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arXiv:hep-th/0006187 · Physical review. B, Condensed matter

Nonasymptotic critical behavior from field theory atd=3. II. The ordered-phase case

C. Bagnuls, Claude Bervillier, Daniel I. Meiron, Bernie G. Nickel

Published Mar 1, 1987• 193 citations
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Abstract

We present the first detailed calculations in the ordered phase using the massive ${\ensuremath{\varphi}}^{4}$ field theory directly at d=3. It is shown that an adapted expansion allows the renormalization functions of the symmetric theory to be kept unchanged. Extending results in a previous paper [C. Bagnuls and C. Bervillier, Phys. Rev. B 32, 7209 (1985)] (noted I), we obtain, for Ising-type systems, nonasymptotic functions of temperature for the spontaneous magnetization, the susceptibility, and the specific heat along the critical isochore, which include all the quantitative universal characteristics of critical behavior in the real preasymptotic critical domain ${D}_{\mathrm{preas}}$. All universal leading- and first-correction amplitude combinations (including the new one ${R}_{{B}_{{\mathrm{cr}}^{\mathrm{\ensuremath{-}}})}}$ are accurately estimated and are compared with previous theoretical and experimental estimates. We also show that the functions are well adapted to a suitable comparison with experiment and we describe how the adjustable parameters, limited to only three (the same as in I), enter in nonasymptotic critical behavior. Together with I, this work provides experimentalists with an efficient and coherent method which will facilitate the experimental test of the renormalization-group predictions.

DOI
10.1103/physrevb.35.3585

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The theoretical framework and the details of calculations of nonasymptotic critical behavior, above Tc{T}_{c}Tc​, previously published for n=1n=1n=1, are presented and extended to the cases n=2and3n=2 \mathrm{and} 3n=2and3. The complete description of the real preasymptotic critical domain Dpreas{\mathcal{D}}_{\mathrm{preas}}Dpreas​ needs only three adjustable parameters. We show that the ${\ensuremath{\varphi}}^{4}$ model at infinite cutoff is sufficient to obtain this description. The higher-transient and finite-cutoff effects in Dpreas{\mathcal{D}}_{\mathrm{preas}}Dpreas​ affect only the adjustable parameters. A precise nonperturbative treatment of the ${\ensuremath{\varphi}}^{4}$ field theory exactly for d=3d=3d=3, in the spirit of the Parisi work, yields nonasymptotic functions of temperature which, including all the quantitative universal characteristics of the critical behavior in Dpreas{\mathcal{D}}_{\mathrm{preas}}Dpreas​, are adapted to a suitable comparison with experiments. We emphasize that this work allows a determination, within the experimental accuracy, of the size of Dpreas{\mathcal{D}}_{\mathrm{preas}}Dpreas​ and an appreciation of the effects of the higher corrections to scaling. We also give the estimates for n=1,2,and3n=1, 2, \mathrm{and} 3n=1,2,and3 of a new universal combination of amplitudes (RBcr{{R}_{B}}_{\mathrm{cr}}RB​cr​) which concerns the specific heat alone.
We extend our previous determinations of nonasymptotic critical behavior of Phys. Rev. B 32, 7209 (1985) and 35, 3585 (1987) to accurate expressions of the complete classical-to-critical crossover (in three-dimensional field theory) in terms of the temperaturelike scaling field (i.e., along the critical isochore) for (1) the correlation length, the susceptibility, and the specific heat in the homogeneous phase for the n-vector model (n=1 to 3) and (2) the spontaneous magnetization (coexistence curve), the susceptibility, and the specific heat in the inhomogeneous phase for the Ising model (n=1). The present calculations include the seventh-loop order of Murray and Nickel and closely account for the up-to-date estimates of universal asymptotic critical quantities (exponents and amplitude combinations) provided by Guida and Zinn-Justin [J. Phys. A 31, 8103 (1998)].
We present a field-theoretic renormalization-group treatment of the critical behavior of the cphi4{\mathit{cphi}}^{4}cphi4 model within the minimal-subtraction scheme at fixed dimension d4. We show that correlation functions above and below Tc{\mathit{T}}_{\mathit{c}}Tc​ can be calculated within this renormalization scheme without using the \ensuremath{\epsilon}=4-d expansion. This is demonstrated for the specific heat, for the order parameter, and for the superfluid density (stiffness constant). Various asymptotic amplitudes and correction amplitudes are determined, and some of their universal ratios are calculated in three dimensions. Our result for ${\mathit{a}}_{\mathit{c}}^{\mathrm{\ensuremath{-}}}$/${\mathit{a}}_{\mathrm{\ensuremath{\rho}}\mathit{s}}$ eliminates a previous error of the \ensuremath{\epsilon} expansion regarding the sign of this ratio for d=3 and n=2.
Crossover functions are obtained to three-loop order in an expansion around dimension d=4 for the ${\ensuremath{\varphi}}^{4}$ theory. Expressions for the internal energy, susceptibility, and correlation length are given for the general n-component system in zero magnetic field and above the critical temperature. The full Helmholtz free energy, magnetization equation of state, and susceptibility are given for the Ising-like n=1 case. Corrected values for the critical amplitude ratios as well as for the correction-to-scaling amplitude ratios are given to three-loop order.
We estimate, at d=3d=3d=3, from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above Tc{T}_{c}Tc​. Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.