Scaling of the localization length in linear electronic and vibrational systems with long-range correlated disorder

The localization lengths $\lambda$ of one-dimensional disordered systems are studied for electronic wave functions in the Anderson model and for vibrational states. In the first case, the site energies $\epsilon$ and in the second case, the fluctuations of the vibrating masses m at distance l from each other are long-range correlated and described by a correlation function $C\left(l\right)\mathrm{}\sim l^{-\gamma }$ with $0<\mathrm{}\gamma <~1.$ In the Anderson model, we focus on a scaling theory that applies close to the band edge, i.e., at energies E close to 2. We show that $\lambda$ can be written as $\lambda ={\lambda }_0{f}_{\gamma }\mathrm{}\left(x\right),$ with ${\lambda }_0=〈{\epsilon }^2{〉}^{-1/\left(4-\gamma \right)}\sim \lambda (E=2,〈{\epsilon }^2〉),$ $x={\lambda }_0^2\left(2\mathrm{}-E\right),$ and the scaling function $f_{\gamma }\mathrm{}\left(x\right)=\mathrm{const}$ for $x\ll 1$ and $f_{\gamma }\mathrm{}\left(x\right)\mathrm{}\sim x^{\left(3\mathrm{}-\gamma \right)/2\mathrm{}}$ for $x\gg 1.$ Mapping the Anderson model onto the vibrational problem, we derive the vibrational localization lengths for small eigenfrequencies $\omega ,$ $\lambda \sim 〈m{〉}^{\left(3\mathrm{}-\gamma \right)/2\mathrm{}}〈m^2{〉}^{-1}{\omega }^{-(1+\gamma )},$ where $〈m〉$ is the mean mass and $〈m^2〉$ the variance of the masses. This implies that, unexpectedly, at small $\omega ,$ $\lambda$ is larger for uncorrelated than for correlated chains.