Quasi-Long-Range Order in Random-Anisotropy Heisenberg Models
  Monte Carlo simulations have been used to study a discretized Heisenberg ferromagnet (FM) with random uniaxial single-site anisotropy on $L \times L \times L$ simple cubic lattices, for $L$ up to 64. The spin variable on each site is chosen from the twelve [110] directions. The random anisotropy has infinite strength and a random direction on a fraction $x$ of the sites of the lattice, and is zero on the remaining sites. In many respects the behavior of this model is qualitatively similar to that of the corresponding random-field model. Due to the discretization, for small $x$ at low temperature there is a [110] FM phase. For $x>0$ there is an intermediate quasi-long-range ordered (QLRO) phase between the paramagnet and the ferromagnet, which is characterized by a $|k|^{-3}$ divergence of the magnetic structure factor S$(k)$ for small $k$, but no true FM order. At the transition between the paramagnetic and QLRO phases S$(k)$ diverges like $|k|^{-2}$. The limit of stability of the QLRO phase is somewhat greater than $x=0.5$. For $x$ close to 1 the low temperature form of S$(k)$ can be fit by a Lorentzian, with a correlation length estimated to be $11 \pm 1$ at $x=1.0$ and $25 \pm 5$ at $x=0.75$. 
Ronald Fisch