Topological invariants of adiabatic cycles 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 and is physically the pumped charge per cycle. In this talk, we introduce a generalization of the Thouless pump to quantum spin systems in any dimension with any onsite finite group symmetry. We show a simple model with Z2 onsite symmetry and how it is nontrivial via boundary degrees of freedom. Using the injective matrix product state representation, one can construct the topological invariant of adiabatic cycles, similar to the Berry phase. For generic space dimensions, the U(1) phase winding of the group cycle of ground states serves as the topological invariant of adiabatic cycles.