Localization as a breakdown of extended states

A way of understanding localization as a breakdown of extended states is presented by considering a three-dimensional disordered cluster of length $L_1$ and finite cross section $L_2$ $L_3$, which is repeated periodically along the $L_1$ direction. The exact Green’s function is calculated in the tight-binding scheme and the complex self-energy Σ(ε)=Δ(ε)+iΓ(ε) is studied as a function of the cell length; this procedure of finding Γ guarantees that one is always looking at the properties coming from the eigenvalues. The singular spectrum of an infinite isolated cell is then obtained as a limit from the absolutely continuous spectrum of periodic systems with finite cell length. For the repeating cell the measure of the spectrum support approaches zero as $L_1$ increases and this is reflected by the law 〈Γ$L_1$(ε)${〉}_{\mathrm{av}}$∼e-$L_1$ λ for the ensemble average. Theoretical arguments allow us to show that the localization length λ of the eigenfunction of the isolated cells is the same as that calculated by previous authors from the exponential decay of the transmittance and by ourselves from convergence of self-energies in the realm of real numbers. A numerical calculation of the dependence of λ on cross section and disorder parameter W shows a behavior similar to that found by MacKinnon and Kramer.