Two- and three-point functions at criticality: Monte Carlo simulations of the improved three-dimensional Blume-Capel model

TL;DR

This paper uses Monte Carlo simulations of the improved three-dimensional Blume-Capel model to accurately compute two- and three-point functions at criticality, providing results consistent with bootstrap methods but with different precision.

Contribution

It introduces a variance reduced estimator for N-point functions in Monte Carlo simulations of the 3D Ising universality class.

Findings

Computed operator product expansion coefficients with high precision.

Validated finite size correction behavior with lattice size.

Compared results with bootstrap method, noting differences in precision.

Abstract

We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to $L=1600$. As check also simulations of the spin-1/2 Ising model are performed. We find $f_{\sigma \sigma \epsilon} = 1.051(1)$ and $f_{\epsilon \epsilon \epsilon} =1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of $N$-point functions. Finite size corrections vanish with $L^{-\Delta_{\epsilon}}$, where $L$ is the linear size of the lattice and $\Delta_{\epsilon}$ is the scaling dimension of the leading $Z_2$-even scalar $\epsilon$.