From O(3) to Cubic CFT: Conformal Perturbation and the Large Charge Sector

Abstract

The Cubic CFT can be understood as the O(3) invariant CFT perturbed by a slightly relevant operator. In this paper, we use conformal perturbation theory together with the conformal data of the O(3) vector model to compute the anomalous dimension of scalar bilinear operators of the Cubic CFT. When the $Z_2$ symmetry that flips the signs of ${\varphi }_i$ is gauged, the Cubic model describes a certain phase transition of a quantum dimer model. The scalar bilinear operators are the order parameters of this phase transition. Based on the conformal data of the O(3) CFT, we determine the correction to the critical exponent as ${\eta }_{\ast }^{Cubic}-{\eta }_{\ast }^{O(3)}\approx -0.0215(49)$. The O(3) data is obtained using the numerical conformal bootstrap method to study all four-point correlators involving the four operators: $v={\varphi }_i$, $s={\sum }_i{\varphi }_i{\varphi }_i$ and the leading scalar operators with O(3) isospin $j=2$ and 4. According to large charge effective theory, the leading operator with charge $Q$ has scaling dimension ${\Delta }_Q=c_{3/2}{Q}^{3/2}+c_{1/2}{Q}^{1/2}$. We find a good match with this prediction up to isospin $j=6$ for spin 0 and 2 and measured the coefficients $c_{3/2}$ and $c_{1/2}$.

15 pages, 1 figure