Frequency-dependent conductivity and mean-square hopping displacement in a one-dimensional percolation model

The frequency-dependent conductivity $\sigma \left(\omega \right)$ and the mean-square displacement in time $〈X^2\mathrm{}\left(t\right)\mathrm{}〉$ in a one-dimensional lattice perturbed by random interruptions of nearest-neighbor transfer coupling are calculated exactly. At low frequencies the real and imaginary parts of $\sigma \left(\omega \right)$ vary as ${\omega }^2$ and $\omega$, respectively, while saturating at constant values for $\omega \to \infty$. The solution also exhibits the disorder-induced transition from pure dc behavior in the ordered limit to ac behavior, with $\sigma \mathrm{}\left(0\right)=0\mathrm{}$, when disorder is present. The mean-square displacement, which is relevant to spectral diffusion and exciton migration in one-dimensional disordered systems, increases linearly at short times while rising as $\sim {\left(\lambda t\right)}^{\frac{5}{6}}\exp [-(\frac{3}{2}\left){\left(\lambda t\right)}^{\frac{1}{3}}\right]$ towards a constant value at long times. The dependence of both $\sigma \left(\omega \right)$ and $〈X^2\mathrm{}\left(t\right)\mathrm{}〉$ on the disorder is also obtained in closed form. The above frequency dependence of $\sigma \left(\omega \right)$ might be observable in quasi-one-dimensional ionic or organic conductors with a sufficient concentration of large barrier defects producing interruptions.