Transport in Luttinger Liquids

We compute the transport properties of one dimensional interacting electrons, also known as a Luttinger liquid. We show that a renormalization group study allows to obtain the temperature dependence of the conductivity in an intermediate temperature range. In this range the conductivity has a power-law like dependence in temperature. At low temperatures, the motion proceed by tunnelling between localized configurations. We compute this tunnelling rate using a bosonization representation and an instanton technique. We find a conductivity $\sigma(T) \propto e^{-\beta^{1/2}}$, where $\beta$ is the temperature. We compare this results with the standard variable range hopping (VRH) formula.