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=∑j[〈σiσj⟩T-〈σi⟩T〈σj⟩T]h and the structure factor, G=∑j[〈σiσj⟩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 hi=±h0 and a Gaussian distribution in which each hi has variance h02. We obtained series for χ and G in the form ∑n=1,15an(g,d)(J/T)n, where J is the exchange constant and the coefficients an(g,d) are polynomials in g≡h02/J2 and in d. We assume that as T approaches its critical value, Tc, one has χ~(T-Tc)−γ and G~(T-Tc)−γ. 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/χ2)(T2)/(gJ2). We find that A approaches a constant value as T→Tc (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.