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Physical review. B, Solid state

Localization and delocalization in the one-dimensional Tomonaga model

S. T. Chui

Published Jul 15, 1977• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.16.617

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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∥{U}_{\ensuremath{\parallel}}U\ensuremath∥​) 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⊥{U}_{\ensuremath{\perp}}U\ensuremath⊥​. 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.
A one-dimensional interacting electron gas in a random potential exhibits a localized-delocalized transition for increasingly attractive interactions. We develop here a renormalization-group approach to study this transition. Our treatment allows us to obtain the phase diagram and the exponents of the correlation functions in the delocalized regime. The boundary between the two regimes is found to depend both on disorder and the strength of the interactions. For (nearly) spin-isotropic interactions the delocalized phase is dominated by superconducting fluctuations of either singlet or triplet type. The temperature dependence of the conductivity in the delocalized phase is also obtained and a nonuniversal power-law behavior is found. A description of the crossover towards the localized phase is given and the localization length is computed. An analogous description is developed for the localized-superfluid transition of a one-dimensional boson gas. In this case the transition to the localized regime occurs for increasingly repulsive interactions. We suggest a phase diagram with two different localized phases. Finally, we discuss some possible implications of our model for real quasi-one-dimensional metals.
A one-dimensional Coulomb gas is considered in which the particles represent both charges and dipoles. The values of the charges and dipole moments and the fugacities of that gas have been obtained in the preceding paper by mapping the dynamics of a two-level system (TLS) interacting with a degenerate electron gas onto the present problem. The scaling procedure of Anderson, Yuval, and Hamann is applied to eliminate the close pair which corresponds to the short-time behavior of the TLS. The main contribution arises from pairs of zero total charge, but the role of other pairs is considered as well. The scaling equations show that the electron-assisted transitions of the TLS play a very important role if their value is different from zero for the starting Hamiltonian. In that case localization cannot occur in the case of a spin-(1/2 fermionic system. Localization may occur only if the fermion spin has an artificially large value with spin degeneracy Ns{N}_{s}Ns​>2. That clearly demonstrates that a particle moving in a double potential well may have different dissipative behavior depending upon whether it is coupled to a fermionic or a bosonic heat bath. The screening of the TLS by electrons is described by an arbitrarily large phase shift \ensuremath{\delta} which renormalizes towards the value \ensuremath{\delta}=\ensuremath{\pi}/2Ns{N}_{s}Ns​, but the scaling equations lose their validity as the fugacities corresponding to the TLS transitions become of the order of unity. The method developed is a very general one, and thus it can be applied to other similar one-dimensional gases as well.