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Journal of Physical Studies

A three-dimensional random Ising model: Resummation of five-loop series

Viktoria Blavatska, Yurij V. Holovatch

Published Jan 1, 2001• 5 citations
Publisher

Abstract

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.

DOI
10.30970/jps.05.261

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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 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 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 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 leading correction-to-scaling exponent ω for the three-dimensional (3D) dilute Ising model is calculated in the framework of the field theoretic renormalization group approach. Both in the minimal subtraction scheme and in the massive field theory (resummed four-loop expansion) excellent agreement with recent Monte Carlo calculations (H. G. Ballesteros et al., Phys. Rev. B 58, 2740 (1998)) is achieved. The expression of ω as a series in a ε\sqrt \varepsilon ε​ expansion up to O(ɛ2) does not permit a reliable estimate for d=3.
We analyse the critical properties of a weakly diluted (random) Ising model with the long-range interaction decaying with distance x as ∼ x - d - σ in a d-dimensional space. It is known to belong to a new long-range random universality class for certain values of the decay parameter σ. Exploiting the field-theoretic renormalization group approach within the minimal subtraction scheme, we compute the three-loop renormalization group functions. On their basis, with the help of asymptotic series resummation methods, we estimate the correlation length critical exponent ν characterising the new universality class for d = 3 and for those values of σ for which long-range interactions are relevant for the critical behaviour.