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 the infinite correlation radius at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $< m(x) m(y) >\sim |x-y|^{-0.62\epsilon}$. The magnetic susceptibility diverges at low fields as $\chi \sim H^{-1+0.15\epsilon}$. In the random field O(N) model the correlation radius is found to be finite at the arbitrarily weak disorder for any $N>3$. The random field case is studied with a new simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization group equations.