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Journal of Physics C Solid State Physics

Polarons in one-dimensional systems

Marcos H. Degani, Oscar Hipólito, RicardoP.S. M. Lobo, G. A. Farias

Published Jun 10, 1986• 16 citations
Publisher

Abstract

The Feynman path integral formalism is used to calculate the energy and the effective mass of one-dimensional optical and acoustic polarons. It is shown that the optical polaron yields a continuous change as the coupling goes from the weak to the strong limit. On the other hand an abrupt change of the acoustic polaron state at alpha ac =0.5 is observed.

DOI
10.1088/0022-3719/19/16/012

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François Maria Peeters, M. A. Smondyrev

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.

Physical review. B, Condensed matter•Feb 15, 1991•28 citations
The Feynman path integral formalism is used to calculate the energy and the effective mass of one-dimensional optical and acoustic polarons. It is shown that the optical polaron yields a continuous change as the coupling goes from the weak to the strong limit. On the other hand an abrupt change of the acoustic polaron state at alpha ac =0.5 is observed.
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 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.
The ground-state energy of an acoustical bipolaron in three dimensions (3D) is studied within the Feynman path integral method. The acoustical polaron exhibits a transition from the quasi-free state to the self-trapped state. Depending on the value of the Debye cut-off in phonon wavevector space this transition can be continuous or discontinuous. If two electrons are present a second transition becomes possible: the transition to the bipolaron state in which the two electrons are self-trapped around the same position. The corresponding phase diagram is calculated as function of the cut-off , the electron - phonon coupling strength , and the repulsion (U) between the electrons. We found that: (i) the bipolaron transition is a first-order transition; (ii) it is also possible between two polarons which are in the quasi-free state, i.e. are not self-trapped; and (iii) the effective particle mass increases substantially at the transition point.
The description of an impurity atom in a Bose-Einstein condensate can be cast in the form of Fr\"ohlich's polaron Hamiltonian, where the Bogoliubov excitations play the role of the phonons. An expression for the corresponding polaronic coupling strength is derived, relating the coupling strength to the scattering lengths, the trap size and the number of Bose condensed atoms. This allows to identify several approaches to reach the strong-coupling limit for the quantum gas polarons, whereas this limit was hitherto experimentally inaccessible in solids. We apply Feynman's path-integral method to calculate for all coupling strengths the polaronic shift in the free energy and the increase in the effective mass. The effect of temperature on these quantities is included in the description. We find similarities to the acoustic polaron results and indications of a transition between free polarons and self-trapped polarons. The prospects, based on the current theory, of investigating the polaron physics with ultracold gases are discussed for lithium atoms in a sodium condensate.
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.