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CRC Press eBooks

Linear Polaron Model

N. N. Bogolubov Jr.

Published May 17, 2000• 0 citations
Publisher

Abstract

This chapter is mainly a pedagogical introduction to a modern method in equilibrium finite-temperature polaron theory based on the T-product operator technique. As an example of the application of this method, Bogolubov's exactly solvable linearized polaron model, as well as Feynman's two-body oscillator model, are considered.
DOI
10.1201/9781482296884-7

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The main objective of this chapter is the derivation of Bogolubov's inequality for the polaron reduced free energy by means of the algebraic T-product method. This inequality is a source of various upper bounds for the polaron ground-state energy. Feynman's well-known inequality is reproduced as a particular case of Bogolubov's inequality. A weak-interaction systematic finite-temperature perturbation scheme, based on the same T-product formalism, as well as the adiabatic perturbation approach, valid for the strong-coupling case, are also outlined.

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International Journal of Modern Physics B•Sep 30, 2009•0 citations
The main objective of this chapter is the derivation of Bogolubov's inequality for the polaron reduced free energy by means of the algebraic T-product method. This inequality is a source of various upper bounds for the polaron ground-state energy. Feynman's well-known inequality is reproduced as a particular case of Bogolubov's inequality. A weak-interaction systematic finite-temperature perturbation scheme, based on the same T-product formalism, as well as the adiabatic perturbation approach, valid for the strong-coupling case, are also outlined.
Despite the apparent simplicity of the formulation, the polaron problem has not yet been solved, and continues to attract much attention. It plays an important role in statistical mechanics and quantum field theory because it can be considered as the simplest example of a nonrelativistic quantum particle interacting with a quantum field. Therefore many sophisticated mathematical techniques have been tested for the first time using this problem as a model. A shining example of this is Feynman's functional integration method, which was applied first to the polaron problem, before becoming one of the main methods used in statistical mechanics and quantum field theory. Moreover, polaron theory is an expanding field of investigation in solid state physics because polarons are not only theoretical constructs but practically observable physical objects (see e.g. [6]).
This chapter generalizes works by N.N. Bogolubov and N.N. Bogolubov, Jr. [1,6], and \nis written with the aim of formulating methods of investigation of the electron-phonon \nsystem and the elimination of phonon operators from the corresponding kinetic equations. In particular, a kinetic equation is derived for the interaction of an electron \nwith a phonon field. Moreover, under the proper approximation, the exact Boltzmann equation for the polaron system follows from this kinetic equation. Methods of \ncalculation of the response functions (impedance and admittance), based on the "approximating" Hamiltonian with linear interaction, are also proposed. The equilibrium \ndensity probability function of the particle is also calculated.
The polaron model in ionic crystal is studied in the Bogolubov representation using a special RPA-approximation.A new exactly solvable approximated polaron model is derived and described in detail.Its free energy at finite temperature is calculated analytically.The polaron free energy in the constant magnetic field at finite temperature is also discussed.Based on the structure of the Bogolubov unitary transformed polaron Hamiltonian there is stated a very important new result: the full polaron model is exactly solvable.
The polaron model in ionic crystal is studied in the Bogolubov representation using a special RPA-approximation. A new exactly solvable approximated polaron model is derived and described in detail. Its free energy at nite temperature is calculated analytically. The polaron free energy in the constant magnetic eld at nite temperature is also discussed. Based on the structure of the Bogolubov unitary transformed polaron Hamiltonian there is stated a very important new result: the full polaron model is exactly solvable.