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THE BOGOLUBOV REPRESENTATION OF THE POLARON MODEL AND ITS COMPLETELY INTEGRABLE RPA-APPROXIMATION PART 1

Nikolai N. Bogolubov, Russian Federation, Abdus Salam, Yarema A. Prykarpatsky

Published Jan 1, 2009• 0 citations
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Abstract

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.

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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.
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.
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.
New results for the free energy of a polaron in a strong magnetic field are presented. The method used here is the extended version of Feynman's pathintegral formalism in which the most general quadratic form containing an infinite number of variational parameters is employed to simulate the variational trial action. A set of nonlinear integral equations is established to determine the upper bound for the free energy in the entire range of electron-phonon coupling constants, temperatures and magnetic fields. Analytical solutions are provided for some limiting cases of interest, and are compared critically with existing calculations of the ground-state energy at zero temperature. The free energy thus obtained is exact to the lowest order in the coupling constant and provides the good interpolation up to the strong coupling regime.