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Journal of Physics A Mathematical and General

On the crossover to universal criticality in dilute Ising systems

H. K. Janssen, Klaus Oerding, E Sengespeick

Published Nov 7, 1995• 57 citations
Publisher

Abstract

Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.

DOI
10.1088/0305-4470/28/21/012

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Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.
The author has investigated the critical behaviour of site-disordered Ising systems over a large range of dilution (0.6<or=p<or=1.0) investing about 10 15 single spin flips with conventional local dynamics. Critical temperatures have been determined with high-precision using the cumulant method. The finite-size simulations lead to concentration-dependent exponents which seem to violate universality. The author shows that the concentration dependence is induced by a crossover phenomenon which governs a large interval of the critical region. The author finds that the scaling functions of systems with different disorder converge asymptotically to universal behaviour consistent with the fixed point of weak disorder.
Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional weakly site-diluted Ising model with spin concentrations p=0.95 and 0.8 at criticality. In contrast to studies of the critical behavior of the pure systems by the short-time dynamics method, our investigations of site-diluted Ising model have revealed three stages of the dynamic evolution characterizing a crossover phenomenon from the critical behavior typical for the pure systems to behavior determined by the influence of disorder. The static and dynamic critical exponents are determined with the use of the corrections to scaling for systems starting separately from ordered and disordered initial states. The obtained values of the exponents demonstrate a universal behavior of weakly site-diluted Ising model in the critical region. The values of the exponents are compared to results of numerical simulations which have been obtained in various works and, also, with results of the renormalization-group description of this model.
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.
Four-loop expansions of the coefficients of the Callan-Symanzik equation for 3D and 2D dilute Ising models are calculated. Fixed-point coordinates and critical exponents are estimated by two methods of summation of double series: a generalisation of the Pade-Borel approximation and the first confluent form of the in algorithm of Wynn. Summation of the double series for the 2D dilute Ising model gives exponents very close to exponents of the pure Ising model, in accordance with the exact solution. The two methods are also applied to summation of single-variable series of pure Ising and polymer models, both in 3D and 2D cases. Both methods are shown to provide close agreement between present estimates and results obtained earlier either exactly in conformal invariant theories (2D) or numerically with high accuracy (3D). An additional test is provided by estimation of critical exponents of the 2D O(n) model for n=-1. The methods tested this way are used to calculate critical exponents of the 3D dilute Ising model. The values obtained are consistent with recent experimental results.
We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent α\alphaα . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.
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.