Renormalization group approach to unified description of continuous and the first order phase transitions: application to the Blume-Capel model

Abstract: The renormalization group (RG) equation in the self-consistent local potential approximation (SC-LPA) suggested earlier for the description of continuous phase transitions in lattice models of the Landau-Ginzburg type has been applied to the solution of the spin-1 Blume-Capel model on the simple cubic lattice. The calculated transition temperatures of both continuous and the first-order phase transitions (FOPTs) in zero external field have been found to be in excellent agreement with the best available estimates. It has been argued that the SC-LPA RG equation may give more accurate and complete description of the FOPTs than those reported in alternative approaches. It has been shown that the SC-LPA RG equation can be cast in the form of the generalized Burgers' equation (GBE). In this formulation of the RG the FOPTs have been shown to assume the form of the shock-wave solutions of GBE in the inviscid limit. Universality of the RG flow in the vicinity of the fixed point describing FOPTs has been discussed.