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The Cubic CFT can be understood as the O(3) invariant CFT perturbed by aslightly relevant operator. In this paper, we use conformal perturbation theorytogether with the conformal data of the O(3) vector model to compute theanomalous dimension of scalar bilinear operators of the Cubic CFT. When the$Z_2$ symmetry that flips the signs of $\phi_i$ is gauged, the Cubic modeldescribes a certain phase transition of a quantum dimer model. The scalarbilinear operators are the order parameters of this phase transition. Based onthe conformal data of the O(3) CFT, we determine the correction to the criticalexponent as $\eta_{*}^{Cubic}-\eta_{*}^{O(3)}\approx -0.0215(49)$. The O(3)data is obtained using the numerical conformal bootstrap method to study allfour-point correlators involving the four operators: $v=\phi_i$, $s=\sum_i\phi_i\phi_i$ 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 finda good match with this prediction up to isospin $j=6$ for spin 0 and 2 andmeasured the coefficients $c_{3/2}$ and $c_{1/2}$.

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