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arXiv:cond-mat/0010069 · Physical review. B, Condensed matter

Friedel oscillations in a Luttinger liquid with long-range interactions

Victoria Fernández, Carlos Naón

Published Jun 8, 2001• 13 citations
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Abstract

We introduce a path-integral approach that allows us to compute charge-density oscillations in a Luttinger liquid with impurities. We obtain an explicit expression for the envelope of Friedel oscillations in the presence of arbitrary electron-electron potentials. As examples, in order to illustrate the procedure, we show how to use our formula for contact and Coulomb potentials.

DOI
10.1103/physrevb.64.033402

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We introduce a path-integral approach that allows us to compute charge-density oscillations in a Luttinger liquid with impurities. We obtain an explicit expression for the envelope of Friedel oscillations in the presence of arbitrary electron-electron potentials. As examples, in order to illustrate the procedure, we show how to use our formula for contact and Coulomb potentials.
We study the density disturbance of a correlated 1D electron liquid in the presence of a scatterer or a barrier. The 2k_F-periodic density profile away from the barrier (Friedel oscillation) is computed for arbitrary electron--electron interaction and arbitrary impurity strength. We find that in presence of correlations, the Friedel oscillation decays slower than predicted by Fermi liquid theory. In the case of a spinless Luttinger liquid characterized by an interaction constant g<1, the asymptotic decay of the Friedel oscillation is x^-g. For a weak scatterer, the decay is even slower at small-to-intermediate distances from the impurity, with a crossover to the asymptotic x^-g law.
We study Friedel oscillations in one-dimensional electron liquids for arbitrary electron-electron interaction and arbitrary impurity strength. Explicit results for spinless as well as spin- 12\frac{1}{2}21​ electrons are given. In the case of Luttinger-liquid leads, the Friedel oscillations decay as ${x}^{\ensuremath{-}g}$ far away from the impurity where ggg is the interaction constant. For a weak scatterer, a slower decay is found at small-to-intermediate distances from the impurity, with a crossover to the asymptotic ${x}^{\ensuremath{-}g}$ decay.
We use a functional integral formalism developed earlier for the pure Luttinger liquid (LL) to find an exact representation for the electron Green function of the LL in the presence of a single backscattering impurity in the low-temperature limit. This allows us to reproduce results (well known from the bosonization techniques) for the suppression of the electron local density of states (LDOS) at the position of the impurity and for the Friedel oscillations at finite temperature. In addition, we have extracted from the exact representation an analytic dependence of LDOS on the distance from the impurity and shown how it crosses over to that for the pure LL.
Using a bosonization and variational Gaussian wave-functional method we analytically calculate the Friedel oscillations induced by a single impurity in one-dimensional electron systems with long-range Coulomb interaction. It is shown that the Friedel oscillations are drastically different from those in electron systems with short-range interaction. First, in the long-range case the oscillatory part of the Friedel oscillations is not periodic as that in the short-range case. Second, in the long-range case the envelope of the Friedel oscillations decays as $\mathrm{exp}[\ensuremath{-}\mathrm{const}(\mathrm{ln}{x)}^{1/2}]$ far away from the impurity, which is much slower than the power law in the short-range case.
A single impurity in the one-dimensional Luttinger model creates a local modification of the charge density analogous to the Friedel oscillations. In this paper, we present an exact solution of the case g=1/2 (the equivalent of the Toulouse point) at any temperature T and impurity coupling, expressing the charge density in terms of a hypergeometric function. We find in particular that at T=0 the oscillatory part of the density goes as lnx at small distance and ${\mathit{x}}^{\mathrm{\ensuremath{-}}1/2}$ at large distance. \textcopyright{} 1996 The American Physical Society.