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arXiv:cond-mat/0010498 · Physical review. B, Condensed matter

Transport in a Luttinger liquid in the presence of a time-dependent impurity

D. Schmeltzer

Published Mar 13, 2001• 17 citations
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Abstract

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, ${L}_{T}/L>1,$ the zero-mode method introduced here becomes essential for computing the current.

DOI
10.1103/physrevb.63.125332

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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.
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.
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.
Tunneling between two Luttinger liquids driven by a time-dependent field with a frequency fff is investigated using the zero-mode bosonization. We show that inclusion of the zero modes is essential in order to obtain correct results in the limit ${L}_{T}/L\ensuremath{\gg}1$ ( LLL is the channel length and LT{L}_{T}LT​ is the thermal length). We find that the tunneling current is quantized in units of `` ef\mathrm{ef}ef'' and takes the form $I\ensuremath{\sim}\mathrm{ef}\ensuremath{\Sigma}{n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}\ensuremath{\delta}[{\ensuremath{\mu}}_{F}\ensuremath{-}\frac{2\ensuremath{\pi}}{L}v\ensuremath{\Elzxh}(n\ensuremath{-}\frac{1}{2})]$.
We study theoretically the transport properties of an interacting quantum wire in the presence of a time-dependent point impurity. Using the Keldysh formalism within the framework of Luttinger liquid model, we calculate and analyze the dc contribution of the backscattered current and the shot noise of the system with arbitrary Luttinger interaction strength and impurity position. From some numerical examples, we find that the time-dependent point impurity in the wire can enhance the total current in the strongly interaction regime, and the total current enhancement is dependent on both the impurity potential frequency and the equivalent frequency in relation to the drain-source voltage VVV. The dc contribution of the backscattered current oscillates as a function of impurity frequency with the period depending on the Luttinger interaction strength. Furthermore, the oscillation of the dc contribution of the backscattered current is not a finite-length effect, which is different from that in the case of system with a static point impurity, and the oscillation characteristic is also connected with VVV. However, the final current-noise relation is similar to that in the case of a static point impurity.
We study theoretically the transport through a single impurity in a one-channel Luttinger liquid coupled to a dissipative (Ohmic) bath. For nonzero dissipation, the single impurity is always a relevant perturbation which suppresses transport strongly. At zero temperature, the current voltage relation of the link is $I\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}{E}_{0}/eV)$, where ${E}_{0}\ensuremath{\sim}\ensuremath{\eta}/\ensuremath{\kappa}$ and $\ensuremath{\kappa}$ denotes the compressibility. At nonzero temperature TTT, the linear conductance is proportional to $\mathrm{exp}(\ensuremath{-}\sqrt{\mathcal{C}{E}_{0}/{k}_{B}T})$. The decay of Friedel oscillation saturates for a distance larger than ${L}_{\ensuremath{\eta}}\ensuremath{\sim}1/\ensuremath{\eta}$ from the impurity.