Dimensional reduction with correlated random fields. A superspace renormalization-group calculation

Abstract

We consider phase transitions in the presence of random magnetic fields, with long-range correlations such that $〈h\left(0\right)h\left(r\right)\mathrm{}〉\sim \frac{1}{r^{d-\sigma }}$. The upper critical dimension is $6+\sigma$. A superspace renormalization-group calculation is carried out to order ${\epsilon }^2$, where $\epsilon =6+\sigma -d$. To order $\epsilon$, the critical properties are that of a pure system in $d-2-\sigma$ dimensions. This dimensional reduction is also justified by physical arguments concerning the lower critical dimensionality, and the hyperscaling relationship. However, it fails at order ${\epsilon }^2$.