Hopping conductivity in one dimension with asymmetric transfer rates

The frequency-dependent conductivity is calculated for a one-dimensional disordered system in the presence of an external biased electric field. Four classes of transfer-rate distributions are discussed. The singular distribution for which the inverse first moment does not exist gives the most interesting nonanalytic behavior. For the class-(a) distribution such that all inverse moments exist, numerical results are in complete agreement with the analytic expansions for weak disorder, as discussed in our previous work. For the class-(c) distribution such that no inverse moments exist, one finds nonuniversal crossover behavior as well as nonanalytic frequency dependence of the ac conductivity. We find a crossover region ${\omega }_1$≤ω≤${\omega }_2$ such that for ω<${\omega }_1$, the conductivity σ∼${\omega }^{\alpha }$ (0<α<1), and for ω>${\omega }_2$, σ∼${\omega }^{\alpha /\left(2\mathrm{}\mathrm{-}\alpha \right)}$ with behavior the same as for the unbiased case. For the class-(b’) distribution such that only one inverse moment exists, nonanalytic leading corrections in the conductivity are also obtained. Finally, for the bond-percolation model, we find no crossover behavior in the frequency-dependent conductivity even in the strong biased case.