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Journal of Physical Studies

Five-loop critical exponents of the weakly diluted Ising model: 3D approach versus √ε-expansion

R. Folk, Yurij V. Holovatch, Taras Yavors’kii

Published Jan 1, 1998• 13 citations
Publisher

Abstract

Based on the recently obtained ve{loop renormalization group functions of the 4 {theory with O(n){symmetric and cubic interactions, we analyse these functions for the weakly diluted Ising model case n = 0 both by means of the p "{expansion and in the xed d = 3 technique.We apply dierent resummation procedures and report numerical values for the critical exponents dependent on the order of approximation.We compare the convergence of the results obtained in the framework of the two schemes.

DOI
10.30970/jps.02.213

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Five-loop critical exponents of the weakly diluted Ising model: 3d approach versus √{\varepsilon}-expansion (in English)

R. Folk, Yurij V. Holovatch, Taras Yavors’kii

Journal of Physical Studies•Jan 1, 1998•0 citations
Based on the recently obtained ve{loop renormalization group functions of the 4 {theory with O(n){symmetric and cubic interactions, we analyse these functions for the weakly diluted Ising model case n = 0 both by means of the p "{expansion and in the xed d = 3 technique.We apply dierent resummation procedures and report numerical values for the critical exponents dependent on the order of approximation.We compare the convergence of the results obtained in the framework of the two schemes.
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.
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.