Exact renormalization group equation for lattice Ginzburg-Landau models adapted to the solution in the local potential approximation

Abstract

The Wilson Green's function approach and, alternatively, Feynman's diffusion equation and the Hori representation have been used to derive an exact functional RG equation (EFRGE) that in the course of the RG flow interpolates between the interaction part of the lattice Ginzburg-Landau Hamiltonian and the logarithm of the generating functional of the S-matrix. Because the S-matrix vertices coincide with the amputated correlation functions of the fluctuating field, it has been suggested that in the critical region the amputation of the long-range tails makes the S-matrix functional more localized and so more suitable to the local potential approximation than the renormalized free energy functional used in Wilson's EFRGE. It has been shown that the S-matrix EFRGE can be transformed into EFRGE for the effective action (EA) by means of a Legendre transform which means that the EA and the S-matrix EFRGEs are formally equivalent. Their structures, however, are very different. In particular, the S-matrix equation could be straightforwardly transformed into a generalized Burgers' equation. This has made possible to identify the first order phase transitions with the shock wave solutions of the RG equation. Besides, the transparent RG structure of the S-matrix RG equation makes possible to use different RG techniques at different stages of the RG flow.