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arXiv:2509.05195 · Physical review. E

Constraint effective action and critical correlation functions at fixed magnetization

Félix Rose, Adam Rançon, Ivan Balog

Published Mar 24, 2026• 1 citation
arXivHTMLPDF

Abstract

We present an extension of the functional renormalization group (FRG) framework developed to compute critical probability distributions of the order parameter to momentum-dependent observables. Focusing on the constraint effective action at fixed magnetization for the Ising universality class, we derive its exact flow equations and solve them at the second order of the derivative expansion (DE2). We solve these flow equations numerically for two- and three-dimensional systems, extract universal rate functions and momentum-dependent correlation functions, and benchmark them against Monte Carlo simulations. In three dimensions, we recover the rate function and accurately reproduce the first few Fourier modes of the constrained correlation function and demonstrate the convergence of the method. In two dimensions, the lowest-order approximations such as local potential approximation fail, and it is required to consider at least the DE2 to describe the critical point. Our results are in qualitative agreement with the numerics. We confirm the robustness of the FRG approach for calculating both zero- and finite-momentum critical observables at fixed magnetization.

DOI
10.1103/yqjz-kdlj

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We present an extension of the functional renormalization group (FRG) framework developed to compute critical probability distributions of the order parameter to momentum-dependent observables. Focusing on the constraint effective action at fixed magnetization for the Ising universality class, we derive its exact flow equations and solve them at the second order of the derivative expansion (DE2). We solve these flow equations numerically for two- and three-dimensional systems, extract universal rate functions and momentum-dependent correlation functions, and benchmark them against Monte Carlo simulations. In three dimensions, we recover the rate function and accurately reproduce the first few Fourier modes of the constrained correlation function and demonstrate the convergence of the method. In two dimensions, the lowest-order approximations such as local potential approximation fail, and it is required to consider at least the DE2 to describe the critical point. Our results are in qualitative agreement with the numerics. We confirm the robustness of the FRG approach for calculating both zero- and finite-momentum critical observables at fixed magnetization.
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 deviation 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 ξ_{∞} diverge, with the ratio ζ=L/ξ_{∞} held fixed. It compares very accurately with numerical simulations.
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We show that the functional renormalization group (FRG) allows for the generalization of the central limit theorem to 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 LLL and the correlation length of the infinite system xiinfty\\xi_{\\infty}xiinfty​ diverge, with the ratio zeta=L/xiinfty\\zeta=L/\\xi_{\\infty}zeta=L/xiinfty​ held fixed. It compares very accurately with numerical simulations.
We present a truncation scheme of the effective-average-action approach of the nonperturbative renormalization group that allows for an accurate description of the critical regime as well as of correlation functions at finite momenta. The truncation is a natural modification of the standard derivative expansion that includes both all local correlations and two-point and four-point irreducible correlations to all orders in the derivatives. We discuss schemes for both the symmetric and the symmetry broken phase of the O(N) model and present results for D=3. All approximations are done directly in the effective average action rather than in the flow equations of irreducible vertices. The approach is numerically relatively easy to implement and yields good results for all N both for the critical exponents and for the momentum dependence of the two-point function.
Since its introduction, the Renormalization Group has been the theoretical tool of choice for understanding and describing collective behavior characterized by an emergent scale invariance as in the vicinity of a continuous phase transition. It provides a powerful conceptual framework but, exact results being scarce, the search for generic and efficient approximation schemes has been very active from the very beginning. One recent line of research starts from an exact formulation of the Renormalization Group, in the form of a functional Renormalization Group (FRG) for scale-dependent generating functionals of correlation functions, and introduces potentially {it nonperturbative} approximations through ansatzes for the scale-dependent generating functional under study. The question that we address is to what extent such generic approximation schemes are able to describe specific problems in which the long-distance behavior involves strongly nonuniform configurations with, {it e.g.}, localized excitations.An example of such an approximation scheme within the FRG is the so-called derivative expansion of the effective average action (coarse-grained Gibbs free energy in the language of magnetic systems) which amounts to truncating the functional form of the latter in powers of the external momenta or equivalently in gradients of the fields. The derivative-expansion approximation focuses on the long wavelength properties and, in terms of coarse-grained configurations of the system, works as an expansion about uniform configurations. One may therefore wonder if such a scheme is able to capture the physics associated with nonuniform configurations containing, {it e.g.}, domain walls, spin waves, or localized defects such as the kinks and anti-kinks found in the instanton analysis of the 1-dimensional Ising model.In this work we assess the ability of the derivative expansion of the FRG to quantitatively describe the long-distance physics of a model across the whole range of space dimensions {it without a priori knowledge} of the relevant real-space coarse-grained configurations. To do so we first investigate how low orders of the derivative expansion describe the approach to the lower critical dimension d→dlc of the Ising-like scalar φ4 theory which is expected to be controlled by the proliferation of localized excitations (instantons). We show that the convergence of the fixed-point effective potential describing the critical point of the model is nonuniform in the field when the dimension d approaches dlcdlc, with the emergence of a boundary layer around the minimum of the potential. At the lowest nontrivial approximation level (known as LPA’) this allows us to make analytical predictions for the value of the lower critical dimension dlcdlc and for the behavior of the critical temperature as d→dlc, which are both found in fair agreement with the known exact results. We also consider for the same Ising-like scalar φ4 model the ordered phase in the limit where d→dlc. We focus on the way the effective potential in the FRG becomes convex as the proper spatial fluctuations, which now involve nonuniform configurations with a domain wall separating two phases, are included in the calculation.
Abstract There exists an entire family of universal probability distribution functions (PDFs) of the magnetization mode of the three dimensional Ising model parameterized by ζ = lim L , ξ ∞ L / ξ ∞ which is the ratio of the system size L to the bulk correlation length ξ ∞ with both the thermodynamic limit and the critical limit being taken simultaneously at fixed ζ . Recently, the PDFs of the magnetization mode of the three-dimensional Ising model has been computed at one-loop in the ϵ = 4 − d expansion (2024 arXiv preprint arXiv:2407.12603 ). 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.