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Local Noether theorem for quantum lattice systems and topological invariants of gapped states

Kapustin, Anton and Sopenko, Nikita (2022) Local Noether theorem for quantum lattice systems and topological invariants of gapped states. Journal of Mathematical Physics, 63 (9). Art. No. 091903. ISSN 0022-2488. doi:10.1063/5.0085964. https://resolver.caltech.edu/CaltechAUTHORS:20221013-48351800.2

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Abstract

We study generalizations of the Berry phase for quantum lattice systems in arbitrary dimensions. For a smooth family of gapped ground states in d dimensions, we define a closed d + 2-form on the parameter space, which generalizes the curvature of the Berry connection. Its cohomology class is a topological invariant of the family. When the family is equivariant under the action of a compact Lie group G, topological invariants take values in the equivariant cohomology of the parameter space. These invariants unify and generalize the Hall conductance and the Thouless pump. A key role in these constructions is played by a certain differential graded Fréchet–Lie algebra attached to any quantum lattice system. As a by-product, we describe ambiguities in charge densities and conserved currents for arbitrary lattice systems with rapidly decaying interactions.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1063/5.0085964DOIArticle
ORCID:
AuthorORCID
Kapustin, Anton0000-0003-3903-5158
Sopenko, Nikita0000-0002-8479-1924
Additional Information:This paper is part of the Special Topic on Mathematical Aspects of Topological Phases. We would like to thank Karl-Hermann Neeb and Bruno Nachtergaele for reading a preliminary draft of the paper and Bowen Yang for discussions. We are especially grateful to Bruno Nachtergaele for pointing out to us the relevance of the improved Lieb–Robinson bounds from Refs. 25 and 29 and for informing us about his forthcoming work.30 This research was supported, in part, by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632. A.K. was also supported by the Simons Investigator Award.
Funders:
Funding AgencyGrant Number
Department of Energy (DOE)DE-SC0011632
Simons FoundationUNSPECIFIED
Issue or Number:9
DOI:10.1063/5.0085964
Record Number:CaltechAUTHORS:20221013-48351800.2
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20221013-48351800.2
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:117394
Collection:CaltechAUTHORS
Deposited By: Research Services Depository
Deposited On:14 Oct 2022 21:11
Last Modified:14 Oct 2022 21:12

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