Variational Approximations for Renormalization Group Transformations

Approx imate recurs ion relations which give upper and lower bounds on the free energy are described. Opt imal calculations of the free energy can then be obtained by treating parameters within the renormal izat ion equa-tions variationally. As an example, a particularly s imple lower bound approx imat ion which preserves the symmetry of the Hami l ton ian (the one-hypercube approximat ion) is described. The approx imat ion is applied to both the Ising model and the Wi l son-F i sher model. At the fixed point a parameter is set variat ional ly and critical indices are calculated. For the Ising model the agreement with the exact results at d = 2 is surprisingly good, 0.1~o, and is good at d = 3 and even d = 4. For the Wi l son-F isher model the recursion relation is reduced to a one-dimensional integral equa-t ion which can be solved numerical ly giving v = 0.652 at d = 3, or by E expansion in agreement with the results of Wi lson and Fisher to leading order in ~. The method is also used to calculate thermodynamic funct ions for the d = 2 Ising model; excellent agreement with the Onsager solut ion is found. KEY WORDS: Variational approximations; renormalization group eigen-values; thermodynamic functions