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

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.