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Canadian Journal of Physics

Fock approximation to the large polaron in a magnetic field

Yves Lépine, D. Matz

Published Oct 1, 1976• 37 citations
Publisher

Abstract

We study the large polaron ground state energy in the presence of a constant and uniform magnetic field within the Fock approximation. By use of a new trial spectrum we find a new upper bound to the ground state energy for all magnetic fields and electron–phonon coupling constants. The trial spectrum has the novel feature of keeping cylindrical symmetry for certain values of coupling, even in the absence of magnetic field.

DOI
10.1139/p76-236

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physica status solidi (b)•Dec 1, 1979•23 citations
We study the large polaron ground state energy in the presence of a constant and uniform magnetic field within the Fock approximation. By use of a new trial spectrum we find a new upper bound to the ground state energy for all magnetic fields and electron–phonon coupling constants. The trial spectrum has the novel feature of keeping cylindrical symmetry for certain values of coupling, even in the absence of magnetic field.
Abstract The properties of the ground state of the bound large polaron within the Fock approximation are investigated by using a new model spectrum which is known to describe better the polaron at low external fields. The results are compared with those of Matsuura who used the same approximation, obtained from a path integral formulation, with the usual Gaussian and Coulombic trial spectra, Both formulations are variational. It is found that the new spectrum gives better results (lower variational energies) than those of Matsuura for intermediate or large values of electron–phonon coupling and small values of the electrostatic coupling of the electron to the defect. This is related to that fact that the polaron seems to prefer a cylindrical symmetry to a spherical one in the absence of external fields.
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.
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.
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.
Abstract New results for the free polaron ground state energy are presented, based on a non‐perturbative Fock approximation mean field theory. From variational calculations a new energy upper bound for all couplings is obtained, whose newness lies in its cylindrical symmetry, thu sbreaking the spherical symmetry of the Fröhlich Hamiltonian. For α (the electron–phonon coupling constant)≦3 a translational invariant state is found, for 3 ≦ α ⪅ 3.8 a two‐dimensional localized state while for α ⪆ 3.8, a three‐dimensional cylindrical localized state is obtained. By linking the variational approach to a dynamical approximation of the Fröhlich Hamiltonian, the possibility is suggested of second‐order phase transitions for the structure of the polaron, in the framework of the Fock approximation.