Localization, mobility edges, and metal-insulator transition in a class of one-dimensional slowly varying deterministic potentials

Abstract

We study the localization properties of the one-dimensional nearest-neighbor tight-binding Schrödinger equation, un+1+un-1+Vnun=Eun, where the on-site potential Vn is neither periodic (the Bloch case) nor random (the Anderson case), but is aperiodic or pseudorandom. In particular, we consider in detail a class of slowly varying potential with a typical example being Vn= cos(n) with 0<<1. We develop an asymptotic semiclassical technique to calculate exactly (in the large-n limit) the density of states and the Lyapunov exponent for this model. We also carry out numerical work involving direct diagonalization and recursive transfer-matrix calculations to study localization properties of the model. Our theoretical results are essentially in exact agreement with the numerical results. Our most important finding is that, for <2, there is a metal-insulator transition in this one-dimensional model (<1) with the mobility edges located at energies Ec=2-. Eigenstates at the band center (E Ec) are all localized. Another interesting finding is that, in contrast to higher-dimensional random-disorder situations, the density of states, D(E), in this model is not necessarily smooth through the mobility edge, but may diverge according to D(E)E-Ec-. The Lyapunov exponent (or, the inverse localization length) behaves at Ec as (E)E-Ec, with =1-. We solve the exact critical behavior of the general model, deriving analytic expressions for D(E), (E), and the exponents and . We find that , , and are all irrelevant variables in the renormalization-group sense for the localization critical properties of the model. We also give detailed numerical results for a number of different forms of Vn. © 1990 The American Physical Society.