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Physical review. B, Condensed matter

Exact and approximate results for the polaron in one dimension

François Maria Peeters, M. A. Smondyrev

Published Feb 15, 1991• 28 citations
Publisher

Abstract

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}' $^{2}$, the mass ${\mathit{m}}^{\mathrm{*}}$/${\mathit{m}}_{\mathit{b}}$=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}' $^{2}$, and in (b) the strong-coupling limit E=-0.333 088\ensuremath{\alpha}' $^{2}$,${\mathit{m}}^{\mathrm{*}}$/${\mathit{m}}_{\mathit{b}}$=2.1254\ensuremath{\alpha}' $^{4}$, and N=0.666 176\ensuremath{\alpha}' $^{2}$. Here, \ensuremath{\alpha}' is the normalized electron-phonon coupling constant, which was chosen such that E=-\ensuremath{\alpha}' within second-order perturbation theory.

DOI
10.1103/physrevb.43.4920

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Physical review. B, Condensed matter•May 1, 1996•17 citations
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 ground-state energy of a two-dimensional (2D) Fr\"ohlich polaron is calculated to second order in the coupling constant (\ensuremath{\alpha}) and gives E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-(\ensuremath{\pi}/2)\ensuremath{\alpha}-0.063 97${\ensuremath{\alpha}}^{2}$ with \ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$ the surface optical-phonon energy. In the strong-coupling limit the adiabatic approximation is used and E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-0.4047${\ensuremath{\alpha}}^{2}$ is found to leading order in \ensuremath{\alpha}. The Feynman path-integral approximation, the Gaussian approximation, and the modified Lee-Low-Pines unitary transformation approximation to the polaron ground-state energy all satisfy the scaling relation E2D{E}_{2\mathrm{D}}E2D​(\ensuremath{\alpha})=(2/3)E3D{E}_{3\mathrm{D}}E3D​(( 3\ensuremath{\pi}/4)\ensuremath{\alpha}), where E2D{E}_{2\mathrm{D}}E2D​ (E3D{E}_{3\mathrm{D}}E3D​) is the ground-state energy of the 2D (3D) polaron.
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.
The polaron mass of a two-dimensional Fr\"ohlich polaron is calculated to second order in the coupling constant (\ensuremath{\alpha}) and is found to be equal to m∗{m}^{\mathrm{*}}m∗/mb{\mathrm{m}}_{\mathrm{b}}mb​=1+(\ensuremath{\pi}/8)\ensuremath{\alpha}+0.127 23${\mathrm{\ensuremath{\alpha}}}^{2}$, with mb{m}_{b}mb​ the band mass of the electron. In the strong-coupling limit we found the exact result m∗{m}^{\mathrm{*}}m∗/mb{m}_{b}mb​=0.733${\ensuremath{\alpha}}^{4}$. A comparison is made with the polaron mass as obtained from the Feynman polaron theory, and the validity of a scaling relation is discussed.
The Fr\"ohlich Hamiltonian is generalized to the case of an electron moving in n space dimensions. For n=2 and n=3 the familiar Fr\"ohlich Hamiltonian is reobtained. The polaron ground-state energy is calculated up to fourth order in perturbation theory. We found that within the Feynman two-particle polaron model approximation the polaron ground-state energy satisfies the scaling relation ${E}_{n\mathrm{D}(\mathrm{\ensuremath{\alpha}})}$=(n/3)E3D{\mathrm{E}}_{3\mathrm{D}}E3D​({\ensuremath{\Gamma}[ (n-1)/2]3 \ensuremath{\surd}\ensuremath{\pi} /2n\ensuremath{\Gamma}(n/2)}\ensuremath{\alpha}), where EnD{E}_{n\mathrm{D}}EnD​ is the Feynman polaron ground-state energy for the polaron in n dimensions and E3D{E}_{3\mathrm{D}}E3D​ the energy in three dimensions.
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 properties of two- and three-dimensional polarons are investigated with the use of extended coherent states. In the weak-coupling limit, the obtained expansions for the ground-state energy and effective mass to the second order in the coupling constant \ensuremath{\alpha} are the same as the previous exact ones. Moreover, in the intermediate-coupling regime, the numerical results for the ground-state energy and effective mass agree with the results by the Feynman path-integral method, and perhaps closer to the exact results than the Feynman ones. It is suggested that the present approach should be an effective one in polaron physics. \textcopyright{} 1996 The American Physical Society.