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An operator algebraic approach to symmetry defects and fractionalization

Kyle Kawagoe, Siddharth Vadnerkar, Daniel Wallick

TL;DR

The work provides a rigorous operator-algebraic construction of symmetry defects in 2+1D SET orders, showing that the defect sectors form a $G$-crossed braided ${\mathrm C}^*$-tensor category whose $G$-graded components encode both ordinary anyon sectors and symmetry defects. It establishes a general framework with broad applicability to general $G$-SPTs, including explicit results for the Levin-Gu ${\mathbb Z}_2$ SPT and a ${\mathbb Z}_2$-SET toric code, and shows how to construct defects via a finite-depth quantum circuit that commutes with the symmetry. For SPTs, defect categories reduce to ${\mathsf{Vec}}(G,\nu)$ with a 3-cocycle $\nu$, while for the SET toric code the category is a $G$-crossed braided fusion category with nontrivial symmetry fractionalization and tractable skeletal data. The paper also provides a practical defect-construction algorithm and demonstrates the computation of $F$- and $R$-symbols and fractionalization in key examples, bridging bulk operator-algebraic methods with lattice-model topological order. Collectively, this yields a rigorous, scalable route to classify and manipulate symmetry defects in 2D SET phases and paves the way for computing their braiding and fusion data in infinite-volume settings.

Abstract

We provide a superselection theory of symmetry defects in 2+1D symmetry enriched topological (SET) order in the infinite volume setting. For a finite symmetry group $G$ with a unitary on-site action, our formalism produces a $G$-crossed braided tensor category $G\mathsf{Sec}$. This superselection theory is a direct generalization of the usual superselection theory of anyons, and thus is consistent with this standard analysis in the trivially graded component $G\mathsf{Sec}_1$. This framework also gives us a completely rigorous understanding of symmetry fractionalization. To demonstrate the utility of our formalism, we compute $G\mathsf{Sec}$ explicitly in both short-range and long-range entangled spin systems with symmetry and recover the relevant skeletal data.

An operator algebraic approach to symmetry defects and fractionalization

TL;DR

The work provides a rigorous operator-algebraic construction of symmetry defects in 2+1D SET orders, showing that the defect sectors form a -crossed braided -tensor category whose -graded components encode both ordinary anyon sectors and symmetry defects. It establishes a general framework with broad applicability to general -SPTs, including explicit results for the Levin-Gu SPT and a -SET toric code, and shows how to construct defects via a finite-depth quantum circuit that commutes with the symmetry. For SPTs, defect categories reduce to with a 3-cocycle , while for the SET toric code the category is a -crossed braided fusion category with nontrivial symmetry fractionalization and tractable skeletal data. The paper also provides a practical defect-construction algorithm and demonstrates the computation of - and -symbols and fractionalization in key examples, bridging bulk operator-algebraic methods with lattice-model topological order. Collectively, this yields a rigorous, scalable route to classify and manipulate symmetry defects in 2D SET phases and paves the way for computing their braiding and fusion data in infinite-volume settings.

Abstract

We provide a superselection theory of symmetry defects in 2+1D symmetry enriched topological (SET) order in the infinite volume setting. For a finite symmetry group with a unitary on-site action, our formalism produces a -crossed braided tensor category . This superselection theory is a direct generalization of the usual superselection theory of anyons, and thus is consistent with this standard analysis in the trivially graded component . This framework also gives us a completely rigorous understanding of symmetry fractionalization. To demonstrate the utility of our formalism, we compute explicitly in both short-range and long-range entangled spin systems with symmetry and recover the relevant skeletal data.

Paper Structure

This paper contains 61 sections, 106 theorems, 321 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. General Assumptions
  4. Main results
  5. General SPTs
  6. Levin-Gu SPT
  7. An SET Toric Code
  8. Constructing defect automorphisms
  9. Symmetry Defects
  10. Category of $G$-defect sectors
  11. Category of homogeneous $G$-defect sectors
  12. Direct sums and subobjects of $G$-defect sectors
  13. $G$-crossed monoidal and braiding structure on $G\mathsf{Sec}$
  14. $G$-crossed monoidal structure on $G\mathsf{Sec}$
  15. $G$-crossed braided structure on $G\mathsf{Sec}$
  16. ...and 46 more sections

Key Result

Theorem 2.10

The category of defect sectors with respect to $\pi_0$ is a $G$-crossed braided ${\mathrm C}^*$-tensor category.

Figures (20)

  • Figure 1: An example symmetry action $\beta_g^S$ on the triangular lattice, with $S$ being the region colored in red. On all sites $s$ in the red region, the symmetry acts as $U^g_s$, and $\mathds{1}_s$ otherwise.
  • Figure 2: Defining the symmetry action $\beta_g^{r(\Lambda)}$ for different cones.
  • Figure 3: An example of a finite depth quantum circuit in 1 dimensions. Each block is an entangling unitary $U$ with support of 2 sites, so $N = 2$. The depth of this circuit is $D = 3$. We have $|\operatorname{supp}(A)| = 2$ and after the circuit, $|\operatorname{supp}(\alpha(A))| = 8$.
  • Figure 4:
  • Figure 5: Heuristic of a symmetry defect and its interpretation as being implemented by a string-operator.
  • ...and 15 more figures

Theorems & Definitions (249)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4: 2410.21454
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Theorem 2.10
  • Definition 2.11
  • ...and 239 more