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Physical review. B, Condensed matter

Influence of backward scattering on correlation exponents in a one-dimensional system

Yu-Liang Liu

Published Jul 1, 2000• 8 citations
Publisher

Abstract

Using bosonization and phase shift representation, we rigorously treat backward scattering of electrons on an impurity in a one-dimensional interacting spinless electronic system, and demonstrate that correlation exponents of the system depend on a phase shift induced by the backward scattering, and usual exponent duality between ultraviolet and infrared fixed points comes from the phase shift dependence of the correlation exponents.

DOI
10.1103/physrevb.62.1212

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Using bosonization and phase shift representation, we rigorously treat backward scattering of electrons on an impurity in a one-dimensional interacting spinless electronic system, and demonstrate that correlation exponents of the system depend on a phase shift induced by the backward scattering, and usual exponent duality between ultraviolet and infrared fixed points comes from the phase shift dependence of the correlation exponents.
Using bosonization and phase shift representation, we rigorously treat backward scattering of electrons on an impurity in a one-dimensional interacting electronic system, and demonstrate that correlation exponents of the system depend on a phase shift induced by the backward scattering, and usual exponent duality of the correlation functions between ultraviolet and infrared fixed points comes from the phase shift dependence of the correlation exponents. Finally, we study the tunnelling conductance of the system at zero temperature and obtain a modified Landauer-Bütiker formula.
Using an elementary method, we demonstrate that (1) the duality of the correlation exponents near impurity site x=0x=0x=0 between ultraviolet and infrared fixed points is also valid for channel number larger than one; (2) the infrared fixed points ${\ensuremath{\delta}}_{c}^{i}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ do not correspond to two real half-infinity Tomonaga-Luttinger liquids even if the conductance is zero; (3) the electron's density of state at x=0x=0x=0 is enhanced compared with its neighbor ones; (4) as channel number is larger than (or equal to) two, the Fermi-edge singularity induced by backward scattering will be swept off by even small repulsive interaction of the electrons.
The problem of the Fermi-edge singularity in a one-dimensional Tomonaga-Luttinger liquid is reconsidered. The backward scattering of the conduction band electrons on the impuritylike hole in the valence band is analyzed by mapping the problem onto a Coulomb gas theory. For the case when the electron-electron interaction is repulsive, the obtained exponent of the one-dimensional Fermi-edge singularity appears to be different from the exponent found in previous studies. It is shown that the infrared physics of the Fermi-edge singularity in the presence of backward scattering and electron-electron repulsion resembles the physics of the Kondo problem. \textcopyright{} 1996 The American Physical Society.
We study electronic transport in a Luttinger liquid with an embedded impurity, which is either a weak scatterer (WS) or a weak link (WL), when interacting electrons are coupled to one-dimensional massless bosons (e.g., acoustic phonons). We find that the duality relation, ΔWSΔWL=1, between scaling dimensions of the electron backscattering in the WS and WL limits, established for the standard Luttinger liquid, holds in the presence of the additional coupling for an arbitrary fixed strength of boson scattering from the impurity. This means that at low temperatures such a system remains either an ideal insulator or an ideal metal, regardless of the scattering strength. On the other hand, when fermion and boson scattering from the impurity are correlated, the system has a rich phase diagram that includes a metal-insulator transition at some intermediate values of the scattering.
We present a detailed numerical study of the orthogonality catastrophe exponent for a one-dimensional lattice model of spinless fermions with nearest-neighbor interaction using the density-matrix renormalization group algorithm. Keeping up to 2000 states per block we achieve a very great accuracy for the overlap, which is needed to extract the orthogonality exponent reliably. We discuss the behavior of the exponent for three different kinds of a localized impurity. For comparison we also discuss the noninteracting case. In the weak impurity limit our results for the overlap confirm scaling behavior expected from perturbation theory and renormalization-group calculations. In particular we find that a weak backward scattering component of the orthogonality exponent scales to zero for attractive interaction. In the strong impurity limit and for repulsive interaction we demonstrate that the orthogonality exponent cannot be extracted from the overlap for systems with up to 100 sites, due to finite-size effects. Nevertheless we find indirect evidence that the backward scattering contribution to the exponent scales to 1/16 based on predictions of boundary conformal field theory. [S0163-1829(98)01415-5].
We present a detailed numerical study of the orthogonality catastrophe exponent for a one-dimensional lattice model of spinless fermions with nearest-neighbor interaction using the density-matrix renormalization-group algorithm. Keeping up to 200020002000 states per block we achieve a very great accuracy for the overlap, which is needed to extract the orthogonality exponent reliably. We discuss the behavior of the exponent for three different kinds of a localized impurity. For comparison we also discuss the noninteracting case. In the weak impurity limit our results for the overlap confirm scaling behavior expected from perturbation theory and renormalization-group calculations. In particular we find that a weak backward scattering component of the orthogonality exponent scales to zero for attractive interaction. In the strong impurity limit and for repulsive interaction we demonstrate that the orthogonality exponent cannot be extracted from the overlap for systems with up to 100 sites, due to finite-size effects. Neverthless we find indirect evidence that the backward scattering contribution to the exponent scales to 1/161/161/16 based on predictions of boundary conformal field theory.