Magnets with random anisotropies: The random cubic model and other models with p-fold spin interactions

In this paper we investigate in detail models with random anisotropies and p-fold spin interactions. We construct the random cubic model (p = 4) and show that when N > 2 (N being the number of spin components) its properties are similar to the random uniaxial anisotropy model, since quartic interactions, characteristic of the random uniaxial model, are generated through renormalization even if they vanish to start with. Similar conclusions apply to a random p-fold interaction model of the form $\text{?D∑}{}_{\mathrm{i}}\text{(}\text{n}\text{?}\text{,}{}_{\mathrm{i}}\text{·}\text{S}{}_{\mathrm{i}}\text{)}{}^{\mathrm{p}}$, when p is even. In the case of odd p, a random field interaction is generated. Other models are also discussed.