Disordered system withn orbitals per site: Lagrange formulation without replica trick, and scaling law for the density of states

Abstract

A Lagrange formulation of the gauge invariantn orbital model of disordered electronic systems is given for the one-particle Green's function. The replica trick is avoided by starting from a formulation on a Grassmann algebra. A vector model of a real, two component vector is derived.

Fluctuations around the saddle point solution of the model are studied. A non-linear transformation allows the consideration of all the important fluctuations. In contrast to the 1/n-expansion of Oppermann and Wegner it is possible to take then=∞ band edges into account.

In a vicinity of these band edges a scaling law for the density of states is found:

$$\rho ({\rm E}) = n^{ - \user2{\xi }} \bar \rho (n^{2\user2{\xi }} (|E| - E_0 ))$$

with an exponent ξ=2/(6−d) ford<2 and large values ofn.