Symmetry-Enriched Quantum Spin Liquids in $(3+1)d$

TL;DR

This work provides a comprehensive framework to classify symmetry-enriched phases in (3+1)d by leveraging intrinsic higher-form symmetries and their interaction with ordinary 0-form symmetries. The authors encode the enrichment data in triplets (η2, ν3, ξ) constrained by a higher-group structure and analyze resulting ’t Hooft anomalies, SPT absorption, and correlations via defect junctions and correlation functions. They develop explicit constructions for symmetry defects arising from permuting non-local operators and from higher-form defects, derive general coupling rules to fixed higher-form backgrounds, and map how these couplings modify junctions and selection rules. The framework is applied to Abelian finite-group gauge theories, adjoint QCD4, and U(1) theories with time-reversal, yielding numerous non-anomalous SET phases and revealing tensions with proposed dualities; it also shows how gauging higher-form symmetries can realize Gu-Wen-type fermionic SPTs. Overall, the paper extends symmetry-enrichment taxonomy to (3+1)d, providing a systematic toolset to classify, analyze anomalies, and connect SETs with higher-form symmetry dynamics.

Abstract

We use the intrinsic one-form and two-form global symmetries of (3+1)$d$ bosonic field theories to classify quantum phases enriched by ordinary ($0$-form) global symmetry. Different symmetry-enriched phases correspond to different ways of coupling the theory to the background gauge field of the ordinary symmetry. The input of the classification is the higher-form symmetries and a permutation action of the $0$-form symmetry on the lines and surfaces of the theory. From these data we classify the couplings to the background gauge field by the 0-form symmetry defects constructed from the higher-form symmetry defects. For trivial two-form symmetry the classification coincides with the classification for symmetry fractionalizations in $(2+1)d$. We also provide a systematic method to obtain the symmetry protected topological phases that can be absorbed by the coupling, and we give the relative 't Hooft anomaly for different couplings. We discuss several examples including the gapless pure $U(1)$ gauge theory and the gapped Abelian finite group gauge theory. As an application, we discover a tension with a conjectured duality in $(3+1)d$ for $SU(2)$ gauge theory with two adjoint Weyl fermions.

Figures (7)

• Figure 1: When a line (surface) operator of type $a$ crosses a codimension-one symmetry defect of type ${\bf g}\in G$, it leaves as a line (surface) operator of a different type $a'$. If the line (surface) operator generates a higher-form global symmetry, this represents the action of 0-form symmetry $G$ on the higher-form symmetries by automorphisms $a'=\rho_{\bf g}(a)$.
• Figure 2: Sliding the 0-form symmetry defect (\ref{['eqn:symmetrydefectnotpermute']}) (denoted by the solid black lines) across the junction of three surface operators (denoted by the red dashed lines) produces an additional symmetry line defect at the junction (denoted by the blue dot). The 0-form symmetry defect acts on the surface operators on the lower side of the solid black line in the figure as in (\ref{['eqn:transformthreegroupdefect']}).
• Figure 3: The junction of two $\mathbb{Z}_2$ 0-form symmetry defects (\ref{['eqn:exception']}) (indicated by the black solid line) fusing into the trivial defect (indicated by the dashed line). The 0-form symmetry defects meet at the codimension-two red surface. The black arrows indicate their orientations. Intersecting the junction with the green surface $\oint v_2$ (that fills the plane in the figure) emits an additional symmetry line defect $\oint u$ (indicated by the blue line) on one of the two branches.
• Figure 4: The red operator pierces the codimension-one symmetry defect $L\times L'$ where $L$ supports the symmetry defect not local with respect to the red operator. Then by shrinking the circumference of the cylinder we find the red operator leaves the codimension-one symmetry defect with (a power of) additional symmetry defect $L'$ attached to it.
• Figure 5: A junction (in red) where three codimension-one 0-form symmetry defects of type ${\bf g}_1,{\bf g}_2,{\bf g}_1{\bf g}_2\in G$ meet in codimension two, with one additional dimension suppressed in the figure. The junction can be modified by inserting codimension-two symmetry defects of one-form symmetry $\eta({\bf g}_1,{\bf g}_2)\in{\cal A}$.