Large violation of the Wiedemann-Franz law in Luttinger liquids

We show that in weakly disordered Luttinger liquids close to a commensurate filling the ratio of thermal conductivity kappa and electrical conductivity sigma can deviate strongly from the Wiedemann-Franz law valid for Fermi liquids scattering from impurities. In the regime where the umklapp scattering rate Gamma(U) is much larger than the impurity scattering rate Gamma(imp), the Lorenz number L = kappa/(sigmaT) rapidly changes from very large values L approximately Gamma(U)/Gamma(imp) >> 1 at the commensurate point to very small values L approximately Gamma(imp)/Gamma(U) << 1 for a slightly doped system. This surprising behavior is a consequence of approximate symmetries existing even in the presence of strong umklapp scattering.