The Operator Product Expansion for Radial Lattice Quantization of 3D $\phi^4$ Theory

Abstract

At its critical point, the three-dimensional lattice Ising model is described by a conformal field theory (CFT), the 3d Ising CFT. Instead of carrying out simulations on Euclidean lattices, we use the Quantum Finite Elements method to implement radially quantized critical $\phi^4$ theory on simplicial lattices approaching $\mathbb{R} \times S^2$. Computing the four-point function of identical scalars, we demonstrate the power of radial quantization by the accurate determination of the scaling dimensions $\Delta_{\epsilon}$ and $\Delta_{T}$ as well as ratios of the operator product expansion (OPE) coefficients $f_{\sigma \sigma \epsilon}$ and $f_{\sigma \sigma T}$ of the first spin-0 and spin-2 primary operators $\epsilon$ and $T$ of the 3d Ising CFT.