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International Journal of Modern Physics B

THE LINEARIZED POLARON MODEL SYSTEM IN A MAGNETIC FIELD

A. A. Ghazaryan

Published Nov 10, 2008• 0 citations
Publisher

Abstract

This article is devoted to the calculation of the free energy on electron-phonon interaction in the external magnetic field according to Green's function method. It should be noted that creation of structures with predicted features occurred (with the use of experimental techniques). To sum up, it should be noted that the given result coincides with the result of Devreese and Brosens, which was obtained by Feynman's path integral method.

DOI
10.1142/s0217979208049194

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Soviet physics. Doklady•Oct 1, 1984•0 citations
This article is devoted to the calculation of the free energy on electron-phonon interaction in the external magnetic field according to Green's function method. It should be noted that creation of structures with predicted features occurred (with the use of experimental techniques). To sum up, it should be noted that the given result coincides with the result of Devreese and Brosens, which was obtained by Feynman's path integral method.
The free energy of an ideal polaron gas in a static magnetic field is calculated using Feynman's path-integral formalism. The trial action, used in Feynman's polaron theory, is extended to take into account the anisotropy of the effective electron-phonon interaction. This results in a free-energy expression with four variational parameters. According to Feynman's conjecture, the resulting free energy is an upper bound to the exact result. The approximate free energy is expected to provide accurate results for arbitrary electron-phonon coupling strength ($\ensuremath{\alpha}$), temperature (TTT), and magnetic field strength (H\mathcal{H}H). The free energy per polaron is evaluated for limiting values of $\ensuremath{\alpha}$, TTT, and H\mathcal{H}H.
The ground-state energy of the free polaron is calculated, using the Fock approximation. A new model Hamiltonian involving an electron in a harmonic potential of cylindrical symmetry in the presence of a variational magnetic field, internal in origin, is used. This magnetic field, which has no physical existence, introduces the possibility of symmetry breaking. This approach is shown to be equivalent to that of Peters and Devreese (1982), who used path integrals to study the properties of the polaron in a magnetic field, if the variational parameters of their model action are chosen to match those of the author's model and if the magnetic field in the model action is used as a variational parameter whose origin is internal. Using that approach, a strong-coupling ground-state energy much lower than the usual one is found (E 0 =-0.1658 alpha 2 ). It is shown that such a low value cannot be obtained from the adiabatic approximation because a model spectrum having a degenerate ground state has been used. His results call into question the assumption that the energies obtained from the Fock approximation or from the Feynman integrals are upper bounds to the polaron ground-state energy in the case of model actions or Hamiltonians involving magnetic fields.
Energy levels of two-dimensional Fr\"ohlich polarons in a magnetic field are calculated variationally by use of a generalization of the Lee-Low-Pines ansatz for the variational wave function allowing the phonon distribution function to depend on the electronic coordinates. It is shown that this choice leads to the correct interpolation between the weak- and strong-field limits. Comparing our ground-state energies with those obtained within second-order perturbation theory and the path-integral method, we argue that neither of these methods gives a satisfactory description of the dynamics of the electron-phonon interaction under the influence of an external field.
Abstract It is shown that the variational principle connected with a quadratic trial Lagrangian in Feynman's path‐integral representation can be applied for calculating the energy eigenvalues and thermodynamic functions of systems in a uniform arbitrary external magnetic field. As an example the ground‐state energy of a hydrogen‐like atom in a magnetic field is calculated. A theory of shallow F‐centres in the presence of an arbitrary external magnetic field is developed. The results are valid for arbitrary magnitudes of the polaron coupling constant.
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.
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.