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Physical review. B, Solid state

Perturbation Theory for the Bound Polaron

P. K. Dubey, V. V. Paranjape, M. M. L. Anand

Published Feb 15, 1973• 6 citations
Publisher

Abstract

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 $O〈f|O|g〉=〈g|{O}^{*}|f〉$. 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.

DOI
10.1103/physrevb.7.1718

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Direct path-integral treatment of the polaron problem

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Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that $n\mathrm{th}$-order terms in the polaron coupling constant are obtained from $n\mathrm{th}$-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a $2n\mathrm{th}$-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.

Physical review. B, Condensed matter•Apr 15, 1983•23 citations
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.
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 one-dimensional optical polaron is treated on the basis of perturbation theory in the weak-coupling limit. A special matrix diagrammatic technique is developed. It is shown how to evaluate all terms of the perturbation theory for the ground-state energy of a polaron to any order by means of this technique. The ground-state energy is calculated up to eighth order of perturbation theory. The effective mass of an electron is obtained up to sixth order of perturbation theory. The radius of convergence of the series obtained is estimated.
Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that nthn\mathrm{th}nth-order terms in the polaron coupling constant are obtained from nthn\mathrm{th}nth-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a 2nth2n\mathrm{th}2nth-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.