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

Abstract

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 ${R}^{\ensuremath{-}(d+\ensuremath{\sigma})}$, critical exponents are calculated to first order in $d\ensuremath{-}2\ensuremath{\sigma}$. They are identical to those in a $d\ensuremath{-}\ensuremath{\sigma}$ expansion of the nonrandom model. However, the hyperscaling law becomes $(d+{\ensuremath{\lambda}}_{T})\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$ (${\ensuremath{\lambda}}_{T}$ is the eigenvalue associated with the dangerous irrelevant operator $T$), and, for $m$-component spins, $\ensuremath{-}{\ensuremath{\lambda}}_{T}=\ensuremath{\sigma}+\frac{(d\ensuremath{-}2\ensuremath{\sigma})}{m}$.