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Journal of Physics Conference Series

Impurity Scattering in Luttinger Liquid with Electron-Phonon Coupling

Alexey Galda, I. V. Yurkevich, Igor V. Lerner

Published Mar 1, 2011• 1 citation
Publisher

Abstract

We study the influence of electron-phonon coupling on electron transport through a Luttinger liquid with an embedded weak scatterer or weak link. We derive the renormalization group (RG) equations which indicate that the directions of RG flows can change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. This results in the rich phase diagram with up to three fixed points: an unstable one with a finite value of conductance and two stable ones, corresponding to an ideal metal or insulator.

DOI
10.1088/1742-6596/286/1/012049

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We study the influence of electron-phonon coupling on electron transport through a Luttinger liquid with an embedded weak scatterer or weak link. We derive the renormalization group (RG) equations which indicate that the directions of RG flows can change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. This results in the rich phase diagram with up to three fixed points: an unstable one with a finite value of conductance and two stable ones, corresponding to an ideal metal or insulator.
We study the influence of electron-phonon coupling on electron transport through a Luttinger liquid with an embedded weak scatterer or weak link. We derive the renormalization group (RG) equations, which indicate that the directions of RG flows can change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. This results in a rich phase diagram with up to three fixed points: an unstable one with a finite value of conductance and two stable ones, corresponding to an ideal metal or insulator.
This thesis addresses a theoretical study of the problem of a single impurity embedded in a one-dimensional system of interacting electrons in presence of electron-phonon coupling. First we consider a system with a featureless point-like potential impurity, followed by the case of a resonant level hybridised with a Luttinger Liquid. The stress is made on a more fundamental problem of a featureless scatterer, for which two opposite limits in the impurity strength are considered: a weak scatterer and a weak link. We have found that, regardless of the transmission properties of phonons through the impurity, the scaling dimensions of the conductance in these limits obey the duality condition, \\( \\triangle_{WS}\\) \\( \\triangle_{WL}\\) = 1, known for the Luttinger Liquid in the absence of phonons. However, in the case when the strength of phonon scattering is correlated with electron scattering by the impurity, we find a nontrivial phase diagram with up to three fixed points and a possibility of a metal-insulator transition. We also consider the case of a weakly interacting electron-phonon system in the presence of a single impurity of an arbitrary scattering potential. In the problem of a resonant level attached to the Luttinger Liquid we show that the electron-phonon coupling significantly modifies the effective energy-dependent width of the resonant level in two different geometries, corresponding to the resonant and anti-resonant transmission in the Fermi gas.
We study electronic transport in a Luttinger liquid with an embedded impurity, which is either a weak scatterer (WS) or a weak link (WL), when interacting electrons are coupled to one-dimensional massless bosons (e.g., acoustic phonons). We find that the duality relation, ΔWSΔWL=1, between scaling dimensions of the electron backscattering in the WS and WL limits, established for the standard Luttinger liquid, holds in the presence of the additional coupling for an arbitrary fixed strength of boson scattering from the impurity. This means that at low temperatures such a system remains either an ideal insulator or an ideal metal, regardless of the scattering strength. On the other hand, when fermion and boson scattering from the impurity are correlated, the system has a rich phase diagram that includes a metal-insulator transition at some intermediate values of the scattering.
We have studied one dimensional acoustic phonon linearly coupled to the electron density. The phonon spectrum is linear with cutoff at the Debye frequency, it is straightforward modification of three dimensional optical phonons. Phonon field results in substituting the dynamical coupling for the screened coulomb interaction. We have shown that the electron-phonon coupling quantitatively changed the phase diagram of Luttinger liquid with a single impurity. The existence of slow and fast polaron modes with different weights in different regimes resulted in different renormalization group flow for a weak scatterer and a weak tunneling link. The resulting phase diagram depending on the parameters of the problem, regimes corresponding to purely metallic or purely insulating behavior and an intermediate regime with two stable fixed points and one unstable, finite conductance fixed point. we have found that the direction of renormalization group flow change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. The obtained results were found in good agreement with previously obtained results.