Disordered system withn orbitals per site: Lagrange formulation, hyperbolic symmetry, and goldstone modes

Abstract

We give a Lagrange formulation of the gauge invariantn-orbital model for disordered electronic systems recently introduced by Wegner. The derivation proceeds analytically without use of diagrams, and it identifies the previously discussedn→∞ limit as the saddle-point approximation of the Lagrangian formulation. We discover that the Lagrangian model crucially depends on the position with respect to the real axis of the energies involved. If the energies occur on both sides of the real axis as is the case in the calculation of the conductivity, then the order parameter field takes values in a set of complex non-hermitean matrices. If all energies are on the same side of the real axis then a hermitean matrix model emerges. This difference reflects a difference in the symmetries. Whereas in the latter case normal unitary symmetry holds, the symmetry in the former case is of hyperbolic nature. The corresponding symmetry group is not compact and this might be a source of singularities also in the region of localized states. Eliminating massive modes in tree approximation we derive an effective Lagrangian for the Goldstone modes. The structure of this Lagrangian resembles the non-linear σ-model and is a very general consequence of broken isotropic symmetry. We also consider the first correction to the tree approximation which is related to the invariant measure of the generalized non-linear σ-model.