Conformal Bootstrap Dashing Hopes of Emergent Symmetry

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 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.

Comments: 4+5 pages, v2: improved bound, reference added, appendix added