Critical Chaotic Spectra of One and Two Interacting Electrons in Quasiperiodic Chains

Abstract

In the last years we have seen a growing interest in the description of the quantum motion for few interacting electrons in a disordered potential [1, 2, 3, 4, 5]. Although this is by no means a true many-body problem, in the sense that it is a finite density and not just a few electrons which is required for a complete understanding of the combined interaction-disorder effects, it has shed some light on this very hard problem. On the other hand, the statistics of energy levels has become a valuable tool for extracting the essential information of the localization of eigenfunctions (without actually computing them) for non-interacting electrons in disordered media, making also possible to identify the localization-delocalization transition [6, 7]. The level-statistics is a general tool since the spectra are always independent of the choice of basis while localization due to disorder or quasiperiodicity is usually expected to occur in the “position” basis. In this area a lot of emphasis was focused on understanding the critical level-statistics at the Anderson transition. This statistics turned out to be intermediate between Wigner (extended states) and Poisson (localized states) [6, 7] being intimately connected to the fractal structure of the corresponding critical wave functions [8].