Large Violation of the Wiedemann-Franz Law in Luttinger Liquids

Abstract

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