Scalar CFTs and Their Large N Limits

We study scalar conformal field theories whose large $N$ spectrum is fixed by the operator dimensions of either Ising model or Lee-Yang edge singularity. Using numerical bootstrap to study CFTs with $S_N\otimes Z_2$ symmetry, we find a series of kinks whose locations approach $({\mathrm{\Delta }}_{\sigma }^{\text{Ising}},{\mathrm{\Delta }}_{ϵ}^{\text{Ising}})$ at $N\to \mathrm{\infty }$. Setting $N=4$, we study the cubic anisotropic fixed point with three spin components. As byproducts of our numerical bootstrap work, we discover another series of kinks whose identification with previous known CFTs remains a mystery. We also show that “minimal models” of ${\mathcal{W}}_3$ algebra saturate the numerical bootstrap bounds of CFTs with $S_3$ symmetry.