Generalized symmetry enriched criticality in (3+1)d

TL;DR

This work identifies two families of continuous phase transitions between generalized-symmetry phases in (3+1)d PSU(N) adjoint QCD with $N_f$ odd Majorana fermions: a SET-to-SET transition between $\mathbb{Z}_N$ and $\mathbb{Z}_{N/2}$ topological orders, and a SET-to-non-invertible SSB transition between a topologically ordered phase and a phase that spontaneously breaks a non-invertible time-reversal symmetry. The analysis combines PSU$(N)$ gauging, one-form symmetry structure, and non-invertible symmetry operators to derive the bulk topological orders, symmetry fractionalization patterns, and anomaly constraints; explicit lattice constructions and twisted BF/TQFT descriptions verify the continuous transitions under suitable $N$, $N_f$, and mass deformations. A key result is the string-tension critical exponent $\mu=4N_f/11$ in the SET transition and the Landau-type exponents for the non-invertible symmetry order parameter, indicating rich, beyond-Landau universality. The findings illuminate how generalized symmetries orchestrate criticality in 3+1 dimensions and open avenues for lattice realizations, dualities, and boundary phenomena in SETs and non-invertible-symmetry contexts.

Abstract

We construct two classes of continuous phase transitions in 3+1 dimensions between gapped phases that break distinct generalized global symmetries. Our analysis focuses on $SU(N)/\mathbb{Z}_N$ gauge theory coupled to $N_f$ flavors of Majorana fermions in the adjoint representation. For $N$ even and sufficiently large odd $N_f$, upon imposing time-reversal symmetry and an $SO(N_f)$ flavor symmetry, the massless theory realizes a quantum critical point between a gapped phase in which a $\mathbb{Z}_N$ one-form symmetry is completely broken and a phase where it is broken to $\mathbb{Z}_2$, leading to $\mathbb{Z}_{N/2}$ topological order. We characterize the possible patterns of symmetry fractionalization in these phases and provide an explicit lattice model that exhibits the transition. The critical point has an enhanced symmetry, which includes non-invertible analogues of time-reversal symmetry. Enforcing a non-invertible time-reversal symmetry and the $SO(N_f)$ flavor symmetry, for $N$ and $N_f$ both odd, we demonstrate that this critical point can appear between a topologically ordered phase and a phase that spontaneously breaks the non-invertible time-reversal symmetry, furnishing an analogue of deconfined quantum criticality for generalized symmetries.

Figures (3)

• Figure 1: Schematic depiction of the phase diagram of $SU(N)$ adjoint QCD, Eq. \ref{['eq: su(n) adjoint qcd']}, for $N$ even and $N_f$ odd flavors of Majorana fermions, as a function of the mass $m$. For $m>0$, the theory flows to $SU(N)$ gauge theory at $\theta=0$, which is adiabatically connected to a trivial product state. For $m<0$, the state is a nontrivial fermionic SPT protected by the $\mathbb{Z}_N^{(1)}$ electric one-form symmetry and $G=SO(N_f)\times\mathbb{Z}_2^T$, which consists of a flavor symmetry and time-reversal.
• Figure 2: Schematic depiction of the phase diagram of $PSU(N)=SU(N)/\mathbb{Z}_N$ adjoint QCD, Eq. \ref{['eq: psu(n) adjoint qcd']}, for $N$ even and $N_f$ odd flavors of Majorana fermions as a function of the mass $m$. The $m>0$ phase spontaneously breaks the full $\mathbb{Z}_N^{(1)}$ magnetic one-form symmetry at low energies, giving rise to $\mathbb{Z}_N$ topological order described by BF theory at level $N$, Eq. \ref{['eq: BF theory']}. For $m<0$, the low energy physics is described by Eq. \ref{['eq: psu(N) negative mass phase']}. The $\mathbb{Z}_N^{(1)}$ magnetic one-form symmetry, is spontaneously broken to $\mathbb{Z}_2^{(1)}$, resulting in $\mathbb{Z}_{N/2}$ topological order. Additionally, the $m<0$ phase is stacked with a nontrivial fermionic SPT for the unbroken $\mathbb{Z}_2^{(1)}$ and $G=SO(N_f)\times\mathbb{Z}_2^T$ symmetries. The $G$ symmetry may be fractionalized on the point-like anyon excitations of the bulk topological order in the $m>0$ phase. These different choices of symmetry fractionalization in the $m>0$ phase are correlated with the particular $G\times \mathbb{Z}_2^{(1)}$ SPT order realized in the $m<0$ phase.
• Figure 3: Schematic phase diagram of $PSU(N)=SU(N)/\mathbb{Z}_N$ adjoint QCD with a theta term $\theta=\pi n$, Eq. \ref{['eq: non-invertible psu(n) adjoint qcd']}, for $N_f$ odd flavors of Majorana fermions as a function of the fermion mass $m$. Here, we take $N=k^2$ and $n=2k$ with $k>1$ odd. In the $m>0$ phase, described by the TQFT in Eq. \ref{['eq: non-invertible SET TQFT']}, the $\mathbb{Z}_N=\mathbb{Z}_{k^2}$ magnetic one-form symmetry is spontaneously broken to $\mathbb{Z}_k$, resulting in $\mathbb{Z}_k$ topological order. The unbroken $\mathbb{Z}_k$ one-form symmetry also has nontrivial SPT order. For $m<0$, the $\mathbb{Z}_N$ one-form symmetry is unbroken, but the non-invertible time-reversal symmetry associated with operator $\mathsf{T}_{2k}=\mathsf{D}_{2k}\mathsf{T}$ is spontaneously broken, resulting in two ground states that are distinct $\mathbb{Z}_N$ one-form SPTs with responses given by Eq. \ref{['eq: non-invertible SSB GS']}. The domain walls in this phase obey the non-invertible fusion rules in Eq. \ref{['eq: non-invertible fusion rules']}. The $SO(N_f)$ flavor symmetry remains unbroken throughout the phase diagram.