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Journal of Physics C Solid State Physics

Perturbation theory for the bound polaron

David M. Larsen

Published Nov 1, 1970• 4 citations
Publisher

Abstract

Stoneham (see ibid., vol.3, L131-3, 1970) attempted to evaluate numerically the second order perturbation theory correction, delta ', to the ground state of a bound polaron. The author points out that Stoneham's numerical evaluation stars from a formula which itself is not exact, contrary to his implication.
DOI
10.1088/0022-3719/3/11/001

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arXiv:2011.08638•Journal of Physics Communications•Oct 27, 2022•1 citation
Sak and Wang, Arora, and Matsuura have recently calculated the ground-state energy of a bound polaron in second-order perturbation theory by making use of an identity that for a Hermitian operator O〈f∣O∣g〉=〈g∣O∗∣f〉O〈f|O|g〉=〈g|{O}^{*}|f〉O〈f∣O∣g〉=〈g∣O∗∣f〉. It is the purpose of the present communication to show that the identity is incorrectly used by these authors since one of the integrals involved in the identity is divergent and in this situation the identity does not hold. Fortunately, this does not significantly affect the specific conclusions of their work, but amends the analysis in arriving at their result.
For pt.I see Phys. Lett. A, vol.71, p.385 (1979). A recently developed non-perturbative iterative variational method has been applied to the bound polaron problem. The electron-phonon Hamiltonian is reduced in successive orders resulting in a transformed electronic Hamiltonian which gets an attractive phonon-induced potential from each order of reduction. The transformed electronic Hamiltonian has been solved numerically for a series of values of the parameters alpha , beta . Relation to perturbative solutions is discussed. Interesting localisation criteria derived from the calculated wavefunctions are also discussed.
The Schrodinger-Coulomb Green function is discussed as a tool for evaluating terms in bound-state perturbation theory. As an example the energy shift of the ground state of hydrogen in a uniform electric field is considered. The Green function is used in an evaluation of the second-order perturbation theory expression for this energy shift. For comparison a direct numerical approach to the evaluation of this energy shift is also described. The Green function approach is simpler and more elegant, and leads to an exact result.
Abstract An analytic theory for the spectral function for electrons coupled with phonons is formulated in the adiabatic limit. In the case when the chemical potential is large and negative μ → − ∞ the ground state does not have the adiabatic deformation and the spectral function is defined by the standard perturbation theory. In this limit we use the diagram technique in order to formulate an integral equation for the renormalized vertex. The spectral function was evaluated by solving the Dyson’s equation for the self-energy with the renormalized vertex. The moments of the spectral function satisfy the exact sum rules up to the 7th moment. In the case when the chemical potential is pinned at the polaron binding energy the spectral function is defined by the ground state with a nonzero adiabatic deformation. We calculate the spectral function with the finite polaron density in the adiabatic limit. We also demonstrate how the sum rules for higher moments may be evaluated in the adiabatic limit. Contrary to the case of zero polaron density the spectral function with the finite polaron concentration has some contributions which are characteristic for polarons.