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Physical Review Letters

Critical Exponents of the Random-Field O(N) Model

Dima E. Feldman

Published Apr 15, 2002• 62 citations
Publisher

Abstract

The critical behavior of the random-field Ising model has long been a puzzle. Different methods predict that its critical exponents in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components contrary to the experiments and simulations. We calculate the exponents of the random-field O(N) model with the (4+epsilon)-expansion and obtain values different from the exponents of the pure ferromagnet in 2+epsilon dimensions. An infinite set of relevant operators missed in previous studies leads to a breakdown of the (6-epsilon)-expansion.

DOI
10.1103/physrevlett.88.177202

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The critical behavior of the random-field Ising model has long been a puzzle. Different methods predict that its critical exponents in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components contrary to the experiments and simulations. We calculate the exponents of the random-field O(N) model with the (4+epsilon)-expansion and obtain values different from the exponents of the pure ferromagnet in 2+epsilon dimensions. An infinite set of relevant operators missed in previous studies leads to a breakdown of the (6-epsilon)-expansion.
The critical behavior of the random-field Ising model has been a puzzle for a long time. Different theoretical methods predict that the critical exponents of the random-field ferromagnet in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components. This result contradicts the experiments and simulations. We calculate the critical exponents of the random-field O(N) model with the (4+ε)-expansion and obtain values different from the critical exponents of the pure ferromagnet in 2+εdimensions. In contrast to the previous approaches we take into account an infinite set of relevant operators emerging in the problem. We demonstrate how these previously missed relevant operators lead to the breakdown of the (6-ε)-expansion for the random-field Ising model.
The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D-2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields. We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D=5. We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon. We close our contribution by comparing universal quantities for the random-field problem at dimensions 3≤D<6 to their values in the pure Ising model at D-2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.
Critical behaviors of a ferromagnetic spin system under quenched random magnetic fields are investigated near above four space dimensions. Critical temperature Tc(m, µ, ε) as a function of the number of spin components m(≥2) and strength µ of the random field is obtained in a power series of ε ≡ d-4 (>0) by means of the Migdal-Polyakov renormalization group theory. Critical exponents ν and η are also calculated in powers of ε. It is shown that they are in agreement with those of (2 + ε)-dimensional pure system for sufficiently close to Tc. In the limit m→∞, present results reduce to d = 4 + ε case of our previous results for the spherical model.
This is the third article in a sequence in which we reexamine the calculation of the critical exponents of the N-vector model from field theory. In the first article we have applied summation methods to the ε-expansion for which new terms have been calculated (up to order ε5). In the second article we have incorporated in the summation method, in the case of Ising-like systems (N = 1), the knowledge of the exact values of exponents in d = 2 dimensions. Since exact values are now also known in the case of polymers (N = 0), we apply here the same method to this case. Moreover, since for the exponent η for N = 1 the results coming from the ε-expansion and perturbation series at fixed dimension are slightly different, we have recalculated the exponents in the latter case using the variant of the summation method developed for the ε-expansion. The results do not change significantly showing both the reliability and the limits of the method.
We solve a long-standing puzzle in statistical mechanics of disordered systems. By performing a high-statistics simulation of the D=3 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute the complete set of critical exponents for this class, including the correction-to-scaling exponent, and we show, to high numerical accuracy, that scaling is described by two independent exponents. Discrepancies with previous works are explained in terms of strong scaling corrections.
Random-field ferromagnetism systems are discussed, using an equivalent annealed system. It is shown how the critical exponents of the random system are related to the exponents of the pure system. The critical exponent eta is expressed in terms of a dimensionality reduction d'=d-2+ eta (d'). The lower critical dimension for the Ising model is two and for the continuous models it is four. Very strong arguments are given for a hyperscaling relation 2- alpha =d'v and for all the exponents to be given by a dimensionality reduction.