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 .