Variational renormalization method and the Potts model

The variational principle is presented for a continuous renormalization transformation. It is shown that a strict application of the variational method leads to a singularity in the variational parameter which is related to that in the free energy. The equations governing the behaviour of the variational parameter can, in the continuous formulation (for a single coupling parameter), be cast in a differential form from which it can be seen that the method will predict either a very weak singularity (α < 0) or a first order transition. A numerical study of the q-state Potts model is presented which shows that as q increases to 4 the model just fails to exhibit the expected first-order transition.