Probability distributions of the order parameter of the $O(N)$ model

TL;DR

This paper investigates the probability distribution of the order parameter in the 3D $O(N)$ model at criticality using the functional renormalisation group, deriving universal scaling functions and comparing with Monte Carlo simulations.

Contribution

The study extends the functional renormalisation group method to the $O(N)$ model, providing universal PDFs and scaling functions for different N and critical regimes.

Findings

LPA accurately describes the functional form of the PDFs.

Universal scaling functions are computed for various N and critical conditions.

Results agree well with Monte Carlo simulations after amplitude corrections.

Abstract

We study the probability distribution function (PDF) of the order parameter of the three-dimensional $O(N)$ model at criticality using the functional renormalisation group. For this purpose, we generalize the method introduced in [Balog et al., Phys. Rev. Lett. {\bf 129}, 210602 (2022)] to the $O(N)$ model. We study the large $N$ limit, as well as the cases $N=2$ and $N=3$ at the level of the Local Potential Approximation (LPA), and compare our results to Monte Carlo simulations. 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 $\zeta=L/\xi_\infty$ constant. We also generalize our results to the approach of criticality from the low-temperature phase where another infinite family of universal PDF exists. We find that the LPA describes very well the functional form of the family of PDFs, once we correct for a global amplitude of the (logarithm of the) PDF and of $\zeta$.