Critical probability distributions of the order parameter from the functional renormalization group

TL;DR

This paper demonstrates how the functional renormalization group can be used to calculate the probability distribution functions of the order parameter in the critical three-dimensional Ising model, providing universal scaling functions that match simulations.

Contribution

It introduces a novel application of the FRG to compute the full probability distribution of the order parameter at criticality, including universal scaling functions.

Findings

Accurately computed probability distribution functions using FRG.

Universal scaling functions match numerical simulations.

Applicable to strongly correlated systems at criticality.

Abstract

We show that the functional renormalization group (FRG) allows for the calculation of the probability distribution function of the sum of strongly correlated random variables. On the example of the three-dimensional Ising model at criticality and using the simplest implementation of the FRG, we compute the probability distribution functions of the order parameter or equivalently its logarithm, called the rate functions in large deviations theory. We compute the entire family of universal scaling functions, obtained in the limit where the system size $L$ and the correlation length of the infinite system $\xi_{\infty}$ diverge, with the ratio $\zeta=L/\xi_{\infty}$ held fixed. It compares very accurately with numerical simulations.