Notes on Migdal's recursion formulas

Abstract

A set of renormalization group recursion formulas which were proposed by Migdal are rederived, reinterpreted, and critically analyzed. The new derivation shows the connection between these formulas and previous work on renormalization via decimation and block transformations. The new interpretation which arises from these derivations indicates that Midgal's formulas are best understood as referring to systems in which the couplings are anisotropic. A strong indication of the correctness of this reinterpretation comes from the two-dimensional Ising model: The new interpretation gives an exact (!) expression for the critical couplings in this case for all ratios of J_x to J_y. This paper describes the major failings of this approximation which arise from its source as a decimation approximation, in terms of the well-known inadequacy of the fixed points which result from this type of scheme. Some proposals for improvement of the approximation are described. Finally, a new potential-moving scheme is proposed which is used to show that the Migdal approximation is exact when the potentials are strong and ferromagnetic in sign.