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Journal of Physics C Solid State Physics

Analysis of the polaron problem on the basis of the operator method

И. Д. Феранчук, S I Fisher, L I Komarov

Published Aug 30, 1984• 12 citations
Publisher

Abstract

The equations which define the polaron ground state throughout the entire range of the electron-phonon interaction constant are obtained. The best estimate of the polaron energy in the strong-coupling regime is found. Two qualitatively different polaron states are discovered and their characteristics are investigated. The point of the phase transition between these states is calculated.

DOI
10.1088/0022-3719/17/24/012

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Analytical investigation of the polaron problem

И. Д. Феранчук, S I Fisher, L I Komarov

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Polaron features for long-range electron–phonon interaction

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The polaron features for long-range electron–phonon interaction are investigated by extending a variational approach previously proposed for the study of systems with local coupling. The ground state spectral weight, the average kinetic energy, the mean number of phonons, and the electron–lattice correlation function are discussed for a wide range of model parameters focusing on the adiabatic regime and comparing the results with the short-range case (Holstein model). A strong mixing of electronic and phononic degrees of freedom for small values of the electron–phonon coupling constant is found in the adiabatic case due to the long-range interaction. Finally a polaron ‘phase diagram’ is proposed.

arXiv:cond-mat/0402211•Journal of Physics Condensed Matter•Feb 20, 2004•21 citations
The equations which define the polaron ground state throughout the entire range of the electron-phonon interaction constant are obtained. The best estimate of the polaron energy in the strong-coupling regime is found. Two qualitatively different polaron states are discovered and their characteristics are investigated. The point of the phase transition between these states is calculated.
It is well known that two qualitatively different pictures arise in the description of the motion of an electron in an ionic crystal. In the first case, when the electron interacts weakly with the ionic lattice of the crystal, its motion is the same as the motion of a free band electron whose energy is shifted downward with respect to the bottom of the conduction band, and the effective mass is replaced by the renormalized mass (weak-coupling polaron). In the other limiting case, when the electron interacts strongly with the ionic crystal, an entire series of different self-consistent states of the electron and lattice, each of which has its own effective mass and radius (strong-coupling polaron), arises. Although the motion of an electron in an ionic crystal is described in both limiting cases with the help of the Pekar-Frohlich Hamiltonian, each case requires its own special methods of analysis. Thus, for example, in the strong-coupling theory a special form of adiabatic perturbation theory has been developed in which the trans-lational degeneracy is removed before the perturbation expansion is made. It has been possible to relate both descriptions within the framework of Feynman's varia-tional approach, in which a smooth upper bound was obtained for the energy of the ground state of the polaron for all values of the electron-phonon coupling constant a (see Fig. 1). We call attention in this connection to Ref. 6, where the thermodynamic properties are calculated with the help of a special method for performing the calculations and for finding averages of T-products. The approach developed there is more convenient for performing specific calculations than is the use of path integrals. It seems to us that it has definite advantages in the construction of general proofs concerning the choice of the approximating Hamiltonians (this is illustrated in Ref. 6 for the example of the linear Bogolyubov model), in the investigation of the characteristics of the behavior of a polaron in external fields, etc.
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 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 polaron features for long-range electron–phonon interaction are investigated by extending a variational approach previously proposed for the study of systems with local coupling. The ground state spectral weight, the average kinetic energy, the mean number of phonons, and the electron–lattice correlation function are discussed for a wide range of model parameters focusing on the adiabatic regime and comparing the results with the short-range case (Holstein model). A strong mixing of electronic and phononic degrees of freedom for small values of the electron–phonon coupling constant is found in the adiabatic case due to the long-range interaction. Finally a polaron ‘phase diagram’ is proposed.