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

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.

💡 Research Summary

The paper extends the two‑loop calculation of critical order‑parameter probability distribution functions (PDFs) from the Ising (n = 1) case to the full O(n) universality class. Starting from the definition of the normalized total spin (\hat s_i = L^{-d}\int d^dx,\hat\phi_i(x)), the authors replace the delta‑function constraint by a sharply peaked Gaussian, which allows the PDF to be expressed as a functional integral with a modified Hamiltonian (H_{M,s}