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Physical review. B, Solid state

Calculation of the Critical Exponent η via Renormalization-Group Recursion Formulas

Geoffrey R. Golner

Published Jul 1, 1973• 36 citations
Publisher

Abstract

This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.

DOI
10.1103/physrevb.8.339

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Renormalization-group calculation of critical exponents in three dimensions

Geoffrey R. Golner, Eberhard K. Riedel

A scaling‐field representation of Wilson’s exact renormalization‐group equation is derived and used to calculate critical exponents for the continuous‐spin Ising model in three dimensions. Here the results for two successive truncations of the infinite hierarchy of scaling‐field equations are presented. The truncation that retains only the two most relevant scaling fields yields the critical exponents ν?0.55 and η=0; the truncation retaining the four most relevant fields yields the improved values ν?0.60 and η?0.06. Approximations to the full hierarchy of scaling‐field equations introduce a weak dependence of the critical exponents on redundant parameters in the renormalization‐group Hamiltonian. That question is also discussed.

AIP conference proceedings•Jan 1, 1975•0 citations
This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent \ensuremathη\ensuremath{\eta}\ensuremathη. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero \ensuremathη\ensuremath{\eta}\ensuremathη. They are intended to demonstrate, qualitatively, how nonzero values for \ensuremathη\ensuremath{\eta}\ensuremathη are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents \ensuremathν\ensuremath{\nu}\ensuremathν and \ensuremathγ\ensuremath{\gamma}\ensuremathγ. The equations are solved both by \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion about four dimensions and by numerical integration in three dimensions. To order \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2 we obtain \ensuremathη=0.05\ensuremathϵ2\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}\ensuremathη=0.05\ensuremathϵ2. Numerical results in three dimension are \ensuremathη=0.058\ensuremath{\eta}=0.058\ensuremathη=0.058, \ensuremathν=0.588\ensuremath{\nu}=0.588\ensuremathν=0.588, and \ensuremathγ=1.14\ensuremath{\gamma}=1.14\ensuremathγ=1.14. The relation \ensuremathγ=(2\ensuremath−\ensuremathη)\ensuremathν\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}\ensuremathγ=(2\ensuremath−\ensuremathη)\ensuremathν is confirmed.
A scaling-field representation of Wilson's exact renormalization-group equation is derived and used for the approximate calculation of critical exponents for the continuous-spin Ising model in three dimensions. The truncation of the hierarchy of scaling-field equations that retains only the four most relevant scaling fields yields the critical exponents $\ensuremath{\nu}\ensuremath{\approx}0.60$, $\ensuremath{\eta}\ensuremath{\approx}0.06$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\approx}\ensuremath{-}0.47$ (where ${\ensuremath{\Delta}}_{1}$ denotes Wortis's correction-to-scaling exponent).
The "subcritical" exponent ${\ensuremath{\Delta}}_{2s}$ for corrections to asymptotic scaling in the renormalization-group theory of critical phenomena, has been calculated for the continuous-spin versions of the classical, d=3d=3d=3 Ising, $X\ensuremath{-}Y$, and Heisenberg models by numerical integrations of Wilson's approximate recursion formula. The results are ${\ensuremath{\Delta}}_{2s}(\mathrm{Ising})=\ensuremath{-}0.640$, ${\ensuremath{\Delta}}_{2s}(X\ensuremath{-}Y)=\ensuremath{-}0.644$, and ${\ensuremath{\Delta}}_{2s}(\mathrm{Heisenberg})=\ensuremath{-}0.647$. Comparison is made with results from perturbation expansions in $\ensuremath{\epsilon}=4\ensuremath{-}d$ (ddd being the dimensionality), high-temperature series expansions, and experimental measurements.
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.
The $\ensuremath{\epsilon}$-expansion formula for the critical exponent $\ensuremath{\eta}$ derived by Shukla and Green from Wilson's exact renormalization-group equation is found to yield the standard result $\ensuremath{\eta}=\frac{1}{54}{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, independent of the choice of the incomplete-integration function. The result is obtained analytically in the limit $a\ensuremath{\rightarrow}\ensuremath{\infty}$ and numerically for arbitrary aaa, where aaa denotes the normalization constant of the Gaussian fixed-point functional.
A scaling‐field representation of Wilson’s exact renormalization‐group equation is derived and used to calculate critical exponents for the continuous‐spin Ising model in three dimensions. Here the results for two successive truncations of the infinite hierarchy of scaling‐field equations are presented. The truncation that retains only the two most relevant scaling fields yields the critical exponents ν?0.55 and η=0; the truncation retaining the four most relevant fields yields the improved values ν?0.60 and η?0.06. Approximations to the full hierarchy of scaling‐field equations introduce a weak dependence of the critical exponents on redundant parameters in the renormalization‐group Hamiltonian. That question is also discussed.