Kohn’s localization in the insulating state: One-dimensional lattices, crystalline versus disordered

Abstract

The qualitative difference between insulators and metals stems from the nature of the low-lying excitations, but also--according to Kohn's theory [W. Kohn, Phys. Rev. 133, A171 (1964)]--from a different organization of the electrons in their ground state: electrons are localized in insulators and delocalized in metals. We adopt a quantitative measure of such localization, by means of a "localization length" lambda, finite in insulators and divergent in metals. We perform simulations over a one-dimensional binary alloy model, in a tight-binding scheme. In the ordered case the model is either a band insulator or a band metal, whereas in the disordered case it is an Anderson insulator. The results show indeed a localized/delocalized ground state in the insulating/metallic cases, as expected. More interestingly, we find a significant difference between the two insulating cases: band versus Anderson. The insulating behavior is due to two very different scattering mechanisms; we show that the corresponding values of lambda differ by a large factor for the same alloy composition. We also investigate the organization of the electrons in the many body ground state from the viewpoint of the density matrices and of Boys' theory of localization.