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Physical review. B, Condensed matter

Two-impurity scattering in one-dimensional interacting fermion systems

Yu-Liang Liu

Published Oct 24, 2003• 15 citations
Publisher

Abstract

Using eigenfunctional theory, we study two-impurity scattering in one-dimensional interacting fermion systems, and clearly show the evolution of the systems from ultraviolet fixed point to infrared critical fixed point as increasing the impurity scattering potential. For the single-impurity scattering problem, we obtain the same fermion Green's function as that by previous methods. For two-impurity scattering, we demonstrate that the nonlocal interaction of the fermions produced by impurity scattering depends on not only the scattering potential, but also the distance between these two impurities. As two impurities far away from each other, the fermions are completely reflected by the impurities, and in the region between these two impurities there appear stationary waves; while as these two impurities are very close, the fermions with some discrete momentums can easily transmit through them.

DOI
10.1103/physrevb.68.155116

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Using eigenfunctional theory, we study two-impurity scattering in one-dimensional interacting fermion systems, and clearly show the evolution of the systems from ultraviolet fixed point to infrared critical fixed point as increasing the impurity scattering potential. For the single-impurity scattering problem, we obtain the same fermion Green's function as that by previous methods. For two-impurity scattering, we demonstrate that the nonlocal interaction of the fermions produced by impurity scattering depends on not only the scattering potential, but also the distance between these two impurities. As two impurities far away from each other, the fermions are completely reflected by the impurities, and in the region between these two impurities there appear stationary waves; while as these two impurities are very close, the fermions with some discrete momentums can easily transmit through them.
We first introduce the basic ingredients of the eigenfunctional theory, and show that a D-dimensional quantum many-particle system is mapped into a (D+1)-dimensional time-depending single-particle problem, and in the representation of the eigenfunctionals of the particle propagator, the particles become free. Then using this method, we study five kinds of quantum many-particle systems: interacting boson system, repulsive, attractive interacting fermion systems, Hubbard model and single-impurity scattering in one-dimensional fermion system, and demonstrate that the microscopic Bogoliubov theory and the phenomenological Bijl–Feynman theory of the bosons are closely related, and apart from an anti-symmetry factor Det ‖eikj·xl‖ the ground state wave function of the repulsive interacting fermion system has a similar form to that of the interacting boson system. Moreover, we show that the attractive interacting fermion system has a sound-type excitation spectrum like that in the interacting boson system. For one-dimensional Hubbard model we calculate the electron Green's function, and charge and spin density–density correlation functions which are consistent with the exact ones obtained by the Bethe ansatz and numerical calculations, and show that the ground state energy is increasing with U, and the electrons has single-occupied constraint in the large U limit. Finally, we demonstrate clearly the evolution of the system from its ultraviolet fixed point to infrared critical fixed point as the impurity potential increases. At the infrared critical fixed point, the fermion Green's function shows that the fermions are completely reflected on the impurity site.
We studied low-energy effects of impurities in several chiral one-dimensional (1D) electron systems that contain only excitations moving in one direction. We first considered single-impurity scattering between two branches of 1D chiral Luttinger liquids. The general form of the impurity-scattering matrix was found which, sometimes, allows the chemical potentials on the outgoing branches to be higher (or lower) than the maximum (or minimum) chemical potentials on the incoming branches. We also studied the effects of many impurities (with finite density) on two branches of 1D chiral Fermi liquids (that appear on the edge of the \ensuremath{\nu}=2 integral quantum Hall liquid). We found that the impurities drive the system to a fix point at low energies which has an SU(2) symmetry even when the two branches originally have different velocities. [Note that a clean system is SU(2) symmetric only when the two branches have the same velocity.] The probability distribution and correlation of scattering matrices at different energies are described by universal functions which are calculated exactly for this fixed point.
We study interference patterns and Friedel oscillations (FO) due to scattering from two or more localized impurities and scattering from extended inhomogeneities in the two-dimensional lattice systems of interacting fermions. Correlations between particles are accounted for by using an approximate method based on the real-space dynamical mean-field theory and a homogeneous self-energy approximation (HSEA), where the site-dependent part of the self-energy is neglected. We find that the interference maxima and minima change systematically as we vary the relative distance between the two impurities. At the same time, the increase of the interaction does not shift the position of interference fringes but only reduces their intensities. A comparison with the single impurity cases clearly shows complex patterns in FO fringes induced by additional multiple scattering processes. In the case of an extended steplike potential the system becomes more homogeneous when the interaction increases. FO and interference patterns are not present in the Mott insulating phase in both single and many impurity models.