Renormalization group solution of Ising spin models

Abstract

An excellent series of lectures on the renormalization group theory for critical phenomena have been given at this school by Professor Wegner and I will assume in my discussion that the basic ideas of this theory are known to you. I would like to discuss some new developments based on recent work done in collaboration with a graduate student, Bernard Nienhuis, at the University of Utrecht. We have extended the renormalization group approach to evaluate the complete free energy for general Ising spin models, which give a concrete realization of the scaling operators introduced by Professor Wegner. In particular we can evaluate not only the critical exponents and critical temperature but also the coefficients of the singular terms which had not been determined previously except for the special case of a logarithmic singularity. Two basic assumptions of renormalization group theory, the existence of a fixed point Hamiltonian and the analyticity of the renormalization group transformations have been verified for planar Ising models in a cell cluster approximation of Niemeijer and van Leeuwen, but the third assumption introduced by Professor Wegner, the continuity of the renormalization transformations as functions of the dimension of the Kadanoff cells, cannot be justified in this model.