Nonasymptotic critical phenomena

C. Bagnuls, Claude Bervillier

The theoretical framework and the details of calculations of nonasymptotic critical behavior, above ${T}_{c}$, previously published for $n=1$, are presented and extended to the cases $n=2 \mathrm{and} 3$. The complete description of the real preasymptotic critical domain ${\mathcal{D}}_{\mathrm{preas}}$ 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 ${\mathcal{D}}_{\mathrm{preas}}$ affect only the adjustable parameters. A precise nonperturbative treatment of the ${\ensuremath{\varphi}}^{4}$ field theory exactly for $d=3$, in the spirit of the Parisi work, yields nonasymptotic functions of temperature which, including all the quantitative universal characteristics of the critical behavior in ${\mathcal{D}}_{\mathrm{preas}}$, are adapted to a suitable comparison with experiments. We emphasize that this work allows a determination, within the experimental accuracy, of the size of ${\mathcal{D}}_{\mathrm{preas}}$ and an appreciation of the effects of the higher corrections to scaling. We also give the estimates for $n=1, 2, \mathrm{and} 3$ of a new universal combination of amplitudes (${{R}_{B}}_{\mathrm{cr}}$) which concerns the specific heat alone.

John F. Nagle

Yoshitsugu Oono

C. Bagnuls, Claude Bervillier, Daniel I. Meiron +1 more

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.

C. Bagnuls, Claude Bervillier

The obtention (up to five or six loop orders) of nonasymptotic critical behavior, above and below Tc, from the field theoretical framework is presented and discussed.

M. P. Kawatra, J. I. Budnick