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, chi=Sigma(j)[[sigma(i) sigma(j)](T)-[sigma(i)](T)[sigma(j)](T)](h) and the structure factor, G = Sigma(j)[[sigma(i) sigma(j)](T)](h), where [](T) indicates a canonical average at temperature T for an arbitrary configuration of random fields and [](h) indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each h(i)=+/-h(0) and a Gaussian distribution in which each h(i) has variance h(0)(2). We obtained series for chi and G in the form Sigma(n)=(1,15)a(n)(g,d)(J/T)(n), where J is the exchange constant and the coefficients a(n)(g,d) are polynomials in g equivalent to h(0)(2)/J(2) and in d. We assume that as T approaches its critical value, T-c, one has chi similar to(T-T-c)(-gamma) and G similar to(T-T-c)(-<(gamma)over bar>). 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 <(gamma)over bar>=2 gamma, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A = (G/chi(2))(T-2)/(gJ(2)). We find that A approaches a constant value as T --> T-c (consistent with <(gamma)over bar>=2 gamma) with A approximate to 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.