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 Schrödinger equation, ${\mathit{u}}_{\mathit{n}+1\mathrm{}}$+${\mathit{u}}_{\mathit{n}\mathrm{-}1}$+${\mathit{V}}_{\mathit{n}}$ ${\mathit{u}}_{\mathit{n}}$=${\mathit{Eu}}_{\mathit{n}}$, where the on-site potential ${\mathit{V}}_{\mathit{n}}$ 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 ${\mathit{V}}_{\mathit{n}}$=λ cos(πα${\mathit{n}}^{\nu }$) with 0<ν<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 λ<2, there is a metal-insulator transition in this one-dimensional model (ν<1) with the mobility edges located at energies ${\mathit{E}}_{\mathit{c}}$=±‖2-λ‖. Eigenstates at the band center (‖E‖<‖${\mathit{E}}_{\mathit{c}}$‖) are all extended whereas the band-edge states (‖E‖>‖${\mathit{E}}_{\mathit{c}}$‖) 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-${\mathit{E}}_{\mathit{c}}$ ${\mathrm{\Vert }}^{\mathrm{-}\mathrm{\delta }}$. The Lyapunov exponent γ (or, the inverse localization length) behaves at ${\mathit{E}}_{\mathit{c}}$ as γ(E)∼‖E-${\mathit{E}}_{\mathit{c}}$ ${\mathrm{\Vert }}^{\mathrm{\beta }}$, 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 ${\mathit{V}}_{\mathit{n}}$.