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arXiv:1907.01829 · Physical Review Letters

Convergence of Nonperturbative Approximations to the Renormalization Group

Ivan Balog, Hugues Chaté, Bertrand Delamotte, Maroje Marohnić, Nicolás Wschebor

Published Dec 13, 2019• 133 citations
arXivHTMLPDF

Abstract

We provide analytical arguments showing that the "nonperturbative" approximation scheme to Wilson's renormalization group known as the derivative expansion has a finite radius of convergence. We also provide guidelines for choosing the regulator function at the heart of the procedure and propose empirical rules for selecting an optimal one, without prior knowledge of the problem at stake. Using the Ising model in three dimensions as a testing ground and the derivative expansion at order six, we find fast convergence of critical exponents to their exact values, irrespective of the well-behaved regulator used, in full agreement with our general arguments. We hope these findings will put an end to disputes regarding this type of nonperturbative methods.

DOI
10.1103/physrevlett.123.240604

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With a view to study the convergence properties of the derivative expansion of the exact renormalization group (RG) equation, I explicitly study the leading and next-to-leading orders of this expansion applied to the Wilson–Polchinski equation in the case of the N -vector model with the symmetry O ( N ). As a test, the critical exponents η and ν as well as the subcritical exponent ω (and higher ones) are estimated in three dimensions for values of N ranging from 1 to 20. I compare the results with the corresponding estimates obtained in preceding studies or treatments of other O ( N ) exact RG equations at second order. The possibility of varying N allows the derivative expansion method to be better valued. The values obtained from the resummation of high orders of perturbative field theory are used as standards to illustrate the eventual convergence in each case. Particular attention is drawn to the preservation (or not) of the reparameterization invariance.
The search for controlled approximations to study strongly coupled systems remains a very general open problem. Wilson's renormalization group has shown to be an ideal framework to implement approximations going beyond perturbation theory. In particular, the most employed approximation scheme in this context, the derivative expansion, was recently shown to converge and yield accurate and very precise results. However, this convergence strongly depends on the shape of the employed regulator. In this paper we clarify the reason for this dependence and justify, simultaneously, the most commonly employed procedure to fix this dependence, the principle of minimal sensitivity.
We compute the critical exponents ν, η and ω of O(N) models for various values of N by implementing the derivative expansion of the nonperturbative renormalization group up to next-to-next-to-leading order [usually denoted O(∂^{4})]. We analyze the behavior of this approximation scheme at successive orders and observe an apparent convergence with a small parameter, typically between 1/9 and 1/4, compatible with previous studies in the Ising case. This allows us to give well-grounded error bars. We obtain a determination of critical exponents with a precision which is similar or better than those obtained by most field-theoretical techniques. We also reach a better precision than Monte Carlo simulations in some physically relevant situations. In the O(2) case, where there is a long-standing controversy between Monte Carlo estimates and experiments for the specific heat exponent α, our results are compatible with those of Monte Carlo but clearly exclude experimental values.