Adiabatic pumps in quantum spin systems

The Thouless pump is a one-parameter cycle of 1-dimensional gapped quantum systems with U(1) symmetry, which is classified by integers. In this talk, I introduce a generalization of the Thouless pump to quantum spin systems in any dimension with any finite group onsite symmetry. I show a simple model with Z_2 onsite symmetry, and how it is nontrivial via boundary degrees of freedom. Using the framework of the injective matrix product state, one can construct the topological invariant in a way similar to the Berry phase. If time allows, I will briefly introduce a group cohomology model by Roy and Harper for generic space dimensions and discuss its properties.