High-temperature series and the random-field Ising model

A formalism is given whereby high-temperature series for the random-field Ising model are obtained by partitioning the vertices of the pure Ising series diagrams; for any given order in the high-temperature variable, the random fields are thereby treated exactly. Series for the susceptibility up to seventh order in the high-temperature variable have been obtained for an arbitrary distribution of the random fields. Series for a bimodal and a Gaussian distribution were analyzed by Dlog-Padé methods. The analysis predicts that, for a bimodal distribution of random fields, the tricritical point predicted by mean-field theory persists in all dimensions above the lower critical dimension, but moves to weaker fields as the dimension is reduced. For the Gaussian distribution, the transition is predicted to be second order over the whole phase boundary for dimension d4; however, for d<4 the analysis suggests that fluctuations drive the transition first order for sufficiently strong fields implying that d=4 is a critical dimension for this model. Above four dimensions the value of the susceptibility exponent determined from the series is consistent with the predictions of the dd-2 rule for both distributions.