Anomalous scaling and generalized Lyapunov exponents of the one-dimensional Anderson model

Abstract

We introduce a family of localization lengths ${\ensuremath{\xi}}_{q}$ (related to generalized Lyapunov exponents in a transfer-matrix approach) for the one-dimensional discrete Schr\"odinger equation with diagonal disorder. We show that, at the band edge of the pure system and with a random bounded potential with zero average, there is a q-dependent crossover in the scaling of ${\ensuremath{\xi}}_{q}$ with the disorder amplitude \ensuremath{\epsilon}: For \ensuremath{\epsilon}\ensuremath{\le}\ensuremath{\epsilon}\ifmmode\bar\else\textasciimacron\fi{}(q)\ensuremath{\sim}${q}^{\mathrm{\ensuremath{-}}6}$, ${\ensuremath{\xi}}_{q}$\ensuremath{\propto}${\ensuremath{\epsilon}}^{\mathrm{\ensuremath{-}}2/3}$; otherwise, ${\ensuremath{\xi}}_{q}$\ensuremath{\propto}${\ensuremath{\epsilon}}^{\mathrm{\ensuremath{-}}1/2}$. The limit therefore reproduces the scaling with exponent -(2/3), whereas deviations from this scaling law appear at each fixed \ensuremath{\epsilon} if q is sufficiently large. These results involve a ``multifractal'' structure of the asymptotic decay of the wave functions.