Physica A 167 (1990) 119-131 North-Holland EFFECTIVE MEDIUM APPROXIMATION IN THE LOCALIZATION THEORY: SADDLE POINT IN A LAGRANGIAN FORMULATION K.B. EFETOV Max-Planck-lnstitut far Festkrrperforschung, Heisenbergstrasse 1, D-7000 Stuttgart 80, Fed. Rep. Germany A new method of studying the metal-insulator transition is proposed. All physical quantities are represented in terms of a field theory model in an extended space which consists of the real space and the space of supermatrices. The minimum of the effective Lagrangian describing the model gives equations coinciding with the corresponding equations derived before on the Bethe lattice and in high dimensionality. A concept of a function order parameter is discussed. In the saddle point approximation the density-density correlation function both in the metal and insulator regions is calculated. 1. Introduction After 30 years of studying the Anderson metal-insulator transition the critical behavior is still a problem. For example, the conventional second order phase transition behavior predicted by 2 + e expansion [1, 2] does not agree with the critical behavior predicted by the exact solution of the non-linear o- model on the Bethe lattices [3]. The main reason for the unusual results obtained on the Bethe lattice or in high dimensionality [4] is the noncompact- ness of the group of the symmetry of the supermatrices Q. In spite of the availability of the exact results on the Bethe lattice the critical behavior on real lattices in 2D or 3D spaces is still not clear. Therefore it is desirable to find a new method which would give the possibility to study the transition on real lattices. In the theory of phase transitions the usual proce- dure is: first one uses the mean field approximation, then one should calculate corrections. These corrections are convergent provided the dimensionality of the space d is high, d > d r. For d < d_c the corrections are divergent. In this case it is necessary to collect all divergencies. The critical di- mensionality d c is called upper critical dimensionality. The usual way to do this procedure is to use a proper Ginzburg-Landau free energy functional [5]. In this approach the mean field theory corresponds to the minimum of the free energy functional. Expansion near the minimum gives corrections. One can pick up all divergent terms using the very well-developed method of the renormalization group. 0378-4371/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)