Long-Range Resonance in Anderson Insulators: Finite-Frequency Conductivity of Random and Incommensurate Systems

The low-frequency conductivity $\sigma \left(\omega \right)$ of an exactly solvable model of an incommensurate potential is calculated for several classes of commensuration parameter and compared with the random case. The conductivity is related to the degree to which the eigenstates of the model occur in resonating multiplets. It is found that $\sigma \left(\omega \right)\sim {\omega }^x$, where $x=2\mathrm{}$ for the random case, $x=\infty$ for generic commensuration, and $-1<x<\mathrm{}\infty$ for special commensurations. There are fewer resonances for large $x$, with no resonance at $x=\infty$.