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Journal of Experimental and Theoretical Physics Letters

Critical exponents for a three-dimensional impure Ising model in the five-loop approximation

D. V. Pakhnin, Aleksandr Ivanovich Sokolov

Published May 1, 2000• 7 citations
Publisher

Abstract

The renormalization-group functions governing the critical behavior of the three-dimensional weakly-disordered Ising model are calculated in the five-loop approximation. The random fixed point location and critical exponents for impure Ising systems are estimated by means of the Padé-Borel-Leroy resummation of the renormalization-group expansions derived. The asymptotic critical exponents are found to be γ=1.325 ± 0.003, η=0.025 ± 0.01, ν= 0.671 ± 0.005, α=−0.0125 ± 0.008, β=0.344 ± 0.006, while for the correction-to-scaling exponent, a less accurate estimate ω=0.32 ± 0.06 is obtained.

DOI
10.1134/1.568366

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The renormalization-group functions governing the critical behavior of the three-dimensional weakly-disordered Ising model are calculated in the five-loop approximation. The random fixed point location and critical exponents for impure Ising systems are estimated by means of the Padé-Borel-Leroy resummation of the renormalization-group expansions derived. The asymptotic critical exponents are found to be γ=1.325 ± 0.003, η=0.025 ± 0.01, ν= 0.671 ± 0.005, α=−0.0125 ± 0.008, β=0.344 ± 0.006, while for the correction-to-scaling exponent, a less accurate estimate ω=0.32 ± 0.06 is obtained.
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.
RG functions are calculated up to 4-loop order in 3 and 2 dimensions to evaluate, within Chisholm-Borel resummation technique, critical exponents for cubic and impure uniaxial crystals and the order parameter critical dimensionality nc . For 3D impure Ising crystals γ = 1.326, v = 0.670, α = -0.011, β = 0.342, η = 0.034, while critical exponents of 3D cubic crystals practically coincide with those of the Heisenberg model; nc = 2.9. For 2D systems the fermion representation enables us to find that their critical behavior is identical to that of pure Ising model, apart some logarithmic factors.
Critical exponents for the three-dimensional O(n)-symmetric model with n>3 are estimated on the basis of six-loop renormalization-group (RG) expansions. A simple Pad\'e-Borel technique is used for the resummation of the RG series and the Pad\'e approximants [L/1] are shown to give rather good numerical results for all calculated quantities. For large n, the fixed point location gc{\mathit{g}}_{\mathit{c}}gc​ and the critical exponents are also determined directly from six-loop expansions without addressing the resummation procedure. An analysis of the numbers obtained shows that resummation becomes unnecessary when n exceeds 28 provided an accuracy of about 0.01 is adopted as satisfactory for gc{\mathit{g}}_{\mathit{c}}gc​ and the critical exponents. Further, results of the calculations performed are used to estimate the numerical accuracy of the 1/n expansion. The same value n=28 is shown to play the role of the lower boundary of the domain where this approximation provides high-precision estimates for the critical exponents.
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 critical behavior of three-dimensional weakly diluted quenched Ising model is examined on the base of six-loop renormalization group expansions obtained within the minimal subtraction scheme in 4-ɛ space dimensions. For this purpose the ϕ^{4} field theory with cubic symmetry was analyzed in the replica limit n→0. Along with renormalization group expansions in terms of renormalized couplings the sqrt[ɛ] expansions of critical exponents are presented. Corresponding numerical estimates for the physical, three-dimensional system are obtained by means of different resummation procedures applied both to the sqrt[ɛ] series and to initial renormalization group expansions. The results given by the latter approach are in a good agreement with their counterparts obtained experimentally and within the Monte Carlo simulations, while resumming of sqrt[ɛ] series themselves turned out to be disappointing.
We investigate the critical behaviour of a d = 3-dimensional weakly diluted quenched Ising model, analyzing the series for the renormalization group functions at d = 3 obtained in a minimal subtraction scheme.Formerly it was established [R.Folk, Yu.Holovatch, T. Yavorskii, Phys.Rev. B 61, 15 114 (2000)℄ that the resummed renormalization group series possess an \optimal truncation" behaviour, provided that a Chisholm{Borel resummation technique is applied.This resulted in a conjecture that a four-loop approximation is a nal one for the random Ising model renormalizationgroup functions in a d = 3 minimal subtraction scheme.We apply the method of subsequent resummation, developed in the context of the d = 0-dimensional random Ising model, discuss the convergence properties of the series and give the results of the critical exponents increasing the order of approximation to 5-loop level.