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arXiv:1510.05709 · Physical review. E

Ordered phase of the O(N) model within the nonperturbative renormalization group

Marcela Peláez, Nicolás Wschebor

Published Oct 26, 2016• 18 citations
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Abstract

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.

DOI
10.1103/physreve.94.042136

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We study the qqq-state Potts model for qqq and the space dimension ddd arbitrary real numbers using the derivative expansion of the nonperturbative renormalization group at its leading order, the local potential approximation (LPA and ${\mathrm{LPA}}^{\ensuremath{'}}$). We determine the curve qc(d){q}_{c}(d)qc​(d) separating the first [q>qc(d)][q>{q}_{c}(d)][q>qc​(d)] and second [q<qc(d)][q<{q}_{c}(d)][q<qc​(d)] -order phase transition regions for $2.8<d\ensuremath{\le}4$. At small $\ensuremath{\epsilon}=4\ensuremath{-}d$ and $\ensuremath{\delta}=q\ensuremath{-}2$ the calculation is performed in a double expansion in these parameters, and we find ${q}_{c}(d)=2+a{\ensuremath{\epsilon}}^{2}$ with $a\ensuremath{\simeq}0.1$. For finite values of $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$, we obtain this curve by integrating the LPA and ${\mathrm{LPA}}^{\ensuremath{'}}$ flow equations. We find that qc(d=3)=2.11(7){q}_{c}(d=3)=2.11(7)qc​(d=3)=2.11(7), which confirms that the transition is of first order in d=3d=3d=3 for the three-state Potts model.
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 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 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.
The behavior of many critical phenomena at large distances is expected to be invariant under the full conformal group, rather than only isometries and scale transformations. When studying critical phenomena, approximations are often required, and the framework of the nonperturbative, or functional, renormalization group is no exception. The derivative expansion is one of the most popular approximation schemes within this framework, due to its great performance on multiple systems, as evidenced in the past few decades. Nevertheless, it has the downside of breaking conformal symmetry at a finite order. This breaking is not observed at the leading order of the expansion, denoted the local potential approximation, and it only appears once one considers, at least, the next-to-leading order of the derivative expansion [O(∂^{2})] when including composite operators. In this work, we study the constraints arising from conformal symmetry for the O(N) models using the derivative expansion at order O(∂^{2}). We explore various values of N and minimize the breaking of conformal symmetry to fix the nonphysical parameters of the approximation procedure. We compare our prediction for the critical exponents with those coming from a more usual procedure, known as the principle of minimal sensitivity.