Low-temperature renormalization-group study of the random-field model

The continuous-spin random-field model is investigated by means of the low-temperature renormalization-group technique with the use of the replica trick. The Wilson-Kogut recursion method is applied. For short-range exchange, the results are in exact agreement with those of the random-axis model studied by Pelcovits. For long-range exchange varying with distance $R$ as $R^{-(d+\sigma )}$, critical exponents are calculated to first order in $d-2\mathrm{}\sigma$. They are identical to those in a $d-\sigma$ expansion of the nonrandom model. However, the hyperscaling law becomes $(d+{\lambda }_T)\nu =2-\alpha$ (${\lambda }_T$ is the eigenvalue associated with the dangerous irrelevant operator $T$), and, for $m$-component spins, $-{\lambda }_T=\sigma +\frac{(d-2\mathrm{}\sigma )}{m}$.