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Physical review. B, Condensed matter

Friedel oscillations in one-dimensional fermion systems with long-range Coulomb interaction

Qingshan Yuan, Hong Chen, Yumei Zhang, Yuguang Chen

Published Jul 15, 1998• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.58.1084

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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.
Using an asymptotic phase representation of the particle density operator in the one-dimensional harmonic trap, the part which describes the Friedel oscillations is extracted. The expectation value with respect to the interacting ground state requires the calculation of the mean square average of a properly defined phase operator. This calculation is performed analytically for the Tomonaga–Luttinger model with harmonic confinement. It is found that the envelope of the Friedel oscillations at zero temperature decays with the boundary exponent ν = ( K +1)/2 away from the classical boundaries. This value differs from that known for open boundary conditions or strong pinning impurities. The soft boundary in the present case thus modifies the decay of Friedel oscillations. The case of two components is also discussed.
The Friedel oscillations caused by an impurity located at one edge of a disordered interacting quantum wire are calculated numerically. The electron density in the system's ground state is determined using the density-matrix renormalization-group method, and the Friedel oscillation data are extracted using the density difference between the case in which the wire is coupled to an impurity and the case where the impurity is uncoupled. We show that the power-law decay of the oscillations occurring for an interacting clean one-dimensional sample described by the Luttinger liquid theory is multiplied by an exponential decay term due to the disorder. Scaling of the average Friedel oscillations by this exponential term collapses the disordered sample data on the clean results. We show that the length scale governing the exponential decay may be associated with the Anderson localization length and thus be used as a convenient way to determine the dependence of the localization length on disorder and interactions. The localization length decreases as a function of the interaction strength, in accordance with previous predictions.
We numerically study Friedel Oscillations and screening effect around a single impurity in one- and two-dimensional interacting lattice electrons. The interaction between electrons is accounted for by using a momentum independent self-energy obeying the Luttinger theorem. It is observed in one-dimensional systems that the amplitude of oscillations is systematically damped with increasing the interaction while the period remains unchanged. The variation of screening charge with the impurity potential is discussed. We see that the screening charge is suppressed by the interactions. In case of two-dimensional systems the surface oscillations around the impurity are more localized with increasing the interactions.