Quasi-long-range order in the random anisotropy Heisenberg model:: Functional renormalization group in 4-ε dimensions

The large-distance behaviors of the random field and random anisotropy O(N) models are studied with the functional renormalization group in 4-epsilon dimensions. The random anisotropy Heisenberg (N=3) model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law [m(r(1))m(r(2))]similar to\r(1)-r(2)\(-0.62 epsilon). The magnetic susceptibility diverges at low fields as chi similar to H-1+0.15 epsilon. In the random field O(N) model the correlation length is found to be finite at the arbitrarily weak disorder for any N>3. The random field case is studied with a simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization-group equations.