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arXiv:cond-mat/9606124 · Physical review. B, Condensed matter

Exact Friedel oscillations in theg=1/2 Luttinger liquid

André LeClair, Frédéric Lesage, Hubert Saleur

Published Nov 15, 1996• 32 citations
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Abstract

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.

DOI
10.1103/physrevb.54.13597

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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 x\ensuremath−1/2{\mathit{x}}^{\mathrm{\ensuremath{-}}1/2}x\ensuremath−1/2 at large distance. \textcopyright{} 1996 The American Physical Society.
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.
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 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.
Abstract In this work, the four-point Green functions relevant to the study of Friedel oscillations are calculated for a Luttinger liquid (LL) with a cluster of impurities around an origin using the recently developed non-chiral bosonization technique (NCBT). Analytical expressions are obtained for the most singular part of the envelope of the oscillatory part of the local density in the form of power laws, providing closed expressions for the governing exponents. A comparison is done with the work of Egger and Grabert and an exact match of the exponent is obtained both for points near and far from the impurity. The two-point functions (most singular part) obtained using the same method are used to calculate the dynamical density of states (DDOS), which exhibits a power law in energy and closed analytical expressions for the DDOS exponent is calculated. These results interpolates between the weak barrier and weak link cases which are typically studied in the literature. The dependence of the DDOS on the nature of interactions and the strength of the impurity clusters are highlighted. Finally, the results of DDOS are favorably compared with existing literature such as density of states far away and near the impurity. We also compare with the often studied special case when the Luttinger parameter g = 1/2 and find agreement with our results for strong barriers.
The paramagnetic metallic phase of the one-dimensional Kondo lattice model is studied by the density matrix renormalization group method. We observe charge and spin Friedel oscillations. They reflect the long- range charge-charge and spin-spin correlation functions. The observed oscillations are consistent with a Tomonaga-Luttinger liquid. From the period of the oscillations it is concluded that the Fermi surface is large, including both the conduction electrons and the localized spins, kF{\mathit{k}}_{\mathit{F}}kF​=\ensuremath{\pi}(1+nc{\mathit{n}}_{\mathit{c}}nc​)/2, where nc{\mathit{n}}_{\mathit{c}}nc​ is the density of conduction electrons. \textcopyright{} 1996 The American Physical Society.