Critical behaviour of the discrete spin cubic model

An n-component spin model, with the nearest-neighbour Hamiltonian H=-J Sigma (S i .S j )-K Sigma (S i .S j ) 2 , where S i is a discrete unit vector pointing only along one of the 2n cubic axes directions, is studied exactly at dimensions d=1 and d=1+ epsilon and approximately (using dedecoration renormalization group recursion relations) at d=2. The model exhibits four competing possible types of critical behaviour, related to the Ising model, to the n-state and 2n-state Potts models and to a 'new cubic' fixed point. For large n, at d=2, the last three types of behaviour show peculiarities which may be related to the transition becoming a first-order one.