Renormalization-group calculation of first- and second-order phase transitions in the Potts model

A renormalization-group calculation has been carried out in d=6-ε dimensions to lowest order in ε, using the continuum generalization of the Potts model. The equation of state and the various thermodynamic functions have been calculated explicitly by integrating up the recursion relations in both the ordered and disordered phases. The transition is found to be first order for $p>2$. For 2<p<10/3 the transition is first order in spite of the fact that there is an accessible stable fixed point for all p<10/3. At the first-order transition, instead of the usual expansion in temperature, the various thermodynamic functions are studied as functions of (p-2). Finally it is shown that the free energy of the Potts model has an essential singularity at the first-order transition temperature Tc, due to instantons, which is of the same form as that of the Ising model at $H=0$.