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arXiv:cond-mat/9601117 · Physical review. B, Condensed matter

Nonequilibrium dc noise in a Luttinger liquid with an impurity

Paul Fendley, Hubert Saleur

Published Oct 15, 1996• 81 citations
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Abstract

We compute exactly the nonequilibrium dc noise in a Luttinger liquid with an impurity and an applied voltage. By generalizing Landauer transport theory for Fermi liquids to interacting, integrable systems, we relate this noise to the density fluctuations of quasiparticles. We then show how to compute these fluctuations using the Bethe ansatz. The nontrivial density correlations from the interactions result in a substantial part of the nonequilibrium noise. The final result for the noise is a scaling function of the voltage, temperature, and impurity coupling. It may eventually be observable in tunneling between edges of a fractional quantum Hall effect device. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.10845

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We compute exactly the nonequilibrium dc noise in a Luttinger liquid with an impurity and an applied voltage. By generalizing Landauer transport theory for Fermi liquids to interacting, integrable systems, we relate this noise to the density fluctuations of quasiparticles. We then show how to compute these fluctuations using the Bethe ansatz. The nontrivial density correlations from the interactions result in a substantial part of the nonequilibrium noise. The final result for the noise is a scaling function of the voltage, temperature, and impurity coupling. It may eventually be observable in tunneling between edges of a fractional quantum Hall effect device. \textcopyright{} 1996 The American Physical Society.
A point contact in a Luttinger liquid couples left- and right-moving channels, producing shot noise. We calculate exactly the dc shot noise at zero temperature in the out-of-equilibrium steady state where current is flowing. Integrability ensures the existence of a quasiparticle basis where scattering is ``one by one'' off the point contact, enabling us to generalize the Landauer approach to shot noise to this interacting model. Experiments in a fractional quantum Hall effect device should test our results, providing a clear signal of the fractional charge of the Laughlin quasiparticles.
We study nonequilibrium noise in chiral Luttinger liquids using the Landauer-Buttiker scattering approach, obtaining the current and voltage noise spectrum for a four-terminal measurement scheme. Experimental consequences of the tunneling of charges are present in the four-terminal measurement of both the low-frequency shot noise (\ensuremath{\omega} near 0), and the high-frequency Josephson noise (\ensuremath{\omega} near ${\mathrm{\ensuremath{\omega}}}_{\mathit{J}}$=e∗{\mathit{e}}^{\mathrm{*}}e∗V/\ensuremath{\Elzxh}). Within perturbation theory, an algebraic singularity is present at the Josephson frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{J}}$=e∗{\mathit{e}}^{\mathrm{*}}e∗V/\ensuremath{\Elzxh}, whose position depends on the charge e∗{\mathit{e}}^{\mathrm{*}}e∗ of the tunneling particles, either electrons or fractionally charged quasiparticles. These two types of tunneling are related by a strong-weak--coupling duality transformation. We show in a nonperturbative calculation for an exactly solvable point that the singularity at the quasiparticle frequency exists only in the limit of vanishing coupling, whereas the singularity at the electron frequency is present for all coupling strengths. The vanishing coupling limit corresponds to perfectly quantized Hall conductance in the case of quasiparticle tunneling between edge states in the fractional quantum Hall regime, and thus tunneling destroys the singularity at the quasiparticle frequency concomitantly with the quantized current. \textcopyright{} 1996 The American Physical Society.
We study non-equilibrium noise in Chiral Luttinger Liquids using the Landauer-Buttiker Scattering approach, obtaining the current/voltage noise spectrum for a four-terminal measurement scheme. Experimental consequences of the tunneling of charges are present in the four-terminal measurement of both the low-frequency shot noise (ωωω near 0), and the high-frequency Josephson noise (ωωω near ωJ=e∗V/ℏω_J=e^*V/\hbarωJ​=e∗V/ℏ). Within perturbation theory, an algebraic singularity is present (to all orders) at the Josephson frequency ωJ=e∗V/ℏω_J=e^*V/\hbarωJ​=e∗V/ℏ, whose position depends on the charge e∗e^*e∗ of the tunneling particles, either electrons or fractionally charged quasiparticles. We show in a non-perturbative calculation for an exactly solvable point that the singularity at the quasiparticle frequency exists only in the limit of vanishing coupling, whereas the singularity at the electron frequency is present for all coupling strengths. The vanishing coupling limit corresponds to perfectly quantized Hall conductance in the case of quasiparticle tunneling between edge states in the fractional quantum Hall regime, and thus tunneling destroys the singularity at the quasiparticle frequency concomitantly with the quantized current.
A Luttinger liquid coupled to a quantum impurity describes a large number of physical systems. The Hamiltonian consists of left- and right-moving fermions interacting among themselves via a density-density coupling and scattering off a localized transmitting and reflecting impurity. We solve exactly the Hamiltonian by means of an incoming-outgoing scattering Bethe basis which properly incorporates all scattering processes. A related model, the weak-tunneling model, wherein the impurity is replaced by a tunnel junction, is solved by the same method. The consistency of the construction is established through a generalized Yang-Baxter relation. Periodic boundary conditions are imposed and the resulting Bethe ansatz equations are derived by means of the off-diagonal Bethe ansatz approach. We derive the spectrum of the model for all coupling constant regimes and calculate the impurity free energy. We discuss the low energy behavior of the systems for both repulsive and attractive interactions.
The quantum physics of one dimensional systems exhibits many remarkable and exotic physical phenomena. These are the result of the strong correlations which exist in such systems due to the reduced dimensionality and in many cases can be described by Luttinger Liquid theory. Likewise quantum impurity systems, where a bath of particles couples to the same impurity, exhibit well known non perturbative phenomena like the Kondo effect. In this thesis we study the physics of quantum impurities in interacting environments - the intersection of these two areas. Such Luttinger-impurity systems can be realized experimentally in a range of different settings from quantum wires and carbon nano tubes to edges of fractional quantum hall systems and cold atom gases. We show that many of the models used to describe these experiments are integrable and can be studied by means of the Bethe Ansatz. This powerful and exact method provides us with the exact eigenstates and energy levels of the model Hamiltonian. The models considered include two which describe a Lutttinger liquid coupled to a back scattering impurity, the Kane-Fisher and weak tunnelling model as well as Luttinger liquids coupled a quantum dot in various geometries: at the boundary, sidecoupled and embedded. To incorporate both the backscattering nature of the impurities and the interacting bulk particles a new formulation of the Bethe Ansatz is developed and implemented. The eigenstates and spectra of the models are found and used to investigate their ground state and thermodynamic properties.
The conductance for tunneling through an impurity in a Luttinger liquid is described by a universal scaling function. We compute this scaling function exactly, by using the thermodynamic Bethe ansatz and a kinetic (Boltzmann) equation. This model has been proposed to describe resonant tunneling through a point contact between two $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3$ quantum Hall edges. Recent experiments on quantum Hall devices agree well with our exact results. We also derive the exact conductance and I(V)I\left(V\right)I(V) curve, out of equilibrium, in this fully interacting system.