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Title:
A local Noether theorem for quantum lattice systems and topological invariants of gapped states
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Abstract:
In field theory, conserved currents have well-known ambiguities. Thanks to the Poincare lemma, these ambiguities are physically harmless. Similar issues arise for lattice systems, but have not been explored previously. I will explain some general results which both ensure the existence of local currents on a lattice and describe the corresponding ambiguities. A starring role in this problem is played by a certain 1-shifted differential graded Lie algebra attached to a quantum lattice system. A similar 1-shifted DG Lie algebra can also be attached to any gapped state of a quantum lattice system. I explain how to use this algebraic structure to extract a topological invariant out of a gaped state of a 2d lattice system invariant under a U(1) symmetry. This invariant is the zero-temperature Hall conductance. This is joint work with Nikita Sopenko.
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