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arXiv:cond-mat/9910327

Quasi-particle characteristics of the one-dimensional polaron

Gurjav Ganbold

Published Oct 21, 1999• 0 citations
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Abstract

The main quasi-particle characteristics of the one-dimensional polaron are estimated within and beyond the most general Gaussian approximation at arbitrary electron-phonon coupling. We have derived explicitly the ground-state energy and the effective mass in the weak- and strong-coupling regimes. For arbitrary coupling, the Gaussian leading-order term of the polaron self energy improves the corresponding Feynman estimate and represents the lowest upper bound available. We have calculated the next non-Gaussian corrections. Taking into account systematically higher-order corrections does not perturb considerably the obtained results.

DOI
10.48550/arxiv.cond-mat/9910327

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The main quasi-particle characteristics of the one-dimensional polaron are estimated within and beyond the most general Gaussian approximation at arbitrary electron-phonon coupling. We have derived explicitly the ground-state energy and the effective mass in the weak- and strong-coupling regimes. For arbitrary coupling, the Gaussian leading-order term of the polaron self energy improves the corresponding Feynman estimate and represents the lowest upper bound available. We have calculated the next non-Gaussian corrections. Taking into account systematically higher-order corrections does not perturb considerably the obtained results.
The ground-state energy, the effective mass and the number of virtual phonons of the optical large polaron confined strictly in one dimension have been estimated by using the generalized Gaussian approximation. The leading-order terms take care of all Gaussian fluctuations in the system and improve the conventional variational estimates at finite coupling. Particularly, the lowest upper bound to the polaron ground-state energy has been obtained. The non-Gaussian contributions systematically correct the leading-order approximations. We have obtained exact analytical solutions in the weak- and strong-coupling limit and reasonable numerical data for intermediate coupling. Our result for the number of excited phonons limits the validity region of the few-phonon approximation methods.
The polaron path integral introduced by Feynman is generalized to the case of an electron moving in d space dimensions. A scheme of systematic calculations of the obtained path integral is developed to estimate the ground-state energy of the polaron in the same way for different values of the electron-phonon coupling constant \ensuremath{\alpha}. The leading-order term in this approach yields an upper bound to the polaron self-energy and improves Feynman's variational estimates for d=2 and d=3. A scaling relation between polaron self-energies for different space dimensions is obtained for this term. The next correction to this estimation is calculated by numerical integration for d=2 and d=3.
The effective mass of the Fröhlich polaron model in d spatial dimensions is obtained within the Gaussian-equivalent representation, that is, within a suitably rearranged perturbative path-integral approach, whose leading order already takes care of all Gaussian fluctuations. In doing so, the authors extend their earlier investigations ( Phys. Rev. B 50 3733 (1994)) of the polaron ground-state energy. Analytical expressions for the effective mass are derived in the weak- and strong-coupling limit. In the intermediate-coupling regime, the effective mass is calculated numerically for the cases of two and three dimensions. The new results are compared with previously known data.
The ground-state characterization of the polaron problem is retrieved within the framework of a variational scheme proposed previously by Devreese et al for the bound polaron. The formulation is based on the standard canonical transformation of the strong coupling ansatz and consists of a variationally determined perturbative extension serving for the theory to interpolate in the overall range of the coupling constant. Specializing our considerations to the bulk and strict two-dimensional polaron models we see that the theory yields significantly improved energy upper bounds in the strong coupling regime and, moreover, extrapolates itself successfully towards the well-established weak coupling limits for all polaron quantities of general interest.
The Fr\"ohlich polaron confined to move in one space dimension and interacting with a three-dimensional lattice is studied. We calculate exact and approximate results for the ground-state energy and the effective mass. The following exact results were obtained. In (a) the weak-coupling limit for the energy E=-\ensuremath{\alpha}'-[(3/2 \ensuremath{\surd}2 )-1]\ensuremath{\alpha}' 2^{2}2, the mass m∗{\mathit{m}}^{\mathrm{*}}m∗/mb{\mathit{m}}_{\mathit{b}}mb​=1/2\ensuremath{\alpha}'+[(5/2 \ensuremath{\surd}2 )-1]/4, and the average number of virtual phonons N=1/2\ensuremath{\alpha}'+[(3/ \ensuremath{\surd}2 )-2]\ensuremath{\alpha}' 2^{2}2, and in (b) the strong-coupling limit E=-0.333 088\ensuremath{\alpha}' 2^{2}2,m∗{\mathit{m}}^{\mathrm{*}}m∗/mb{\mathit{m}}_{\mathit{b}}mb​=2.1254\ensuremath{\alpha}' 4^{4}4, and N=0.666 176\ensuremath{\alpha}' 2^{2}2. Here, \ensuremath{\alpha}' is the normalized electron-phonon coupling constant, which was chosen such that E=-\ensuremath{\alpha}' within second-order perturbation theory.
A refined approach where the wave function is gradually improved is developed in the one-dimensional (1D) large-Frohlich-polaron problem. The results for the ground-state energy, the effective mass, and the average number of virtual phonons are obtained for a wide range of the coupling constant It, and are in agreement with those of the Feynman path-integral formalism. In addition, in the weak-coupling limit, the expansions of these observables in powers of the coupling constant are exactly calculated up to the alpha 3 -term. They are E=- alpha -0.06066 alpha 2 -0.00844 alpha 3 , m*=1+0.5 alpha +0.19194 alpha 2 +0.06912 alpha 3 , and N=0.5 alpha +0.12132 alpha 2 +0.029540 alpha 3 , which agree with the known results of the fourth-order perturbation theory.