Variational principles in renormalization theory

The variational principles introduced by Kadanoff et al. in the renormalization theory are analyzed. It is shown that the values for the specific heat critical exponent α which can be found by a variational method are restricted to α < 0 or α = 1 (first order transition). The reason is the confluence of the singularities in the free energy and in the variational parameters. A full implementation of the variational principle changes for the square Ising lattice the earlier obtained α = 0.001756 to α = −0.123413.