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Journal of Physics A Mathematical and General

Critical indices in three dimensions

J -L Colot, J.A.C. Loodts, R. Brout

Published Apr 1, 1975• 9 citations
Publisher

Abstract

A modification of Wilson's epsilon -expansion scaling procedure is adapted to direct calculation in three dimensions. The parameter log r is replaced by (r - epsilon 2/-1) where r=t gamma (t=temperature) and all calculations are carried out directly at epsilon =1. Several conclusions are drawn: (i) Numerical agreement with the best known data for gamma and eta is excellent (1% in gamma and 10% in eta ) whereas the corresponding values are far less good in the epsilon -expansion where eta is off by a factor of more than two. (ii) The series appears to converge very well when corrections to the leading term are summed two by two. (iii) Extrapolations of the early orders of the epsilon -expansion to epsilon =1 are unjustified.

DOI
10.1088/0305-4470/8/4/021

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A modification of Wilson's epsilon -expansion scaling procedure is adapted to direct calculation in three dimensions. The parameter log r is replaced by (r - epsilon 2/-1) where r=t gamma (t=temperature) and all calculations are carried out directly at epsilon =1. Several conclusions are drawn: (i) Numerical agreement with the best known data for gamma and eta is excellent (1% in gamma and 10% in eta ) whereas the corresponding values are far less good in the epsilon -expansion where eta is off by a factor of more than two. (ii) The series appears to converge very well when corrections to the leading term are summed two by two. (iii) Extrapolations of the early orders of the epsilon -expansion to epsilon =1 are unjustified.
It is useful to compare the results obtained so far with other approaches to the critical exponents. One is a similar field-theoretic approach based on perturbation expansions of φ 4-theories. But instead of working in D = 4−ε dimensions and continuing the results to ε = 1 to obtain critical exponents in the physical dimension D = 3, perturbation expansions can be derived directly in three dimensions, and studied in the regime of small masses. Fundamentally, this approach is less satisfactory than the previous one since there is no small parameter analogous to ε which permits us to prove the existence of a scaling limit at least for small ε. Another disadvantage is that it is impossible to calculate the expansion coefficients analytically, except for one- and two loop-diagrams. In one respect the three-dimensional approach has, however, an advantage: the power series have long been available up to six loops for ω, η, and ν, and there exist recent results up to seven loops for the critical exponents η and ν. Although numerical expansions are not as esthetic as the exact expansions derived in this book, their increased order leads to a higher accuracy in the critical exponents, as we shall see below in Section 20.4. Another approach is based on high-temperature expansions for lattice models of the O(N)symmetric classical Heisenberg model with the energy of Eq. (1.52). The critical exponents will be derived from such expansions in Section 20.8. Finally we take advantage of the fact that the O(N)-symmetric classical Heisenberg model is exactly solvable in the limit of large N, where it reduces to the spherical model [recall the remark after Eq. (1.69)]. It is therefore possible to expand the φ 4-theory around this limit in powers of 1/N. The results will be compared with those from the perturbation expansions in
This is the third article in a sequence in which we reexamine the calculation of the critical exponents of the N-vector model from field theory. In the first article we have applied summation methods to the ε-expansion for which new terms have been calculated (up to order ε5). In the second article we have incorporated in the summation method, in the case of Ising-like systems (N = 1), the knowledge of the exact values of exponents in d = 2 dimensions. Since exact values are now also known in the case of polymers (N = 0), we apply here the same method to this case. Moreover, since for the exponent η for N = 1 the results coming from the ε-expansion and perturbation series at fixed dimension are slightly different, we have recalculated the exponents in the latter case using the variant of the summation method developed for the ε-expansion. The results do not change significantly showing both the reliability and the limits of the method.
The critical exponent $\ensuremath{\eta}$ for three-dimensional systems with an nnn-vector order parameter is evaluated in the framework of the pseudo-$\ensuremath{\epsilon}$ expansion approach. The pseudo-$\ensuremath{\epsilon}$ expansion ($\ensuremath{\tau}$ series) for $\ensuremath{\eta}$ found up to the ${\ensuremath{\tau}}^{7}$ term for nnn = 0, 1, 2, 3 and within the ${\ensuremath{\tau}}^{6}$ order for general nnn is shown to have a structure that is rather favorable for getting numerical estimates. The use of Pad\'e approximants and direct summation of the $\ensuremath{\tau}$ series result in iteration procedures rapidly converging to the asymptotic values that are very close to the most reliable numerical estimates of $\ensuremath{\eta}$ known today. The origin of such an efficiency is discussed and shown to lie in the general properties of the pseudo-$\ensuremath{\epsilon}$ expansion machinery interfering with some peculiarities of the renormalization group expansion of $\ensuremath{\eta}$.
New series expansion data for the three-dimensional Ising model are analysed. A special study of the diamond and face-centred cubic lattices is made. Below the critical temperature convergence is found to be slow. It is concluded that all available data are consistent with beta = lambda 5/16 (magnetization) and not inconsistent with the 'scaling' value gamma '=5/4 (susceptibility) and that with the data and methods available at present it is not possible to draw more precise conclusions.
In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.