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 Schr\"odinger equation, ${\mathit{u}}_{\mathit{n}+1}$+${\mathit{u}}_{\mathit{n}\mathrm{\ensuremath{-}}1}$+${\mathit{V}}_{\mathit{n}}$${\mathit{u}}_{\mathit{n}}$=${\mathit{Eu}}_{\mathit{n}}$, where the on-site potential ${\mathit{V}}_{\mathit{n}}$ is neither periodic (the ``Bloch'' case) nor random (the ``Anderson'' case), but is aperiodic or pseudorandom. In particular, we consider in detail a class of slowly varying potential with a typical example being ${\mathit{V}}_{\mathit{n}}$=\ensuremath{\lambda} cos(\ensuremath{\pi}\ensuremath{\alpha}${\mathit{n}}^{\ensuremath{\nu}}$) with 01. We develop an asymptotic semiclassical technique to calculate exactly (in the large-n limit) the density of states and the Lyapunov exponent for this model. We also carry out numerical work involving direct diagonalization and recursive transfer-matrix calculations to study localization properties of the model. Our theoretical results are essentially in exact agreement with the numerical results. Our most important finding is that, for \ensuremath{\lambda}2, there is a metal-insulator transition in this one-dimensional model (\ensuremath{\nu}1) with the mobility edges located at energies ${\mathit{E}}_{\mathit{c}}$=\ifmmode\pm\else\textpm\fi{}\ensuremath{\Vert}2-\ensuremath{\lambda}\ensuremath{\Vert}. Eigenstates at the band center (\ensuremath{\Vert}E\ensuremath{\Vert}\ensuremath{\Vert}${\mathit{E}}_{\mathit{c}}$\ensuremath{\Vert}) are all extended whereas the band-edge states (\ensuremath{\Vert}E\ensuremath{\Vert}>\ensuremath{\Vert}${\mathit{E}}_{\mathit{c}}$\ensuremath{\Vert}) are all localized. Another interesting finding is that, in contrast to higher-dimensional random-disorder situations, the density of states, D(E), in this model is not necessarily smooth through the mobility edge, but may diverge according to D(E)\ensuremath{\sim}\ensuremath{\Vert}E-${\mathit{E}}_{\mathit{c}}$${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$. The Lyapunov exponent \ensuremath{\gamma} (or, the inverse localization length) behaves at ${\mathit{E}}_{\mathit{c}}$ as \ensuremath{\gamma}(E)\ensuremath{\sim}\ensuremath{\Vert}E-${\mathit{E}}_{\mathit{c}}$${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\beta}}}$, with \ensuremath{\beta}=1-\ensuremath{\delta}. We solve the exact critical behavior of the general model, deriving analytic expressions for D(E), \ensuremath{\gamma}(E), and the exponents \ensuremath{\delta} and \ensuremath{\beta}. We find that \ensuremath{\lambda}, \ensuremath{\alpha}, and \ensuremath{\nu} 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 ${\mathit{V}}_{\mathit{n}}$.