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.
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For two-dimensional lattice systems, it is often assumed that gapped phases are in one-to-one correspondence with unitary topological quantum field theories. However, it is difficult to justify such an assumption, and there is no general way to determine TQFT data for a given gapped Hamiltonian. In higher dimension, the relations between gapped phases and TQFT is even less understood.
In a sense defined below.
It is believed that for invertible states, these classes are quantized. The authors of Ref. 18 argued (non-rigorously) that this is the case for families parameterized by spheres. We plan to address quantization for generic families of invertible states elsewhere.
In the sense that interactions are either finite-range or decay faster than any power of the distance.
In fact, it is a lattice. That is, every two elements of have a join and a meet in .
The existence of roughly means that any restriction of an arbitrary state-preserving Hamiltonian to a region (that in general does not preserve the state) can be modified at the boundary of the region so that the resulting Hamiltonian still preserves the state. A similar property has been considered in Ref. 39.
