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 κ and electrical conductivity σ can deviate strongly from the Wiedemann-Franz law valid for Fermi liquids scattering from impurities. In the regime where the umklapp scattering rate ΓU is much larger than the impurity scattering rate Γimp, the Lorenz number L=κ/(σT) rapidly changes from very large values L∼ΓU/Γimp1 at the commensurate point to very small values L∼Γimp/ΓU1 for a slightly doped system. This surprising behavior is a consequence of approximate symmetries existing even in the presence of strong umklapp scattering.