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arXiv · 2410.19287

A mathematical theory of topological invariants of quantum lattice systems

Adam Artymowicz·Anton Kapustin·Bowen Yang
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Abstract

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.

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Adam Artymowicz, Anton Kapustin, Bowen Yang. 2024-10-25. A mathematical theory of topological invariants of quantum lattice systems. https://doi.org/10.1090/cams%2F74

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