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physica status solidi (b)

Ground State Energy of Bound Polarons

Yves Lépine, G. Badder, D. Matz

Published Sep 1, 1978• 10 citations
Publisher

Abstract

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.

DOI
10.1002/pssb.2220890106

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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.

physica status solidi (b)•Dec 1, 1979•23 citations
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.
Based on the unitary transformation, a general treatment of the bound polaron is proposed. In distinction from earlier calculations, the present method gives explicit expressions for not only the energy but also the wave functions both in the ground state and in the excited states such as the relaxed excited state and the phonon scattering state. These expressions involve a summation on the complete set of eigenstates of a trial Hamiltonian, and in actual calculations the Coulomb Hamiltonian and the harmonic oscillator Hamiltonian are chosen as trial Hamiltonians. For the ground state, the energy expression is the same as that obtained by Matsuura with the path integral method. The present method is valid in the whole region of the electron-phonon coupling constant α and Coulomb force β except for β→0 and α>6.
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.
Using a variational ansatz in which pair correlations between wave vectors of virtually emitted phonons are taken into account, we have obtained better upper bounds to the polaron ground-state energy than have been available heretofore for $\ensuremath{\alpha}\ensuremath{\le}3.5$. The same variational trial functions are also used to obtain lower bounds for $\ensuremath{\alpha}\ensuremath{\le}2.5$. These lower bounds, although not completely rigorous, represent a considerable improvement over the Lieb-Yamazaki values, which are, to the author's knowledge, the only other lower bounds in the literature. The variational ansatz chosen is also suitable for practical calculations of the polaron effective mass.
The ground-state energy of the optical polaron is calculated by the generalized path-integral formulation. The ground-state approximation is made to simplify the complicated expression, and the Ritz variational and direct integration methods are used to obtain the ground-state energy in the whole range of coupling strength. The results agree with previous work, and it is found that there is a transition for coupling constant ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}9.2$. The ground-state energy obtained by harmonic approximation, which is equivalert to taking Gaussian-like trial wave functions, is compared with those obtained by Pekar's trial wave function.
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.