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International Journal of Modern Physics B

IMPURITIES IN A MAGNETIC-FIELD-INDUCED LUTTINGER LIQUID

Shan-Wen Tsai, Dmitrii L. Maslov, L. I. Glazman

Published Aug 30, 2002• 0 citations
Publisher

Abstract

We study the effect of dilute impurities on a system of interacting electrons subject to a quantizing magnetic field. For the case of point impurities, the calculation of the scattering cross section can be mapped onto a 1D problem of tunnelinng conductance through a barrier for interacting electrons. We find that the electron-electron interaction produces temperature-dependent corrections to the cross-section, and thus to the tensor of conductivities. Upon summation of the most diverging corrections in all orders of perturbation theory, a scaling behavior of the conductivities emerges. This behavior is similar to that of 1D Luttinger liquid. The scaling exponents depend on the magnetic field and are calculated explicitly in the weak-coupling limit. The limitations on the scaling behavior due to the formation of a charge-density wave are also discussed.

DOI
10.1142/s0217979202013377

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We study the effect of dilute impurities on a system of interacting electrons subject to a quantizing magnetic field. For the case of point impurities, the calculation of the scattering cross section can be mapped onto a 1D problem of tunnelinng conductance through a barrier for interacting electrons. We find that the electron-electron interaction produces temperature-dependent corrections to the cross-section, and thus to the tensor of conductivities. Upon summation of the most diverging corrections in all orders of perturbation theory, a scaling behavior of the conductivities emerges. This behavior is similar to that of 1D Luttinger liquid. The scaling exponents depend on the magnetic field and are calculated explicitly in the weak-coupling limit. The limitations on the scaling behavior due to the formation of a charge-density wave are also discussed.
In this paper, we study the effect of a weak magnetic field and the charging energy on electron conduction in a Luttinger liquid with two impurities. We consider the situation in which the two potential barriers are different in the symmetric tunneling and asymmetric tunneling limits. We focus on the dependence of the charge conductance on the temperature, the charging energy and the magnetic field. It is shown that the asymmetry of the potential barrier obstructs the resonant conduction in the symmetric tunneling limit. On the other hand, in the asymmetric tunneling limit, we treat the cases of excitation both with and without a gap. For excitation without a gap, the charging energy changes the exponent of the temperature dependence of the conductance, and there appears resonant conduction under zero magnetic field. Contrastingly, for excitation with a gap, the charging energy does not change the exponent of the conductance, and there appears resonant conduction unrelated to the magnetic field. Our results suggest that there is no spin-filter phase in the asymmetric tunneling limit.
We study the low-temperature properties of a spin-1/2 magnetic impurity coupled to a one-dimensional interacting electron system. Using a formalism by Affleck and Ludwig, with a scale invariant boundary condition replacing the impurity, we exploit boundary conformal field theory to deduce the impurity thermal and magnetic response. In the case of only forward electron scattering off the impurity, we predict the same critical scaling as for the two-channel Kondo effect for noninteracting electrons, but with a different Wilson ratio. Backward electron scattering off the impurity destabilizes this behavior and drives the system to a new fixed point. In the case of equal amplitudes for forward and backward scattering (Kondo interaction), we show that there are only two types of scaling behaviors consistent with the symmetries of the problem: either a local Fermi liquid or a critical theory with an anomalous specific heat. The latter case agrees with a recent poor-man's scaling result proposed by Furusaki and Nagaosa. \textcopyright{} 1996 The American Physical Society.
Using bosonization and path-integral methods, we study general low-temperature behavior of nonmagnetic and magnetic impurity scattering in Tomonaga-Luttinger liquid, and calculate the electron Green function for a general backward scattering potential. We demonstrate that the electron density of state near the impurity site is suppressed by backward scattering, but it mainly remains invariant far away from the impurity, and that at zero temperature the electrons are completely reflected on the impurity site; the system breaks into two subsystems, but right- and left-moving electron fields have a twisted boundary condition. We also show that a testing charge can only be partially screened by conduction electrons, and in a strong interaction region the impurity susceptibility has a 1/T1/T1/T-type low-temperature behavior.
This thesis addresses a theoretical study of the problem of a single impurity embedded in a one-dimensional system of interacting electrons in presence of electron-phonon coupling. First we consider a system with a featureless point-like potential impurity, followed by the case of a resonant level hybridised with a Luttinger Liquid. The stress is made on a more fundamental problem of a featureless scatterer, for which two opposite limits in the impurity strength are considered: a weak scatterer and a weak link. We have found that, regardless of the transmission properties of phonons through the impurity, the scaling dimensions of the conductance in these limits obey the duality condition, \\( \\triangle_{WS}\\) \\( \\triangle_{WL}\\) = 1, known for the Luttinger Liquid in the absence of phonons. However, in the case when the strength of phonon scattering is correlated with electron scattering by the impurity, we find a nontrivial phase diagram with up to three fixed points and a possibility of a metal-insulator transition. We also consider the case of a weakly interacting electron-phonon system in the presence of a single impurity of an arbitrary scattering potential. In the problem of a resonant level attached to the Luttinger Liquid we show that the electron-phonon coupling significantly modifies the effective energy-dependent width of the resonant level in two different geometries, corresponding to the resonant and anti-resonant transmission in the Fermi gas.
A quantum Monte Carlo simulation method has been developed and applied to study the critical behavior of a single Kondo impurity in a Luttinger liquid. This numerically exact method has no finite-size limitations and allows to simulate the whole temperature range. Focusing on the impurity magnetic susceptibility, we determine the scaling functions, in particular for temperatures well below the Kondo temperature. In the absence of elastic potential scattering, we find Fermi-liquid behavior for strong electron-electron interactions, gc<12{g}_{c}<\frac{1}{2}gc​<21​, and anomalous power laws for 12<gc<1\frac{1}{2}<{g}_{c}<121​<gc​<1, where gc{g}_{c}gc​ is the correlation parameter of the Luttinger liquid. These findings resolve a recent controversy. If elastic potential scattering is present, we find a logarithmically divergent impurity susceptibility at gc<12{g}_{c}<\frac{1}{2}gc​<21​ that can be rationalized in terms of the two-channel Kondo model.
We study the spin-dependent scattering of charge carriers in a magnetized one dimensional Luttinger liquid from a localized nonhomogeneous magnetic field, which might be brought about by the stray field of magnetic tip near a uniform liquid, or by a transverse domain wall (DW) between two oppositely magnetized liquids. From a renormalization group treatment of the electron interactions we deduce scaling equations for the transmission and reflection amplitudes as the bandwidth is progressively reduced to an energy scale set by the temperature. The repulsive interactions dictate two possible zero temperature insulator fixed points: one in which electrons are reflected in the same spin channel and another where the electron spin is reversed upon reflection. In the latter case, a finite spin current emerges in the absence of a charge current at zero temperature and the Friedel oscillations form a transverse spiraling spin density. Adding a purely potential scattering term has no effect on the fixed points of a uniformly magnetized liquid. For a DW we find that the introduction of potential scattering stabilizes the spin-flip insulator phase even if the single-particle spin-flip scattering produced by the DW is arbitrarily weak. The potential can be induced externally, e.g., by a local gate voltage or a constriction, providing a means for controlling the transport properties of the wire.