Classification of symmetry protected states of quantum spin chains for continuous symmetry groups

TL;DR

This paper extends the classification of symmetry protected states in quantum spin chains from finite groups to compact topological groups, providing a more general understanding of these states.

Contribution

It generalizes the classification of SPT states to compact topological symmetry groups and strengthens existing results for spin chains with bounded local dimensions.

Findings

Classification extended to compact topological groups

Results hold for spin chains with bounded local dimensions

Provides a rigorous framework for broader symmetry groups

Abstract

Symmetry protected states (SPT's) of quantum spin systems were studied by several authors. For one-dimensional systems (spin chains), there is an essentially complete and rigorous understanding: SPT's corresponding to finite on-site symmetry groups $G$ are classified by the second cohomology group $H^2(G,U(1))$, as established by Kapustin et al. [J. Math. Phys. (2021)]. We extend this result to the case of compact topological symmetry groups $G$. We also strengthen the existing results in the sense that our classification results holds within the class of spin chains with locally bounded on-site dimensions.