Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Matthew Corbelli

TL;DR

This work proves that enforcing a symmetry-compatible notion of superselection in 2D quantum spin systems with on-site symmetry $G$ yields a nontrivial sector structure even for phases without long-range entanglement. By developing a $G$-covariant framework and a compatible $\widehat{G}$-grading via the Peter–Weyl decomposition, the authors show that irreducible $G$-covariant representations satisfying the $G$-equivariant superselection criterion relative to a $G$-invariant product reference are classified by the Pontryagin dual $\widehat{G}$. A refined $G\times G$-grading is introduced to handle region-splitting across cones, ensuring intertwiners align with symmetry constraints. Consequently, for a pure $G$-invariant product state, the $G$-equivariant superselection sectors are in one-to-one correspondence with $\widehat{G}$, revealing symmetry-imposed structure where none would be present without symmetry. The results point toward extensions to non-product references and the non-abelian case, where sector labels may correspond to higher-dimensional representations of $G$.

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.

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

TL;DR

This work proves that enforcing a symmetry-compatible notion of superselection in 2D quantum spin systems with on-site symmetry yields a nontrivial sector structure even for phases without long-range entanglement. By developing a -covariant framework and a compatible -grading via the Peter–Weyl decomposition, the authors show that irreducible -covariant representations satisfying the -equivariant superselection criterion relative to a -invariant product reference are classified by the Pontryagin dual . A refined -grading is introduced to handle region-splitting across cones, ensuring intertwiners align with symmetry constraints. Consequently, for a pure -invariant product state, the -equivariant superselection sectors are in one-to-one correspondence with , revealing symmetry-imposed structure where none would be present without symmetry. The results point toward extensions to non-product references and the non-abelian case, where sector labels may correspond to higher-dimensional representations of .

Abstract

We study superselection sectors in two-dimensional quantum spin systems with an on-site action of a compact abelian group . 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 -action. Under this stricter notion of equivalence, the sector structure for a -equivariant product representation becomes nontrivial: the -equivariant superselection sectors are classified by elements of the Pontryagin dual . This shows that even in phases without long-range entanglement, imposing symmetry compatibility can lead to nontrivial sector structure.

Paper Structure

This paper contains 7 sections, 17 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. G-covariant representations
  4. G-hat grading
  5. G-hat grading for G-covariant representations
  6. Refining the G-hat-grading
  7. Classification of Superselection sectors with respect to a Product Representation

Key Result

Lemma 2.3

Let $\omega$ be a $G$-invariant state, and let $(\mathcal{H}_\omega, \pi_\omega, \Omega_\omega)$ be a GNS representation of $\omega$. Then there exists a continuous group homomorphism such that for all $g \in G$,

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 3.1
  • ...and 34 more