A Classification of G-Charge Thouless Pumps in 1D Invertible States

Recently, a theory has been proposed that classifies cyclic processes of symmetry protected topological (SPT) quantum states. For the case of spin chains, i.e. one-dimensional bosonic SPT’s, this theory implies that cyclic processes are classified by zero-dimensional SPT’s. This is often described as a generalization of Thouless pumps, with the original Thouless pump corresponding to the case where the symmetry group is U(1) and pumps are classified by an integer that corresponds to the charge pumped per cycle. In this paper, we review this one-dimensional theory in an explicit and rigorous setting and we provide a proof for the completeness of the proposed classification for compact symmetry groups G.