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arXiv:0903.3054 · Physical Review Letters

Large Violation of the Wiedemann-Franz Law in Luttinger Liquids

Arti Garg, David Rasch, Efrat Shimshoni, Achim Rosch

Published Aug 24, 2009• 63 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevlett.103.096402

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We show that in weakly disordered Luttinger liquids close to a commensurate filling the ratio of thermal conductivity \ensuremathκ\ensuremath{\kappa}\ensuremathκ and electrical conductivity \ensuremathσ\ensuremath{\sigma}\ensuremathσ can deviate strongly from the Wiedemann-Franz law valid for Fermi liquids scattering from impurities. In the regime where the umklapp scattering rate \ensuremathΓU{\ensuremath{\Gamma}}_{U}\ensuremathΓU​ is much larger than the impurity scattering rate \ensuremathΓimp{\ensuremath{\Gamma}}_{\mathrm{imp}}\ensuremathΓimp​, the Lorenz number L=\ensuremathκ/(\ensuremathσT)L=\ensuremath{\kappa}/(\ensuremath{\sigma}T)L=\ensuremathκ/(\ensuremathσT) rapidly changes from very large values L\ensuremath∼\ensuremathΓU/\ensuremathΓimp\ensuremath≫1L\ensuremath{\sim}{\ensuremath{\Gamma}}_{U}/{\ensuremath{\Gamma}}_{\mathrm{imp}}\ensuremath{\gg}1L\ensuremath∼\ensuremathΓU​/\ensuremathΓimp​\ensuremath≫1 at the commensurate point to very small values L\ensuremath∼\ensuremathΓimp/\ensuremathΓU\ensuremath≪1L\ensuremath{\sim}{\ensuremath{\Gamma}}_{\mathrm{imp}}/{\ensuremath{\Gamma}}_{U}\ensuremath{\ll}1L\ensuremath∼\ensuremathΓimp​/\ensuremathΓU​\ensuremath≪1 for a slightly doped system. This surprising behavior is a consequence of approximate symmetries existing even in the presence of strong umklapp scattering.
We consider heat transport in a Luttinger liquid (LL) with weak disorder and study the Lorenz number for this system. We start at a high- T regime, and calculate both the electrical and thermal conductivities using a memory function approach. The resulting Lorenz number L is independent of T but depends explicitly on the LL exponents. Lowering T , however, allows for a renormalization of the LL exponents from their bare values by disorder, causing a violation of the Wiedemann-Franz law. Finally, we extend the discussion to quantum wire systems and study the wire size dependence of the Lorenz number.
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Many metallic quantum materials display anomalous transport phenomena that defy a Fermi liquid description. Here, we use numerical methods to calculate thermal and charge transport in the doped Hubbard model and observe a crossover separating high- and low-temperature behaviors. Distinct from the behavior at high temperatures, the Lorenz number [Formula: see text] becomes weakly doping dependent and less sensitive to parameters at low temperatures. At the lowest numerically accessible temperatures, [Formula: see text] roughly approaches the Wiedemann-Franz constant [Formula: see text], even in a doped Mott insulator that lacks well-defined quasiparticles. Decomposing the energy current operator indicates a compensation between kinetic and potential contributions, which may help to clarify the interpretation of transport experiments beyond Boltzmann theory in strongly correlated metals.
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We study the electrical and thermal transport through the SU(N)(N)(N) Wolff model with the use of bosonization. The Wilson ratio reaches unity as NNN grows to infinity. The electric conductance is dominated by the charge channel, and decreases monotonically with increasing interaction. The thermal conductivity enhances in the presence of a local Hubbard UUU. The Wiedemann-Franz law is violated; the Lorentz number depends strongly on the interaction parameter, which can be regarded as a manifestation of spin-charge separation.