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arXiv:cond-mat/0012467

The polaron confined in one dimension

Gurjav Ganbold

Published Dec 26, 2000• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0012467

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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 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 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 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.
The next lowest order term of the ground state energy has its origin in lowering of frequencies of phonons localized around the polaron. An eigenvalue equation is derived, the solution of which gives phonon eigenmodes and frequencies in the presence of the polaron. Several eigenvalues are obtained by numerical integration. From the results, the ground state energy is estimated as E0/ℏω0=−0.108513α2−2.836+O(1/α2)E_{0}/\hbar\omega_{0}{=}-0.108513\alpha^{2}-2.836+O(1/\alpha^{2})E0​/ℏω0​=−0.108513α2−2.836+O(1/α2), where ω 0 is the frequency of the optical phonon and α is the electron-phonon coupling constant. The numerical value of the effective mass is 0.0227019α 4 .
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 Holstein model often serves as an archetype for electron-phonon interactions and polaron formation in solids. However, precise descriptions of the Holstein polaron are difficult when the phonon frequency is small and the electron-phonon coupling is strong, due to the presence of many phonons in the ground-state. We present a semi-analytical approximation that consists of a variational ansatz with clouds of phonons surrounding the electron in the form of coherent states. This becomes particularly simple and exact in the Lang-Firsov limit. We determine the domain of validity away from this limit, and further explore the improvement achieved with a removal of the requirement that the phonon clouds form coherent states. Both approximations work extremely well for the ground-state energy at strong coupling, and both work surprisingly well also at weak-coupling. For the effective mass, however, both approximations have difficulty, particularly in the crossover region. Moreover, the coherent-state ansatz (CSA) retains an unphysical jump in the crossover region. Nonetheless, the CSA provides a simple and intuitive picture of the polaron ground-state wavefunction, and in addition predicts accurate values for the ground-state energy.