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

The localization lengths λ 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 ε 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(l)~l^-γ with 0<γ<~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 λ can be written as λ=λ₀f_γ(x), with λ₀=〈ε²〉^-1/(4-γ)~λ(E=2,〈ε²〉), x=λ₀²(2-E), and the scaling function f_γ(x)=const for x1 and f_γ(x)~x^(3-γ)/2 for x1. Mapping the Anderson model onto the vibrational problem, we derive the vibrational localization lengths for small eigenfrequencies ω, λ~〈m〉^(3-γ)/2〈m²〉^-1ω^-(1+γ), where $〈m〉$ is the mean mass and 〈m²〉 the variance of the masses. This implies that, unexpectedly, at small ω, λ is larger for uncorrelated than for correlated chains.