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Physical review. D/Physical review. D.

Reanalysis of critical exponents for the O(N) model via a hydrodynamic approach to the functional renormalization group

Fabrizio Murgana, Adrian Koenigstein, Dirk H. Rischke

Published Dec 18, 2023• 9 citations
Publisher

Abstract

We reanalyze some critical exponents of the $O(N)$ model within the functional renormalization group (FRG) approach in the local potential approximation (LPA). We use recent advances which are based on the observation that the FRG flow equation in LPA can be put into the form of an advection-diffusion equation. This allows to employ well-tested hydrodynamical algorithms for its solution to better estimate various sources of errors. Our results complement previous results for the critical exponents obtained within the FRG approach in LPA and compare favorably with those obtained via other methods.

DOI
10.1103/physrevd.108.116016

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We reanalyze some critical exponents of the O(N)O(N)O(N) model within the functional renormalization group (FRG) approach in the local potential approximation (LPA). We use recent advances which are based on the observation that the FRG flow equation in LPA can be put into the form of an advection-diffusion equation. This allows to employ well-tested hydrodynamical algorithms for its solution to better estimate various sources of errors. Our results complement previous results for the critical exponents obtained within the FRG approach in LPA and compare favorably with those obtained via other methods.
We compute the critical exponents of the O(N) model within the Functional Renormalization Group (FRG) approach. We use recent advances which are based on the observation that the FRG flow equation can be put into the form of an advection-diffusion equation. This allows to employ well-tested hydrodynamical algorithms for its solution. In this study we work in the local potential approximation (LPA) for the effective average action and put special emphasis on estimating the various sources of errors. Our results complement previous results for the critical exponents obtained within the FRG approach in LPA. Despite the limitations imposed by restricting the discussion to the LPA, the results compare favorably with those obtained via other methods.
We analyze nonperturbative renormalization group flow equations for the ordered phase of Z_{2} and O(N) invariant scalar models. This is done within the well-known derivative expansion scheme. For its leading order [local potential approximation (LPA)], we show that not every regulator yields a smooth flow with a convex free energy and discuss for which regulators the flow becomes singular. Then we generalize the known exact solutions of smooth flows in the "internal" region of the potential and exploit these solutions to implement an improved numerical algorithm, which is much more stable than previous ones for N>1. After that, we study the flow equations at second order of the derivative expansion and analyze how and when the LPA results change. We also discuss the evolution of the field renormalization factors.
We review the Exact Renormalization Group equations of Wegner and Houghton in an approximation which permits both numerical and analytical studies of nonperturbative renormalization flows. We obtain critical exponents numerically and with the local polynomial approximation (LPA), and discuss the advantages and shortcomings of these methods, and compare our results with the literature. In particular, convergence of the LPA is discussed in some detail. We finally integrate the flows numerically and find a ccc-function which determines these flows to be gradient in this approximation.
We study the probability distribution function (PDF) of the order parameter of the three-dimensional O(N) model at criticality using the functional renormalization group. For this purpose, we generalize the method introduced by Balog et al. [Phys. Rev. Lett. 129, 210602 (2022)0031-900710.1103/PhysRevLett.129.210602] 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 ξ_{∞} diverge, with the ratio ζ=L/ξ_{∞} 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 ζ.
We employ a novel real-time formulation of the functional renormalization group (FRG) to compute universal scaling functions of the thermal diffusivity and the shear viscosity in the vicinity of the liquid-gas critical point, i.e., for the dynamic universality class of Model H from the Halperin-Hohenberg classification. We map out the universal dependence of the transport coefficients on temperature, external magnetic field, and wavenumber, and provide a detailed comparison with the Kawasaki approximation, which is here obtained from a perturbative one-loop approximation to our real-time FRG flow. In contrast to the Kawasaki approximation, the non-perturbative scaling functions from the full real-time FRG flow show a mild dependence on the thermodynamic path towards the critical point. We further compare our FRG results for the universal wavenumber and temperature dependence of the thermal diffusivity with experimental data from critical fluids.
Abstract The critical behavior of the ( n + 1 ) -states Potts model in d -dimensions is studied with functional renormalization group techniques. We devise a general method to derive β -functions for continuous values of d and n and we write the flow equation for the effective potential (LPA’) when instead n is fixed. We calculate several critical exponents, which are found to be in good agreement with Monte Carlo simulations and ϵ-expansion results available in the literature. In particular, we focus on Percolation ( n → 0 ) and Spanning Forest ( n → − 1 ) which are the only non-trivial universality classes in d = 4,5 and where our methods converge faster.