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

We study the long-distance behavior of the O(N) model in the presence of random fields and random anisotropies correlated as similar to 1/x(d-sigma) for large separation x using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double epsilon=d-4 and 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,sigma,N) parameter space and compute the critical exponents to first order in epsilon and 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 He-3-A in aerogel.