Quasi-long-range order in the random anisotropy Heisenberg model

The random field and random anisotropy N-vector models are studied with the functional renormalization group in 4−ε dimensions. The random anisotropy Heisenberg (N=3) model has a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law 〈m(r1)m(r2)〉∼¦r1−r2¦− 0.62ε. The magnetic susceptibility diverges at low fields as χ∼H−1+0.15ε. In the random field N-vector model the correlation length is finite at arbitrarily weak disorder for any N>3.