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 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
Dpreas. 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.
The theoretical framework and the details of calculations of nonasymptotic critical behavior, above
Tc, previously published for
n=1, are presented and extended to the cases
n=2and3. The complete description of the real preasymptotic critical domain
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 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
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 and an appreciation of the effects of the higher corrections to scaling. We also give the estimates for
n=1,2,and3 of a new universal combination of amplitudes (
RBcr) which concerns the specific heat alone.
A semi-microscopic scaling-field theory is developed for crossover phenomena near critical and tricritical points. The theory is based on a renormalization-group description of a model with two competing fixed points (such as a critical and a tricritical fixed point) in terms of scaling fields. The coupled nonlinear differential equations for scaling fields are truncated such as to preserve the physics essential for crossover phenomena. The approach allows the explicit calculation of thermodynamic functions for (i) tricritical systems and (ii) critical systems with an irrelevant scaling field. We obtain, for example, an explicit expression for the scaling function of the susceptibility, which describes the crossover from the tricritical to the critical region. The idea of "flow diagrams" in the scaling-field space is used to characterize crossover phenomena globally in the whole critical region. The concept of asymptotic critical exponents is generalized and effective critical exponents are introduced as logarithmic derivatives of thermodynamic quantities with respect to experimental fields and scaling fields, respectively. By using the method of effective exponents the size of the crossover region between regions of different asymptotic critical behavior is estimated. For the susceptibility, the width of the crossover region in decades of the effective temperature variable is roughly equal to the inverse of the crossover exponent. In the case of a critical system with a slow transient the asymptotic critical exponent is only reached extremely close to the critical point (unless the amplitude of the transient vanishes). It might then be impossible to determine the asymptotic exponent experimentally or by conventional series-expansion techniques, and an analysis of the data in terms of effective exponents is the alternative. The scaling-field approach is applied to three systems with crossover phenomena: (i) the model for tricritical systems with molecular-field tricritical exponents, (ii) the Ashkin-Teller model in three dimensions, and (iii) a model for phase transitions with Fisher exponent renormalization due to a constraint.
Scaling and renormalization-group theories of crossover in criticality induced by random interactions are searchingly tested for Ising (n=1) systems. Extrapolation of 15th-order series in two variables for the susceptibility of the \ifmmode\pm\else\textpm\fi{}J simple-cubic Ising model yields a crossover exponent \ensuremath{\varphi}=0.125\ifmmode\pm\else\textpm\fi{}0.025. This agrees well with the specific-heat exponent, \ensuremath{\alpha}\ensuremath{\simeq}0.11. Consistency between free-energy and susceptibility scaling with \ensuremath{\varphi}=\ensuremath{\alpha} is cross-checked and leading universal scaling-function coefficients and nonuniversal scaling field amplitudes are determined.
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.