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 [〈σᵢ{{{σ}}}ⱼ〉T-〈σᵢ{〉}T$〈${{{σ}}}ⱼ〉T{]}ₕ$ and the structure factor, G=${{∑}}ⱼ$[〈${{{σ}}}ᵢσⱼ{〉}T]ₕ, where 〈〉T indicates a canonical average at temperature T for an arbitrary configuration of random fields and []ₕ indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each hᵢ=±{}h₀ and a Gaussian distribution in which each hᵢ has variance h₀². We obtained series for {χ} and G in the form ∑n=1,15{a}ₙ$(g,d)(J/T${)}ⁿ$, where J is the exchange constant and the coefficients ${a}ₙ(g,d) are polynomials in g≡{h}₀²$/${J}²$ and in d. We assume that as T approaches its critical value, ${T}c, one has χ~(T-{T}c)^-γ and G{~}(T-Tc{)}^{{{-}}{γ}{{}}}$. 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}{}=2{γ}, 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}{}=2{γ}) 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. {} 1996 The American Physical Society.