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arXiv:1402.3894 · Physical Review E

Fisher exponent from pseudo- ε expansion

Aleksandr Ivanovich Sokolov, Margarita Aleksandrovna Nikitina

Published Jul 2, 2014• 12 citations
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Abstract

The critical exponent $\ensuremath{\eta}$ for three-dimensional systems with an $n$-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 $n$ = 0, 1, 2, 3 and within the ${\ensuremath{\tau}}^{6}$ order for general $n$ 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 $\ensuremath{\eta}$.

DOI
10.1103/physreve.90.012102

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The critical exponent \ensuremathη\ensuremath{\eta}\ensuremathη for three-dimensional systems with an nnn-vector order parameter is evaluated in the framework of the pseudo-\ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion approach. The pseudo-\ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion (\ensuremathτ\ensuremath{\tau}\ensuremathτ series) for \ensuremathη\ensuremath{\eta}\ensuremathη found up to the \ensuremathτ7{\ensuremath{\tau}}^{7}\ensuremathτ7 term for nnn = 0, 1, 2, 3 and within the \ensuremathτ6{\ensuremath{\tau}}^{6}\ensuremathτ6 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 \ensuremathτ\ensuremath{\tau}\ensuremathτ series result in iteration procedures rapidly converging to the asymptotic values that are very close to the most reliable numerical estimates of \ensuremathη\ensuremath{\eta}\ensuremathη known today. The origin of such an efficiency is discussed and shown to lie in the general properties of the pseudo-\ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion machinery interfering with some peculiarities of the renormalization group expansion of \ensuremathη\ensuremath{\eta}\ensuremathη.
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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.
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The pseudo-ε-expansions for the coordinate of the fixed point g * , the critical exponents, and the sextic effective coupling constant g 6 are determined for the two-dimensional Ising model on the basis of the five-loop renormalization group series. It is found that the pseudoe-expansions for the coordinate of the fixed point g * , the inverse exponent γ −1 , and the constant g 6 possess a remarkable property, namely, the higher terms of these series are so small that reliable numerical results can be obtained without invoking Borel summation.
A numerical technique, termed the scaling-field method, is developed for solving by successive approximation Wilson's exact renormalization-group equation for critical phenomena in three-dimensional spin systems. The approach uses the scaling-field representation of the Wilson equation derived by Riedel, Golner, and Newman. A procedure is proposed for generating in a nonperturbative and unbiased fashion sequences of successively larger truncations to the infinite hierarchy of scaling-field equations. A "principle of balance" is introduced and used to provide a self-consistency criterion. The approach is then applied to the isotropic NNN-vector model. Truncations to order 13 (10, when N=1N=1N=1) scaling-field equations yield the leading critical exponents, $\ensuremath{\nu}$ and $\ensuremath{\eta}$, and several of the correction-to-scaling exponents, ${\ensuremath{\Delta}}_{m}$, to high precision. Results for N=0,1,2,and3N=0, 1, 2, \mathrm{and} 3N=0,1,2,and3 are tabulated. For the Ising case (N=1N=1N=1), the estimates $\ensuremath{\nu}=0.626\ifmmode\pm\else\textpm\fi{}0.009$, $\ensuremath{\eta}=0.040\ifmmode\pm\else\textpm\fi{}0.007$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\equiv}{\ensuremath{\Delta}}_{400}=0.54\ifmmode\pm\else\textpm\fi{}0.05$ are in good agreement with recent high-temperature-series results, though exhibiting larger confidence limits at the present level of approximation. For the first time, estimates are obtained for the second and third correction-to-scaling exponents. For example, for the Ising model the second "even" and first "odd" correction-to-scaling exponents are ${\ensuremath{\Delta}}_{422}=1.67\ifmmode\pm\else\textpm\fi{}0.11$ and ${\ensuremath{\Delta}}_{500}=1.5\ifmmode\pm\else\textpm\fi{}0.3$, respectively. Extensions necessary to improve the accuracy of the calculation are discussed, while applications of the approach to anisotropic NNN-vector models are described elsewhere. Finally, the scaling-field method is compared with other techniques for the high-precision calculation of critical phenomena in three dimensions, i.e., high-temperature-series, Monte Carlo renormalization-group, and field-theoretic perturbation expansions.