Open ends of thermal transport in a Luttinger liquid: interaction-modified Lorenz numbers, microscopic weak-link parameters, and their experimental test

Kane and Fisher's analysis of thermal transport in a Luttinger liquid (Phys. Rev. Lett. 76, 3192 (1996)) established that the Wiedemann-Franz law fails in one-dimensional interacting electron systems: for a single impurity or weak link the low-temperature Lorenz number is $3L_0/(2g+g^2)$ for $1/2<g<1$, nonuniversal at $g=1/2$, and divergent as $L\sim a^2T^{4-2/g}$ for $g<1/2$, where $a$ is the amplitude of a junction-local density-density perturbation. The analysis was deliberately restricted to an ideal harmonic, SU(2)-symmetric, single-impurity model, leaving explicitly unquantified: (i) the anharmonic and Umklapp interaction contributions to the thermal conductance of translationally invariant nonchiral wires, (ii) the microscopic value of the junction parameter $a$ for realistic weak links, (iii) the temperature windows (set by $\hbar v/\ell$, interchain tunneling $t_\perp$, and three-dimensional crossovers) where the dilute-weak-link power laws apply in real materials, (iv) a direct comparison of the predicted interaction-modified Lorenz numbers with experiment, and (v) spin-anisotropic and spin-flip generalizations of the spinful results. Later literature has developed a substantial theory of interaction-induced thermal relaxation in Luttinger liquids (disorder, Umklapp, plasmon decay, hydrodynamics) and has measured electron thermal conductance in ballistic one-dimensional constrictions and quantized thermal conductance of chiral quantum Hall edges, but no microscopic determination of $a$ and no experimental test of the nonuniversal or divergent weak-link Lorenz numbers is known to date. The problem is therefore open, with its unresolved core in the microscopic junction parameters, the experimental confirmation, and the material-specific crossover windows.

Background

In an ordinary metal the Wiedemann-Franz law states that the electronic thermal conductivity $\kappa$ is proportional to the electrical conductivity $\sigma$, with a universal Lorenz number $L=\kappa/(\sigma T)=L_0=(\pi^2/3)(k_B/e)^2$; this universality is a hallmark of the Fermi liquid. In one spatial dimension, interacting electrons do not form a Fermi liquid: the ground state is a Luttinger liquid, a critical state described by bosonic collective density modes (obtained by bosonization, the exact rewriting of the 1d electron gas in terms of density fields $n_\pm$ for right- and left-movers) rather than electron quasiparticles. A spinless Luttinger liquid is characterized by a single dimensionless parameter $g$ (the ideal-channel conductance in units of $e^2/h$): $g=1$ for noninteracting electrons and $g<1$ for repulsive interactions. Kane and Fisher (Phys. Rev. Lett. 76, 3192 (1996)) analyzed thermal conductance $K$ and thermopower in this model. For a pure, chiral channel — the situation of edge states of the fractional quantum Hall effect (two-dimensional electron systems at rational Landau-level filling, whose gapless edge modes propagate in only one direction and so form chiral Luttinger liquids) — the heat current is conserved and the thermal conductance is quantized, $K=(\pi^2/3)k_B^2T/h$ per channel. For a single impurity they modeled the weak link in two complementary ways: a weak backscatterer $-t_B\cos(\phi_+-\phi_-)\delta(x)$, or, in the strong-backscattering limit, a tunnel junction $-t\cos((\phi_+-\phi_-)/g)\delta(x)$ between two semi-infinite Luttinger liquids (the Kane-Fisher impurity model, introduced in Phys. Rev. B 46, 15233 (1992)); here $\phi_\pm$ are the bosonic phase fields conjugate to the densities, $n_\pm=\pm\partial_x\phi_\pm/2\pi$. By factorizing the thermal current operator into commuting even-density and odd-current pieces they derived an exact relation between the thermal conductance through the link and the frequency-dependent electrical conductance, from which the low-temperature Lorenz number follows: $L=3L_0/(2g+g^2)$ for $1/2<g<1$ (larger than $L_0$ for repulsive interactions), and for $g<1/2$ a Lorenz number that diverges as $T\rightarrow 0$. The divergence is driven by a junction-local density-density perturbation $\mathcal{H}_{\mathrm{pert}}=a\,\delta(x)\,n_+n_-$, an operator that transfers energy but not charge across the link, so that heat is transported much more readily than charge; its amplitude $a$ is nonuniversal (junction-specific). At the solvable point $g=1/2$ (where the ac conductance is known exactly, by the method Fendley, Ludwig and Saleur developed for the $\nu=1/3$ fractional quantum Hall point contact) both electron tunneling and this interaction term scale as $T^3$, so the Lorenz number is nonuniversal but bounded below by $12L_0/5$. For spinful electrons with SU(2) spin symmetry (invariance under rotations in spin space), the same analysis in terms of a charge parameter $g_\rho$ (equal to 2 without interactions) predicts a crossover from $L_0(2/g_\rho)$ to $3L_0(g_\rho+2)^2/(8g_\rho(g_\rho+1))$, with a low-temperature divergence $L\sim T^{3-2/g_\rho}$ for $g_\rho<2/3$. The paper also proposed a many-impurity picture for quasi-one-dimensional conductors such as conducting polymers — dilute weak links between one-dimensional chains — in which $\sigma\sim T^{2/g-2}$, $\kappa\sim T^{2/g-1}$, and the Lorenz number is universal and enhanced, provided tunneling across successive links is incoherent ($k_BT>\hbar v/\ell$, with $v$ the Fermi velocity and $\ell$ the link spacing) and the temperature exceeds the interchain tunneling rate $t_\perp$ so that transport stays effectively one-dimensional. What the 1996 analysis explicitly did not do is: quantify the anharmonic bulk interactions (terms such as $n_+^2n_-$ and Umklapp processes, in which two scattering particles transfer momentum to the lattice and which therefore relax thermal current even in a translationally invariant nonchiral wire, though they freeze out at low temperature) that couple right- and left-moving modes and could modify $K$; determine $a$ microscopically; analyze the crossover regimes and three-dimensional crossovers; compare with experiment; and go beyond SU(2) symmetry in the spinful case.

