Necessary Condition for Emergent Symmetry from the Conformal Bootstrap

We use the conformal bootstrap program to derive the necessary conditions for emergent symmetry enhancement from discrete symmetry (e. g. , Z₍) to continuous symmetry e. g. , U (1) under the renormalization group flow. In three dimensions, in order for Z₂ 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 Δ₁>1. 08. We also obtain the similar necessary conditions for Z₃ symmetry with Δ₁>0. 580 and Z₄ symmetry with Δ₁>0. 504 from the simultaneous conformal bootstrap analysis of multiple four-point functions. As applications, we show that our necessary conditions impose severe constraints on the nature of the chiral phase transition in QCD, the deconfinement criticality in Néel valence bond solid transitions, and anisotropic deformations in critical O (n) models. We prove that some fixed points proposed in the literature are unstable under the perturbation that cannot be forbidden by the discrete symmetry. In these situations, the second-order phase transition with enhanced symmetry cannot happen.