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arXiv:cond-mat/9509068 · Physical Review Letters

Thermal Transport in a Luttinger Liquid

C. L. Kane, Matthew P. A. Fisher

Published Apr 22, 1996• 194 citations
arXivHTMLPDF

Abstract

We study thermal transport in a one-dimensional (1D) interacting electron gas, employing the Luttinger liquid model. Both thermal conductance and thermopower are analyzed for a pure 1D gas and with impurities. The universal ratio of electrical to thermal conductance in a Fermi liquid---the Wiedemann-Franz law---is modified, whereas the thermopower is still linear in temperature. For a single impurity the Lorentz number is given by $L(T\ensuremath{\rightarrow}0){\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3L}_{0}/({2g+g}^{2})$---with ${L}_{0}$ the Fermi liquid value---and the conductance $1/2<g<1$. For $g<1/2$ the Lorentz number diverges as $T\ensuremath{\rightarrow}0$. Possible relevance to thermal transport in conducting polymer systems is discussed.

DOI
10.1103/physrevlett.76.3192

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We study thermal transport in a one-dimensional (1D) interacting electron gas, employing the Luttinger liquid model. Both thermal conductance and thermopower are analyzed for a pure 1D gas and with impurities. The universal ratio of electrical to thermal conductance in a Fermi liquid---the Wiedemann-Franz law---is modified, whereas the thermopower is still linear in temperature. For a single impurity the Lorentz number is given by L(T\ensuremath→0)=3L0/(2g+g2)L(T\ensuremath{\rightarrow}0){\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3L}_{0}/({2g+g}^{2})L(T\ensuremath→0)=3L0​/(2g+g2)---with L0{L}_{0}L0​ the Fermi liquid value---and the conductance 1/2<g<11/2<g<11/2<g<1. For g<1/2g<1/2g<1/2 the Lorentz number diverges as T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0. Possible relevance to thermal transport in conducting polymer systems is discussed.
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.
Thermal transport through one-dimensional systems of strongly interacting electrons is studied in two cases: (i) a finite Luttinger liquid (LL) wire containing a strong impurity, and (ii) a LL ring connected by quantum wires to noninteracting electron reservoirs. It is shown that, for (i), a temperature-induced resonant transport takes place under certain conditions. In case (ii) the heat conductance under resonance conditions does not depend on the electron-electron interaction and the system thermal conductance coincides with the maximal Fermi-liquid thermal value.
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.
The thermopower of a Luttinger liquid (LL), originating from the energy dispersion of electrons at the Fermi level and/or from the backscattering of electrons by impurities, is analytically evaluated. It is shown that in both cases the thermopower is described by a corresponding Fermi-liquid formula renormalized by an interaction-dependent factor. For an infinite LL the renormalization coefficients decrease with an increase of the electron-electron interaction. In a realistic situation, when a LL wire is connected to leads of noninteracting electrons, the dispersion-induced thermopower in the limit of strong repulsive interaction is strongly suppressed, ${S}_{W}^{(d)}\ensuremath{\sim}{g}^{2}{S}_{0}^{(d)}\ensuremath{\ll}{S}_{0}^{(d)}$ (here S0(d){S}_{0}^{(d)}S0(d)​ is the corresponding Fermi-liquid value for the thermopower and $g\ensuremath{\ll}1$ is the LL correlation parameter), while the impurity-induced thermopower ${S}_{W}^{(i)}\ensuremath{\sim}{S}_{0}^{(i)}/g$ is enhanced by the interelectron interaction.
Properties of one-dimensional (1D) transport strongly depend on the proper choice of boundary conditions. It has been frequently stated that the Luttinger-liquid (LL) conductance is renormalized by the interaction as ge2{ge}^{2}ge2/h. To contest this result I develop a model of 1D LL wire with the interaction switching off at the infinities. Its solution shows that there is no renormalization of the universal conductance while the electrons have a free behavior in the source and drain reservoirs.
We present a detailed study of thermal transport in the disordered Fermi liquid with short-range interactions. At temperatures smaller than the impurity scattering rate, i.e., in the diffusive regime, thermal conductivity acquires nonanalytic quantum corrections. When these quantum corrections become large at low temperatures, the calculation of thermal conductivity demands a theoretical approach that treats disorder and interactions on an equal footing. In this paper, we develop such an approach by merging Luttinger's idea of using gravitational potentials for the analysis of thermal phenomena with a renormalization group calculation based on the Keldysh nonlinear sigma model. The gravitational potentials are introduced in the action as auxiliary sources that couple to the heat density. These sources are a convenient tool for generating expressions for the heat density and its correlation function from the partition function. Already in the absence of the gravitational potentials, the nonlinear sigma model contains several temperature-dependent renormalization group charges. When the gravitational potentials are introduced into the model, they acquire an independent renormalization group flow. We show that this flow preserves the phenomenological form of the correlation function, reflecting its relation to the specific heat and the constraints imposed by energy conservation. The main result of our analysis is that the Wiedemann-Franz law holds down to the lowest temperatures even in the presence of disorder and interactions and despite the quantum corrections that arise for both the electric and thermal conductivities.