Symmetry-enriched quantum spin liquids in (3 + 1)d

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 higherform 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.

Discrete theta angles, symmetries and anomalies

“…where δ is the coboundary operator on C * (M , N ) for spacetime M . For instance, we can couple the gauge theory to background gauge fields X 1 for 0-form symmetry G (0) and X 2 for one-form symmetry G (1) by [5,10]…”

“…where the various quotients are explained in section 3.3. The one-form symmetry and the flavor symmetry combines into a two-group symmetry with Postnikov class 5 Θ = pBock(w f 2 ) , (1.9) 4 Unlike the SU(N ), S pin(N ) gauge group discussed here, Sp(N ) gauge group only has a 2 center which does not have any nontrivial proper subgroup. Hence Sp(N ) or Sp(N )/ 2 gauge theories with matters do not have two-group symmetries with nontrivial Postnikov class, and we will not discuss them in this paper.…”

“…Hence Sp(N ) or Sp(N )/ 2 gauge theories with matters do not have two-group symmetries with nontrivial Postnikov class, and we will not discuss them in this paper. 5 For one-form symmetry G (1) and 0-form symmetry G (0) , the Postnikov class Θ ∈ H 3 (G (0) , G (1) ) expresses how the zero-form and one-form symmetries combine in terms of their backgrounds B 1 , B 2…”

“…Some models are also discussed in[4] and the references therein 2. We will also present examples where theories with different discrete theta angles differ in their non-invertible topological defects, see Section 5.4.3 For 4d theories a similar construction is discussed in[5] that studies different symmetry fractionalizations using the TQFT sector.…”

Confinement and moduli locking of Alice strings and monopoles

“…Higher-fom symmetries for the conventional Alice strings were studied in ref. [70]. In our case, it may be the case that there is a certain magnetic 2-form symmetry in our model and the breaking of this symmetry may separate the confined and deconfined phases by a phase transition.…”