Critical Probability Distributions of the order parameter at two loops II: $O(n)$ universality class

TL;DR

The paper develops a systematic two-loop perturbative framework in the $\epsilon=4-d$ expansion to compute the critical PDFs of the order-parameter for the $O(n)$ model, revealing a family of PDFs indexed by $\zeta = L/\xi_\infty$ that captures finite-size scaling effects. The authors formulate the PDFs via a constrained path integral and extract a rate function $I_{\zeta,n}(\tilde{s})$, decomposing it into infinite-volume and finite-size components with explicit $\epsilon$-expansions up to $\epsilon^2$. They show that the rate function exhibits universal large-field behavior $I_{\zeta,n}(x) \sim x^{\delta+1} - \frac{n(\delta-1)}{2}\log x$ with $\delta=3+\epsilon+\frac{n^2+14n+60}{2(n+8)^2}\epsilon^2$, and they compare with Monte Carlo data, achieving notable improvement over one-loop results, though large-field regimes require RG enhancement. The work paves the way for extensions to different boundary conditions, the $1/n$ expansion, and potential analysis at $d=4$, offering a versatile perturbative route to universal finite-size PDFs in $O(n)$ universality.

Abstract

We show how to compute the probability distributions of the order parameter of the $O(n)$ model at two-loop order of perturbation theory generalizing the methods developed for computing the same in case of the Ising model \cite{Sahu:2025bkp}. We show that even for the $O(n)$ model, there exists not one but a family of these probability distribution functions indexed by $ζ$ which is the ratio of system size $L$ to the bulk correlation length $ξ_{\infty}$. We also compare these PDFs to the Monte-Carlo simulations and the existing FRG results for the $O(2)$ and $O(3)$ models.

Figures (7)

• Figure 1: Comparison of the rate function, $I_{\zeta=0,n=2}(x)$ obtained from Monte Carlo (MC) simulations (red) and from one-loop and two-loop order (cyan) of perturbation theory for the $O(2)$ model (Classical XY model).
• Figure 2: Comparison of $I_{\zeta=0.5,n=2}(x)$ obtained from Monte Carlo (MC) simulations (red) and from two-loop order (cyan) of perturbation theory.
• Figure 3: Comparison of $I_{\zeta=1,n=2}(x)$ obtained from Monte Carlo (MC) simulations (red) and from two-loop order (cyan) of perturbation theory.
• Figure 4: Comparison of $I_{\zeta=1.5,n=2}(x)$ obtained from Monte Carlo (MC) simulations (red) and from two-loop order (cyan) of perturbation theory.
• Figure 5: Comparison of $I_{\zeta=0,n=3}(x)$ obtained from Monte Carlo (MC) simulations (red) and from one-loop (blue) and two-loop order (cyan) of perturbation theory for the $O(3)$ model (Heisenberg model).