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arXiv:cond-mat/9503016 · Journal of Physics A Mathematical and General

On the four-dimensional diluted Ising model

Giorgio Parisi A, J J Ruiz-Lorenzo

Published Jul 21, 1995• 16 citations
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Abstract

We give strong numerical evidence that the four dimensional diluted Ising model for a large dilution is not described by the mean-field exponents. These results suggest the existence of a new fixed point with non-Gaussian exponents.
DOI
10.1088/0305-4470/28/14/004

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In this letter we show strong numerical evidence that the four dimensional Diluted Ising Model for a large dilution is not described by the Mean Field exponents. These results suggest the existence of a new fixed point with non-gaussian exponents. PACS numbers: 75.10 Nr. 1 Introduction Random magnetic systems have been the subject of intensive studies over the last 20 years and much progress has been achieved. The simplest model for a random magnetic system is a ferromagnetic system in which the disorder induces fluctuations in the value of the coupling (or equivalently of the temperature). The simplest realization is a randomly diluted Ising system, where sites (site diluted) or bonds (bond diluted) are randomly removed. The equivalent Ginsburg Landau model has the following form Z J = Z d[OE] exp(\\GammaS J [OE]); (1) where S J [OE] = Z d D x ` 1 2 (@ ¯ OE(x)) 2 + 1 2 (m 2 + J(x))OE(x) 2 + g 4! OE(x) 4 ' ; (2) and the quenched random variables J are Gaussian d...
We numerically investigated the critical behavior for the dilute four-dimensional Ising model at different dilution values. For strong dilution, p<0.5p<0.5p<0.5, we found a significant deviation from the pure system, indicating a fixed point.
The N-vector cubic model relevant, among others, to the physics of the randomly dilute Ising model is analyzed in arbitrary dimension by means of an exact renormalization-group equation. This study provides a unified picture of its critical physics between two and four dimensions. We give the critical exponents for the three-dimensional randomly dilute Ising model that are in good agreement with experimental and numerical data. The relevance of the cubic anisotropy in the O(N)O(N)O(N) model is also treated.
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).