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Physical review. B, Condensed matter

Electron localization and delocalization in the one-dimensional Tomonaga model with backscattering and impurities

S. T. Chui, J. W. Bray

Published Apr 15, 1979• 15 citations
Publisher

Abstract

We extend our previous calculation of the effect of impurities on the one-dimensional electron gas in the Tomonaga model by including spin-flip backscattering in the electron-electron interaction. We find that this addition does not alter our conclusions concerning conductivity in this model.
DOI
10.1103/physrevb.19.4020

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To understand what happens in a one-dimensional system when both localization due to impurities and electron-electron interaction (${U}_{\ensuremath{\parallel}}$) are important, we studied the effect of impurities in the Tomonaga model by looking at the perturbation series for the conductivity in the impurity potential ${U}_{\ensuremath{\perp}}$. By making an analogy to the two-dimensional Coulomb gas we were able to relate the static conductivity to the effective potential of two test charges infinitely far apart and understand the localization phenomena in terms of Debye screening. Furthermore we found that as the electron-electron interaction becomes attractive enough the electronic state becomes delocalized. This corresponds to a phase transition in the Coulomb gas when the positive and the negative charges are paired up and Debye screening disappears.
We examine localization and delocalization in the one-dimensional Tomonaga model with impurities. We expand the density autocorrelation function in a perturbation series in the impurity coupling ${U}_{\ensuremath{\perp}}$. We then relate the terms in this series to the partition functions QN{Q}_{N}QN​ for an NNN-particle two-dimensional Coulomb gas. After obtaining new stringent bounds on QN{Q}_{N}QN​, we use these bounds and the analytic properties of the perturbation series to predict localization. In contradiction of earlier results, we find that the electrons are localized (delocalized) for electron-electron interactions $\frac{{U}_{\ensuremath{\parallel}}}{2\ensuremath{\pi}{v}_{F}}$ greater than (less than) -5/13, independent of ${U}_{\ensuremath{\perp}}$ vF{v}_{F}vF​ is the Fermi velocity). This is the first time to our knowledge that a localization transition has been studied by such a bounded-series technique.
Electrons in one-dimensional disordered systems exhibit Anderson localization in the absence of mutual interactions. However the transition is possible between localized and delocalized state as the interaction is turned on. In this paper such interplay between Anderson localization and interactions has been examined for spin dependent Tomonaga model based on the phase Hamiltonian derived by the bosonization of the fermion field. The problem is the extention of the classical impurity pinning of the charge density wave to that of quantum systems with phase variables of not only charge but also spin. The critical value of the interaction constant has been derived by the use of selfconsistent-harmonic approximation to the nonlinearity.
We study the transport properties of a quantum wire, described by the Tomonaga-Luttinger model, in the presence of a backscattering potential provided by several extended time-dependent impurities (barriers). Employing the B\"uttiker-Landauer approach, we first consider the scattering of noninteracting electrons (g=1)(g=1)(g=1) by a rectangularlike barrier and find an exact solution for the backscattering current, as well as a perturbative solution for a weak static potential with an arbitrary shape. We then include electron-electron interactions and use the Keldysh formalism combined with the bosonization technique to study oscillating extended barriers. We show that the backscattering current off time-dependent impurities can be expressed in terms of the current for the corresponding static barrier. Then we determine the backscattering current for a static extended potential, which, in the limit of noninteracting electrons (g=1)(g=1)(g=1), coincides with the result obtained using the B\"uttiker-Landauer formalism. In particular, we find that the conductance can be increased beyond its quantized value in the whole range of repulsive interactions 0<g<10<g<10<g<1 already in the case of a single oscillating extended impurity, in contrast to the case of a pointlike impurity, where this phenomenon occurs only for 0<g<1∕20<g<1∕20<g<1∕2.