Long-range correlated random field and random anisotropyO(N)models: A functional renormalization group study

Abstract

We study the long-distance behavior of the $O(N)$ model in the presence of random fields and random anisotropies correlated as $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation $x$ using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double $\ensuremath{\epsilon}=d\ensuremath{-}4$ and $\ensuremath{\sigma}$ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in $(d,\ensuremath{\sigma},N)$ parameter space and compute the critical exponents to first order in $\ensuremath{\epsilon}$ and $\ensuremath{\sigma}$. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}$ in aerogel.