ANNALS OF PHYSICS 100, 359-394 (1976) Notes on Migdal’s Recursion Formulas*

A set of renormalization group recursion formulas which were proposed by Migdal are rederived, reinterpreted, and critically analyzed. The new derivation shows the con-nection 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 reinter-pretation 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 Jn to JV. This paper describes the major failings of this approximation which arise from its source as a decimation approximation, in terms of the web-known inadequacy of the tixed points which result from this type of scheme. Some proposals for improvement of the approxi-mation 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. I. INTRODU~TI~N The renormalization group recursion formulas proposed by Migdal [l, 21 are a very important new contribution to the calculation of near-critizal properties in many-particle systems. These formulas are important because 1. They can be applied to systems with all kinds of internal symmetries, including local “gauge ” symmetries [3-51. 2. They can be analytically continued in dimensionality (LI) with relative ease, and are apparently asymptotically exact a. As d-+ I for all systems b. As d + 2 for systems with a set of lozal internal symmetries (gauge symmetries