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 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 a model of a quantum Hall system with the electrons confined to a narrow, linear channel. The system is mapped to a one-dimensional (1D) system which in the low-energy approximation has the form of a Luttinger liquid with different interactions between particles of equal and of opposite chiralities. In a previous paper (Part I) [M. Horsdal and J. M. Leinaas, Phys. Rev. B 76, 195321 (2007)], we studied this mapping at the microscopic level, and examined how the interaction in the two-dimensional (2D) system renormalizes 1D Luttinger liquid parameters. We follow up this study here and derive the low-energy form of the 2D electron correlation function by applying the mapping from the 1D correlation function. The derived expression has a natural separation in a bulk part, which is not modified by interactions, and an edge part, which is modified. We study the effect of interactions on the electron density and on the asymptotic form of the correlation function. The asymptotic parameter is found to be consistent with the expected value from the Luttinger liquid theory, but deviations from the asymptotic form are found in a rather wide intermediate interval. We suggest some questions of interest for further study.
We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.
We study a model of a quantum Hall system with the electrons confined to a narrow, linear channel. This system is mapped to a one-dimensional (1D) system which in the low-energy approximation has the form of a Luttinger liquid with different interactions between particles of equal and of opposite chiralities. The mapping is studied at the microscopic level, and we show in detail how the form of the 1D Luttinger liquid is detemined by interactions with particles at the edges and in the bulk of the quantum Hall system. We focus, in particular, on how the parameters are renormalized by the interactions between the two edges and show that the velocity parameter of the current is not modified. The paper is Paper I in a sequence of two papers, where Paper II [M. Horsdal and J. M. Leinaas, Phys. Rev. B 76, 195322 (2007)] gives a derivation of the two-dimensional electron correlation function.
The effects of impurity scattering on a general Abelian fractional quantum Hall (FQH) edge state are analyzed within the chiral-Luttinger-liquid model of low-energy edge dynamics. We find that some disordered edges can have several different phases characterized by different symmetries. The stable impurity edge phases are in general more symmetric than the original clean system and demonstrate the phenomenon of dynamical symmetry restoration at low energies and long length scales. The phase transitions between different disordered phases are characterized by broken symmetries and obey Landau's symmetry-breaking principle for continuous phase transitions. Phase diagrams for various edges are found using a system of coordinates for the interactions between modes in a quantum Hall edge. The temperature dependence of tunneling through a point contact is calculated and is found to be able to distinguish different impurity edge phases of the same FQH state.
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.
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.