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arXiv:cond-mat/9701121 · Europhysics Letters (EPL)

Sum rule for transport in a Luttinger liquid with long-range interaction in the presence of an impurity

Gianaurelio Cuniberti, Maura Sassetti, B. Krämer

Published Feb 20, 1997• 3 citations
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Abstract

We show that the non-linear dc transport in a Luttinger liquid with interaction of finite range in the presence of an impurity is governed by a sum rule which causes the charging energy to vanish.
DOI
10.1209/epl/i1997-00166-9

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The nonlinear transport in a Luttinger liquid with finite-range interaction is considered for a general time-dependent driving electric field, in the presence of an impurity. In the static limit, the nonlinear current-voltage characteristic depends only on the integral of the driving field. For a time-dependent field, a relation between the currents at different sites is derived which depends only on the nonlocal ac conductivity of the Luttinger system without impurity, but is independent of the strength and the shape of the electric field. Experimental consequences are discussed.
The nonlinear charge transfer through a tunnel junction between two Luttinger systems is studied for repulsive, finite-range interaction between electrons on the same side, V11{\mathrm{V}}_{11}V11​, and on different sides, V12{\mathrm{V}}_{12}V12​, of a junction. Features of the Coulomb blockade effect are observed if V12{\mathrm{V}}_{12}V12​=0. We predict an interaction-induced enhancement of the current if V12{\mathrm{V}}_{12}V12​>0. When V12{\mathrm{V}}_{12}V12​=V11{\mathrm{V}}_{11}V11​, the current is suppressed at small bias, but the ``charging energy,'' obtained from the asymptotic behavior at high-bias voltage, vanishes.
We study the transport properties of a Luttinger liquid in the presence of several time-dependent weak pointlike impurities. Our starting point is the bosonized form of the Luttinger liquid Hamiltonian with a potential introduced by the impurities. We find the correction to the total current due to the backscattering of the electrons by the impurities in first order of perturbation theory for a general time-dependent impurity potential. We then apply this result to a single time-dependent impurity, and show that it reproduces the result obtained by Feldman and Gefen [Phys. Rev. B 67, 115337 (2003)], in particular that the current enhancement occurs only for strong repulsive electron-electron interactions. For the case of two impurities oscillating with the same frequency, we find that the dc component of the backscattering current is positive even for weak repulsive interactions, due to the presence of the interference term induced by spatial correlations in the Luttinger liquid. We show that the impurity frequencies at which the enhancement takes place form a bandlike structure. In the case of several impurities, the bands become broader, because of interference effects.
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We show that the macroscopic current and charge can be formulated as a quantum-mechanical zero-mode problem. We find that the current is given by the velocity operator of a particle restricted to move around a circle. As an explicit example we investigated a Luttinger liquid of length L that is perturbed by a time-dependent impurity. Using the statistical mechanics of zero modes we computed the nonequilibrium current. In particular we show that in the low-temperature limit, LT/L>1,{L}_{T}/L>1,LT​/L>1, the zero-mode method introduced here becomes essential for computing the current.