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Physics Today

21-term series yields critical indices

Gloria B. Lubkin

Published Nov 1, 1980• 0 citations
Publisher

Abstract

Discoveries in physics often hinge on a disagreement between theory and experiment of a percent or less. Because of such a discrepancy, many workers in statistical mechanics have been concerned about the complete validity of the scaling and renormalization-group theories of critical phenomena. For the three-dimensional Ising model, the experimentally derived values for the critical exponents in fluids have agreed to three significant figures with those predicted by the renormalization group. However, the predictions disagreed by a couple of percent with exponents calculated by series expansions, and the claimed accuracy of these expansion estimates was much less than a percent.

DOI
10.1063/1.2913819

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Discoveries in physics often hinge on a disagreement between theory and experiment of a percent or less. Because of such a discrepancy, many workers in statistical mechanics have been concerned about the complete validity of the scaling and renormalization-group theories of critical phenomena. For the three-dimensional Ising model, the experimentally derived values for the critical exponents in fluids have agreed to three significant figures with those predicted by the renormalization group. However, the predictions disagreed by a couple of percent with exponents calculated by series expansions, and the claimed accuracy of these expansion estimates was much less than a percent.
It is shown that the conventional theory based on the integral equations for the correlation functions of fluid allows for a reasonable scaling analysis of the critical phenomena. The calculated critical exponents are in rather good agreement with the experimental data for real media, but some of the exponents (first of all, Fisher's exponent eta ) differ from the values predicted by the three-dimensional Ising model. The possibility to obtain Ising-like criticality from the statistical theory of fluids means that the latter does not disagree with the Kadanoff-Wilson-Fisher renormalization group approach.
We present a field-theoretical treatment of the critical behavior of a three-dimensional weakly diluted quenched Ising model. To this end we analyze in the replica limit $\stackrel{\ensuremath{\rightarrow}}{n}0$ the five-loop renormalization-group functions of the ${\ensuremath{\varphi}}^{4}$ theory with O(n)O(n)O(n)-symmetric and cubic interactions [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)]. The minimal subtraction scheme allows one to develop either the $\sqrt{\ensuremath{\varepsilon}}$-expansion series or to proceed within the three-dimensional approach, performing expansions in terms of renormalized couplings. Doing so, we compare both perturbation approaches and discuss their convergence and possible Borel summability. To study the crossover effect we calculate the effective critical exponents. We report resummed numerical values for the effective and asymptotic critical exponents. The results obtained within the three-dimensional approach agree pretty well with recent Monte Carlo simulations. $\sqrt{\ensuremath{\varepsilon}}$ expansion does not allow reliable estimates for d=3.d=3.d=3.
The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pad\'e-Borel-Leroy resummation under n=0.n=0.n=0. These exponents are found to be $\ensuremath{\gamma}=1.325\ifmmode\pm\else\textpm\fi{}0.003,$ $\ensuremath{\eta}=0.025\ifmmode\pm\else\textpm\fi{}0.01,$ $\ensuremath{\nu}=0.671\ifmmode\pm\else\textpm\fi{}0.005,$ $\ensuremath{\alpha}=\ensuremath{-}0.0125\ifmmode\pm\else\textpm\fi{}0.008,$ and $\ensuremath{\beta}=0.344\ifmmode\pm\else\textpm\fi{}0.006.$ For the correction-to-scaling exponent, the less accurate estimate $\ensuremath{\omega}=0.32\ifmmode\pm\else\textpm\fi{}0.06$ is obtained.