Quasi-long-range order in the random anisotropy Heisenberg model: Functional renormalization group in4−εdimensions

The large-distance behaviors of the random field and random anisotropy $O\left(N\right)\mathrm{}$ models are studied with the functional renormalization group in $4-\epsilon$ dimensions. The random anisotropy Heisenberg $\mathrm{}(N=3\mathrm{})\mathrm{}$ 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 $〈\mathbf{m}\left({\mathbf{r}}_1\right)\mathbf{m}\left({\mathbf{r}}_2\right)〉\sim |{\mathbf{r}}_1-{\mathbf{r}}_2{|}^{-0.62\epsilon }.$ The magnetic susceptibility diverges at low fields as $\chi \sim H^{-1+0.15\epsilon }.$ In the random field $O\left(N\right)\mathrm{}$ model the correlation length is found to be finite at the arbitrarily weak disorder for any $N>\mathrm{}3.\mathrm{}$ 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.