A classification of $G$-charge Thouless pumps in 1D invertible states

TL;DR

This paper reviews and rigorously proves a classification scheme for $G$-charge Thouless pumps in 1D invertible states, extending the understanding of topological cyclic processes in symmetry-protected quantum systems.

Contribution

It provides a complete and rigorous classification of $G$-charge Thouless pumps for 1D invertible states with compact symmetry groups, generalizing the original charge pump concept.

Findings

Classification is complete for compact symmetry groups $G$

Provides explicit and rigorous proof of the classification

Extends Thouless pump theory to $G$-charge in 1D invertible states

Abstract

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 $G$.