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

We study the localization properties of the one-dimensional nearest-neighbor tight-binding Schr\"odinger equation, uₙ₊₁+u_n-1+Vₙ{u}ₙ$=${Eu}ₙ$, where the on-site potential ${V}ₙ$ 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={λ} cos({π}{α}n^ν) with 01. 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 {λ}2, there is a metal-insulator transition in this one-dimensional model ({ν}1) with the mobility edges located at energies Ec=±{}{}2-{λ}{}. Eigenstates at the band center ({}E{}{}Ec{}) are all extended whereas the band-edge states ({}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{{{}}}^{{{-}}{{δ}}}. The Lyapunov exponent γ (or, the inverse localization length) behaves at{E}cas γ(E)~E-{E}c^β, with {β}=1-{δ}. We solve the exact critical behavior of the general model, deriving analytic expressions for D(E), {γ}(E), and the exponents {δ} and {β}. We find that {λ}, {α}, and {ν} 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.