Problem Statement

Complete the theory of thermal transport in Luttinger liquids beyond the ideal harmonic, SU(2)-symmetric, single-impurity model of Kane and Fisher, and subject it to experimental test. Concretely, for the one-dimensional interacting electron gas described by a Luttinger parameter $g$ (spinful: charge parameter $g_\rho$): (i) compute, non-perturbatively where needed and for nonchiral (right- and left-moving) wires, the magnitude and temperature dependence of the interaction-induced contributions to the thermal conductance $K$ from anharmonic bulk couplings such as $n_+^2n_-$ and from Umklapp processes, including the regimes where Umklapp is active, determining when these contributions are negligible compared to impurity backscattering and when they dominate; (ii) determine microscopically — from an underlying lattice/electronic model or ab initio input for a realistic weak link or tunnel junction — the amplitude $a$ of the local density-density perturbation $\mathcal{H}_{\mathrm{pert}}=a\,\delta(x)\,n_+n_-$, thereby fixing the predicted nonuniversal low-temperature Lorenz number at $g=1/2$ and the coefficient of the divergence $L\sim a^2T^{4-2/g}$ for $g<1/2$; (iii) establish the temperature windows in real materials — the conditions $k_BT>\hbar v/\ell$ for incoherent successive weak-link tunneling and $T>t_\perp$ for effective one-dimensionality, together with the associated three-dimensional crossovers — within which the dilute-weak-link power laws $\sigma\sim T^{2/g-2}$, $\kappa\sim T^{2/g-1}$ and the interaction-modified Lorenz number actually apply; (iv) perform the direct comparison between the predicted interaction-modified Lorenz numbers and the available thermal-transport measurements in quantum wires, quantum Hall edge states, or conducting polymer systems, and state what measurement would provide a direct test of the nonuniversal ($g=1/2$) and divergent ($g<1/2$) weak-link Lorenz numbers; and (v) extend the spinful results beyond the SU(2)-symmetric case to spin-anisotropic interactions and spin-flip scattering at impurities, determining how the $g_\rho$ thresholds (crossover at $g_\rho=2/3$) and scaling exponents are modified. These five components jointly constitute a single scientific objective — turning the 1996 universal-exponent predictions for interaction-modified thermal transport into quantitative, material-specific, and experimentally testable statements.

References

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