Wegner model in high dimension: U(1) symmetry breaking and a non-standard phase of disordered electronic matter, I. One-replica theory

The Anderson transition between localized and metallic states is traditionally analyzed by assuming a one-parameter scaling hypothesis. Although that hypothesis has been confirmed near two dimensions by epsilon = d-2 expansion of the Wegner-Efetov nonlinear sigma model, there exists mounting evidence that the transition in d=3 or higher may have a second branch and that two relevant parameters are needed in order to describe the universal behavior at criticality. Doubt of the standard hypothesis also comes from field theory. Indeed, increasing the space dimension pushes the Anderson transition towards strong disorder, where a strong-coupling approach very different from the usual weak-coupling analysis of the sigma model is called for. In the present work, we develop a novel field theory of Anderson transitions at strong coupling based on the key observation that the U(1) symmetry which distinguishes retarded from advanced fields may undergo spontaneous symmetry breaking. That symmetry breakdown splits the sigma model coupling into two, thus leading to a natural scenario of two-parameter scaling. While we develop the field theory from the concrete starting point of the Wegner N-orbital model, we believe our results to be of much wider applicability. The first of a series, the present paper offers a pedagogical introduction to the main ideas in the setting of the one-replica theory. Subsequent papers will employ the self-consistent approximation of Abou-Chacra et al. and develop the full supersymmetric theory. The latter establishes the existence of a new renormalization-group fixed point, whose basin of attraction constitutes a third phase of disordered electronic matter.