On Symmetry-Compatible Superselection Structures for Product States in 2D Quantum Spin Systems

TL;DR

This paper investigates how symmetry considerations in 2D quantum spin systems influence superselection sectors, revealing that symmetry-compatible conditions can induce nontrivial sector structures even in simple, non-entangled states.

Contribution

It introduces a symmetry-compatible refinement of superselection sector analysis, demonstrating that such sectors are classified by the Pontryagin dual of the symmetry group in product states.

Findings

Superselection sectors are trivial without symmetry considerations.

Imposing symmetry compatibility yields nontrivial sectors classified by  G.

Symmetry constraints can reveal hidden structures in simple phases.

Abstract

We study superselection sectors in two-dimensional quantum spin systems with an on-site action of a compact abelian group $G$. Naaijkens and Ogata (2022) arXiv:2102.07707 showed that for states quasi-equivalent to a product state, the superselection structure is trivial, reflecting the absence of long-range entanglement. We consider a symmetry-compatible refinement of this setting, in which both the superselection criterion and the notion of equivalence between representations are required to respect the $G$-action. Under this stricter notion of equivalence, the sector structure for a $G$-equivariant product representation becomes nontrivial: the $G$-equivariant superselection sectors are classified by elements of the Pontryagin dual $\widehat{G}$. This shows that even in phases without long-range entanglement, imposing symmetry compatibility can lead to nontrivial sector structure.