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arXiv:cond-mat/9509150 · Physical review. B, Condensed matter

Low-temperature nonequilibrium transport in a Luttinger liquid

Ulrich Weiß, Reinhold Egger, Maura Sassetti

Published Dec 15, 1995• 35 citations
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Abstract

The temperature-dependent nonlinear conductance for transport of a Luttinger liquid through a barrier is calculated in the nonperturbative regime for g=1/2-\ensuremath{\epsilon}, where g is the dimensionless interaction constant. To describe the low-energy behavior, we perform a leading-logarithmic summation of all diagrams contributing to the conductance which is valid for \ensuremath{\Vert}\ensuremath{\epsilon}\ensuremath{\Vert}\ensuremath{\ll}1. With increasing external voltage, the asymptotic low-temperature behavior displays a turnover from the ${\mathit{T}}^{2/\mathit{g}\mathrm{\ensuremath{-}}2}$ to a universal ${\mathit{T}}^{2}$ law.

DOI
10.1103/physrevb.52.16707

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The temperature-dependent nonlinear conductance for transport of a Luttinger liquid through a barrier is calculated in the nonperturbative regime for g=1/2-\ensuremath{\epsilon}, where g is the dimensionless interaction constant. To describe the low-energy behavior, we perform a leading-logarithmic summation of all diagrams contributing to the conductance which is valid for \ensuremath{\Vert}\ensuremath{\epsilon}\ensuremath{\Vert}\ensuremath{\ll}1. With increasing external voltage, the asymptotic low-temperature behavior displays a turnover from the T2/g\ensuremath−2{\mathit{T}}^{2/\mathit{g}\mathrm{\ensuremath{-}}2}T2/g\ensuremath−2 to a universal T2{\mathit{T}}^{2}T2 law.
Transport properties of a single-channel Luttinger liquid impinging on a barrier have been studied for g=1/2−εg=1/2 - εg=1/2−ε, where ggg is the dimensionless interaction constant and ∣ε∣≪1|ε| \ll 1∣ε∣≪1. The relevant diagrams contributing to the conductance are identified and evaluated in all orders. Our approach represents a leading-log summation which is valid for sufficiently low temperature and small voltage. The asymptotic low-temperature corrections exhibit a turnover from the T2/g−2T^{2/g-2}T2/g−2 behavior to a universal T2T^2T2 law as the voltage is increased.
We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g 2 , and subject to a potential barrier of arbitrary strength, as a function of temperature. We first map the Hamiltonian in the basis of scattering states into an effective low-energy Hamiltonian in current algebra form. Analyzing the perturbation theory in the fermionic representation the diagrams contributing to the renormalization group (RG) β-function are identified. A universal part of the β-function is given by a ladder series and summed to all orders in g 2 . First non-universal corrections beyond the ladder series are discussed. The RG-equation for the temperature-dependent conductance is solved analytically. Our result agrees with known limiting cases.
We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g2{g}_{2}g2​, and subject to a potential barrier of arbitrary strength, as a function of temperature. We map the Hamiltonian in the basis of scattering states into an effective low energy Hamiltonian in current algebra form. First the renormalization group (RG) equation for weak interaction is derived in the current operator language both using the operator product expansion and the equation of motion method. To access the strong coupling regime, two methods of deducing the RG equation from perturbation theory, based on the scaling hypothesis and on the Callan-Symanzik formulation, are discussed. The important role of scale-independent terms is emphasized. The latter depend on the regularization scheme used (length versus temperature cutoff). Analyzing the perturbation theory in the fermionic representation, the diagrams contributing to the renormalization group $\ensuremath{\beta}$-function are identified. A universal part of the $\ensuremath{\beta}$-function is given by a ladder series and summed to all orders in g2{g}_{2}g2​. First nonuniversal corrections beyond the ladder series are discussed and are shown to differ from the exact solutions obtained within conformal field theory which use a different regularization scheme. The RG equation for the temperature-dependent conductance is solved analytically. Our result agrees with known limiting cases.
The tunnel current of a Luttinger liquid with a finite density of strong impurities is calculated using an instanton approach. For very low temperatures T or electric fields E the (nonlinear) conductivity is of variable range hopping (VRH) type as for weak pinning. For higher temperatures or fields the conductivity shows power law behavior corresponding to a crossover from multi- to single-impurity tunneling. For even higher T and not too strong pinning there is a second crossover to weak pinning. The determination of the position of the various crossover lines both for strong and weak pinning allows the construction of the global regime diagram.
We study theoretically the transport of a one-channel Luttinger liquid through a weak link. For repulsive electron interactions, the electrons are completely reflected by even the smallest scatterer, leading to a truly insulating weak link, in striking contrast to that for noninteracting electrons. At finite temperature (T) the conductance is nonzero, and is predicted to vanish as a power of T. At T=0 power-law current-voltage characteristics are predicted. For attractive interactions, a Luttinger liquid is argued to be perfectly transmitted through even the largest of barriers. The role of Fermi-liquid leads is also explored.