Absence of long-range order in systems with random p-fold anisotropies

A random p-fold anisotropy, of the form (n.S) p (where S is the m-component spin vector, m>1, and n is a unit vector with random directions), is shown to destroy long-range magnetic order for any value of p and dimensionalities d<4. For p>or=3, this happens in spite of the stability of the non-random fixed point. The ferromagnetic phase is replaced, in these cases, by a phase with algebraically decaying correlations and an infinite zero-field susceptibility.