Conformal Bootstrap Dashing Hopes of Emergent Symmetry

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.