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

• URL: http://arxiv.org/abs/2201.01327v3
• Date: Mon, 29 Aug 2022 06:02:19 GMT
• Title: Local Noether theorem for quantum lattice systems and topological invariants of gapped states
• Authors: Anton Kapustin, Nikita Sopenko
• Abstract summary: We study generalizations of the Berry phase for quantum lattice systems in arbitrary dimensions. A key role in these constructions is played by a certain differential graded Frechet-Lie algebra attached to any quantum lattice system.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 Frechet-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.

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