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arXiv:cond-mat/9608112 · Journal of Physics A Mathematical and General

Form-factors computation of Friedel oscillations in Luttinger liquids

Frédéric Lesage, Hubert Saleur

Published Jul 21, 1997• 19 citations
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Abstract

We show how the form-factors approach can be used to compute correlators of operators with non-trivial anomalous dimension in quantum impurity problems. This is done by a series of regularizations that cure completely the infrared divergencies, without spoiling the exactedness of the method. As an application, we compute for the `Friedel oscillations' of charge density induced by a single impurity in a one-dimensional Luttinger liquid of spinless electrons.

DOI
10.1088/0305-4470/30/14/003

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We show how the form-factors approach can be used to compute correlators of operators with non-trivial anomalous dimension in quantum impurity problems. This is done by a series of regularizations that cure completely the infrared divergencies, without spoiling the exactedness of the method. As an application, we compute for the `Friedel oscillations' of charge density induced by a single impurity in a one-dimensional Luttinger liquid of spinless electrons.
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.
We present an analytical study of impurity-induced density correlations in interacting one-dimensional Luttinger liquids using the Unified Field Bosonization Technique (UFBT). Unlike conventional approaches that treat impurities perturbatively, UFBT incorporates localized impurities exactly at the free-fermion level while treating interactions non-perturbatively. Closed analytical expressions for fermionic Green functions and density-density correlation functions are obtained for strongly inhomogeneous systems, enabling a unified description of the impurity-induced 2kF2k_F2kF​ Friedel oscillations and the 4kF4k_F4kF​ Wigner-crystal like correlations. The corresponding scaling behaviour is derived analytically, revealing a crossover in the dominant long-distance correlations at g=1/3g=1/3g=1/3: Friedel correlations dominate for g>1/3g>1/3g>1/3, whereas Wigner-crystal like correlations become dominant for g<1/3g<1/3g<1/3, corresponding to the regime of sufficiently strong repulsive interactions. A reflection-coupled harmonic analysis is developed to account for the left-right mixing induced by impurity backscattering and to consistently reproduce the resulting density correlations. The spinless limit is also obtained as a direct reduction of the spinful formulation. These results provide an analytically controlled framework for describing impurity-induced and interaction-driven density correlations in strongly inhomogeneous one-dimensional quantum systems.
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.
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.