinitesimal symmetries of a gapped state of a quantum spin system form a local Lie algebra over a site of semilinear sets and use it to construct topological invariants of the state. Our construction applies to lattice systems on arbitrary asymptotically conical subsets of a Euclidean space including those which cannot be studied using field theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19287 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A mathematical theory of topological invariants of quantum lattice systems Artymowicz, Adam Kapustin, Anton Yang, Bowen Mathematical Physics Strongly Correlated Electrons Quantum Algebra We show that Hall conductance and its non-abelian and higher-dimensional analogs are obstructions to promoting a symmetry of a state to a gauge symmetry. To do this, we define a local Lie algebra over a Grothendieck site as a pre-cosheaf of Lie algebras with additional properties and propose that a gauge symmetry should be described by such an object. We show that infinitesimal symmetries of a gapped state of a quantum spin system form a local Lie algebra over a site of semilinear sets and use it to construct topological invariants of the state. Our construction applies to lattice systems on arbitrary asymptotically conical subsets of a Euclidean space including those which cannot be studied using field theory. |
| title | A mathematical theory of topological invariants of quantum lattice systems |
| topic | Mathematical Physics Strongly Correlated Electrons Quantum Algebra |
| url | https://arxiv.org/abs/2410.19287 |