Conformal Bootstrap Dashing Hopes of Emergent Symmetry

TL;DR

This paper uses the conformal bootstrap approach to establish necessary conditions for symmetry enhancement from discrete to continuous groups in three-dimensional conformal field theories, challenging some existing hypotheses.

Contribution

It derives specific bounds on scaling dimensions for symmetry enhancement, providing new constraints that question previous assumptions about emergent symmetries.

Findings

Necessary conditions for symmetry enhancement from $ ext{Z}_n$ to $U(1)$ in 3D.

Constraints on critical phenomena like QCD phase transition and Néel-VBS transitions.

Some proposed emergent symmetries are ruled out by bootstrap bounds.

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 $\Delta_1 > 1.08$. We also obtain the similar conditions for $\mathbb{Z}_3$ symmetry with $\Delta_{1} > 0.580$ and $\mathbb{Z}_4$ symmetry with $\Delta_1 > 0.504$ 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 N\'eel-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.