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/(sigma T) rapidly changes from very large values L similar to Gamma(U)/Gamma(imp)1 at the commensurate point to very small values L similar to 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.