Critical behavior of the random-field Ising model

We study the critical properties of the random field Ising model in general dimension d using high-temperature expansions for the susceptibility, χ=(Formula presented) [〈(Formula presented)(Formula presented)(Formula presented)-〈(Formula presented)(Formula presented)〈(Formula presented)(Formula presented)(Formula presented) and the structure factor, G=(Formula presented)[〈(Formula presented)(Formula presented)(Formula presented)(Formula presented), where 〈(Formula presented) indicates a canonical average at temperature T for an arbitrary configuration of random fields and [(Formula presented) indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each (Formula presented)=±(Formula presented) and a Gaussian distribution in which each (Formula presented) has variance (Formula presented). We obtained series for χ and G in the form (Formula presented)(Formula presented)(g, d)(J/T(Formula presented), where J is the exchange constant and the coefficients (Formula presented)(g, d) are polynomials in g≡(Formula presented)/(Formula presented) and in d. We assume that as T approaches its critical value, (Formula presented), one has χ∼(T-(Formula presented)(Formula presented) and G∼(T-(Formula presented)(Formula presented). For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g. For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that γ¯=2γ, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/(Formula presented))((Formula presented))/((Formula presented)). We find that A approaches a constant value as T→(Formula presented) (consistent with γ¯=2γ) with A≊1. It appears that A is somewhat larger for the bimodal than for the Gaussian model, in agreement with a recent analysis at high d.