Frequency-Temperature Crossover in the Conductivity of Disordered Luttinger Liquids

The temperature ($T$) and frequency ($\omega$) dependent conductivity of weakly disordered Luttinger liquids is calculated in a systematic way both by perturbation theory and from a finite temperature renormalization group (RG) treatment to leading order in the disorder strength. Whereas perturbation theory results in $\omega \mathrm{/}T$ scaling of the conductivity such scaling is violated in the RG traetment. We also determine the non-linear field dependence of the conductivity, whose power law scaling is different from that of temperature and frequency dependence.