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., ${\mathbb{Z}}_n$) 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 ${\mathrm{\Delta }}_1>1.08$. We also obtain the similar necessary conditions for ${\mathbb{Z}}_3$ symmetry with ${\mathrm{\Delta }}_1>0.580$ and ${\mathbb{Z}}_4$ symmetry with ${\mathrm{\Delta }}_1>0.504$ 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.

Figure 1

The upper bound on the scaling dimension ${\mathrm{\Delta }}_2^c$ of the lowest-dimensional charge-two scalar operator appearing in the $O_1\times O_1$ OPE as a function of ${\mathrm{\Delta }}_1$. 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 the $O_1\times O_2$ OPE as a function of ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$. The jump in the bounds appears as soon as they touch the value 3. Note that $1.08<{\mathrm{\Delta }}_2<{\mathrm{\Delta }}_2^c({\mathrm{\Delta }}_1)$ must hold from the assumption that all the charge-four operators are irrelevant. See Fig. 1.