Conformal Bootstrap Dashing Hopes of Emergent Symmetry

TL;DR

This work uses the 3D conformal bootstrap to derive nonperturbative, universal necessary conditions for emergent symmetry enhancement from discrete to continuous groups. By analyzing crossing symmetry of multiple four-point functions and solving a semidefinite program, the authors obtain explicit thresholds such as $Δ_1>1.08$ for Z$_2$, $Δ_1>0.580$ for Z$_3$, and $Δ_1>0.504$ for Z$_4$, which constrain or dash hopes of emergent $U(1)$ symmetry in various critical systems. Applying these bounds to the chiral phase transition in QCD, deconfinement criticality in Néel–VBS transitions, and anisotropic deformations of $O(n)$ models reveals cases where proposed emergent symmetry scenarios are disfavored and others where the outcomes depend sensitively on microscopic details. The results provide a nonperturbative benchmark that informs debates on critical phenomena and lattice realizations, guiding future studies on emergent symmetry in quantum and classical critical systems.

Abstract

We use the conformal bootstrap program to derive necessary conditions for emergent symmetry enhancement from discrete symmetry (e.g. $\mathbb{Z}_n$) to continuous symmetry (e.g. $U(1)$) under the renormalization group flow. In three dimensions, in order for $\mathbb{Z}_2$ symmetry to be enhanced to $U(1)$ symmetry, the conformal bootstrap program predicts that the scaling dimension of the order parameter field at the infrared conformal fixed point must satisfy $Δ_1 > 1.08$. We also obtain the similar conditions for $\mathbb{Z}_3$ symmetry with $Δ_{1} > 0.580$ and $\mathbb{Z}_4$ symmetry with $Δ_1 > 0.504$ from the simultaneous conformal bootstrap analysis of multiple four-point functions. Our necessary conditions impose severe constraints on many controversial physics such as the chiral phase transition in QCD, the deconfinement criticality in Néel-VBS transitions and anisotropic deformations in critical $O(n)$ models. In some cases, we find that the conformal bootstrap program dashes hopes of emergent symmetry enhancement proposed in the literature.

Figures (6)

• Figure 1: The upper bound on the scaling dimension $\Delta^c_2$ of the lowest dimensional charge two scalar operator appearing in $O_1 \times O_1$ OPE as a function of $\Delta _1$. The same bound applies to $O_2\times O_2 \sim O_4$.
• Figure 2: Upper Bounds on the scaling dimension of the lowest dimensional charge three scalar operator appearing in $O_1 \times O_2$ OPE as a function of $\Delta _1$ and $\Delta _2$. The jump in the bounds appears as soon as they touch the value 3. Note that $1.08<\Delta_2 < \Delta^c_2 (\Delta_1)$ must hold from the assumption that all the charge four operators are irrelevant and the bound in Fig.\ref{['fig:charge_2']}.
• Figure 3: Bounds on the scaling dimensions of the second-lowest neutral scalar operator as a function of $\Delta_0$.
• Figure 4: The two-dimensional projection of Fig.\ref{['fig:charge_3']} in the range $\Delta_1 \le 0.535$.
• Figure 5: The two-dimensional projection of Fig.\ref{['fig:charge_3']} in the range $0.540 \le \Delta_1 \le 0.565$.