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

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.

arXiv commentv2: exposition improved, minor errors corrected. Version to appear in Journal of Mathematical Physics. v3: missing pictures added

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• Higher Berry Curvature from the Wave function II: Locally Parameterized States Beyond One Dimension
• Higher Berry Curvature from the Wave Function I: Schmidt Decomposition and Matrix Product States
• Charting the space of ground states with tensor networks
• Remarks on Berry Connection in QFT, Anomalies, and Applications
• Higher Berry curvature from matrix product states
• Exact Chiral Symmetries of 3+1D Hamiltonian Lattice Fermions
• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter
• The index of a pair of pure states and the interacting integer quantum Hall effect