Critical Probability Distributions of the order parameter at two loops I: Ising universality class

There exists an entire family of universal PDFs of the magnetization mode of the three dimensional Ising model parameterized by $ζ= \lim_{L,ξ_{\infty}}L/ξ_{\infty}$ which is the ratio of the system size $L$ to the bulk correlation length $ξ_{\infty}$ with both the thermodynamic limit and the critical limit being taken simultaneously at fixed $ζ$. Recently, the probability distribution functions (PDFs) of the magnetization mode of the three-dimensional Ising model has been computed at one-loop in the $ε=4-d$ expansion [arXiv preprint arXiv:2407.12603 (2024)]. We show how these PDFs or, equivalently, the rate functions which are their logarithm, can be systematically computed at second order of the perturbative expansion. We compute the whole family of universal rate-functions and show that their agreement with the Monte Carlo data improves significantly at this order when compared to their one-loop counterpart.

💡 Research Summary

This paper presents a systematic framework for computing the universal Probability Distribution Functions (PDFs) of the magnetization order parameter in the three-dimensional Ising universality class, extending the calculation to the second order (two-loop) in the perturbative ε = 4-d expansion.

The core objective is to calculate the family of PDFs for the spatially averaged magnetization, which emerges when both the thermodynamic limit (system size L → ∞) and the critical limit (bulk correlation length ξ∞ → ∞) are taken simultaneously while keeping their ratio ζ = L/ξ∞ fixed. This parameter ζ indexes a continuous family of universal distributions, a feature stemming from the breakdown of the Central Limit Theorem in strongly correlated critical systems.

The authors employ a field-theoretic approach based on the φ⁴ theory. The key technical maneuver is to compute the PDF by introducing an auxiliary, infinitely sharp quadratic constraint. Specifically, the delta function δ(ŝ - s) enforcing a fixed magnetization is replaced by an exponential term (M²/2)∫(φ - s)², and the desired PDF is obtained from the effective potential Γ_M