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.
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Garg et al. (2009) studied this question.
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