Phenomenological theory of the metal-insulator transition

We show that the transition of a system from the metallic to the insulating regime is accompanied by a change of the density of the Lyapunov exponents of the transfer matrix of the system. This enables us to construct the distribution of the Lyapunov exponents P(z). It has a form P(z) approximately exp(- beta H), where Hamiltonian H contains the one-particle potential V(z) and the interaction term u(z i ,z j ). In the metallic limit, this distribution has a form proposed previously by random matrix theory: V(z) is quadratic and u(z i ,z j ) converges to the interacting potential found previously by Beenakker and Rajaei. Close to the critical point of the metal-insulator transition, both V(z) and u(z i ,z j ) become dimension-dependent. In particular, V(z) approximately z d (d>2) at the critical point. We also discuss the applicability of the distribution P(z) and of the one-parameter theory of MIT to the description of the insulating regime.