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Physical review. B, Solid state

Localization versus delocalization in the one-dimensional Tomonaga model with impurities

S. T. Chui, J. W. Bray

Published Aug 15, 1977• 41 citations
Publisher

Abstract

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 ${Q}_{N}$ for an $N$-particle two-dimensional Coulomb gas. After obtaining new stringent bounds on ${Q}_{N}$, 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 ${U}_{\ensuremath{\perp}}$ ${v}_{F}$ 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.

DOI
10.1103/physrevb.16.1329

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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⊥{U}_{\ensuremath{\perp}}U\ensuremath⊥​. 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 U\ensuremath∥2\ensuremathπvF\frac{{U}_{\ensuremath{\parallel}}}{2\ensuremath{\pi}{v}_{F}}2\ensuremathπvF​U\ensuremath∥​​ greater than (less than) -5/13, independent of U\ensuremath⊥{U}_{\ensuremath{\perp}}U\ensuremath⊥​ 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.
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.
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.
Localization-delocalization transition has been examined for interacting one-dimensional fermion systems with the backward and the Umklapp scattering by use of the phase Hamiltonian. In this approach the localized state is represented as that where the phase variable of either charge or spin (or both of them) is pinned by impurities and then distorted spatially, i.e. glassy. The transition can be understood as the depinning process due to the quantum fluctuations and its boundary has been determined both from the localized region and from the delocalized region. The two different procedures are shown to yield same criterion for the transition.
We discuss impurities in a one-dimensional Bose gas with arbitrary boson-boson and boson-impurity interactions. To fully account for quantum effects, we employ numerical simulations based on the density-matrix renormalization group (DMRG) and—in the regime of strong boson-boson interactions—the mapping to weakly interacting fermions. A mean-field description of the Bose polaron based on coupled Gross–Pitaevskii Schrödinger equations predicts the existence of a self-localized polaron. We here show that such a solution does not exist and is an artifact of the underlying decoupling approximation. To this end we consider a mobile impurity in a box potential. Our work demonstrates that correlations between the impurity position and the bosons are important even in the limit where mean-field approaches are expected to work well. Furthermore, we derive analytical approximations for the energy of a single polaron formed by a heavy impurity for arbitrary interaction strengths and large but finite boson-boson couplings which accurately reproduce DMRG results. This demonstrates that the polaron problem of a heavy impurity in a one-dimensional Bose gas can be accurately approximated by a proper mean-field description plus a linearized treatment of quantum fluctuations for arbitrary boson-boson and impurity-boson couplings. Finally, we determine the polaron-polaron interaction potential V ( r ) in the Born-Oppenheimer approximation for small and intermediate distances r , which in the Tonks gas limit is oscillatory due to Friedel oscillations in the Bose gas.