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Journal of the Physical Society of Japan

An Improvement of the Feynman Action in the Theory of Polaron. I

Ryuzo Abe, Ken‐ichi Okamoto

Published Nov 1, 1971• 21 citations
Publisher

Abstract

An improvement of the Feynman action is discussed by introducing two additional parameters to Feynman's original polaron theory. Both upper bounds of the ground state energy and corrections to the polaron mass are obtained for some values of the coupling constant. However, corrections are so small that Feynman's trial action is proved to be a very good description of the polaron state.

DOI
10.1143/jpsj.31.1337

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arXiv:cond-mat/9910327•Oct 21, 1999•0 citations
An improvement of the Feynman action is discussed by introducing two additional parameters to Feynman's original polaron theory. Both upper bounds of the ground state energy and corrections to the polaron mass are obtained for some values of the coupling constant. However, corrections are so small that Feynman's trial action is proved to be a very good description of the polaron state.
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.
The polaron path integral introduced by Feynman is generalized to the case of an electron moving in d space dimensions. A scheme of systematic calculations of the obtained path integral is developed to estimate the ground-state energy of the polaron in the same way for different values of the electron-phonon coupling constant \ensuremath{\alpha}. The leading-order term in this approach yields an upper bound to the polaron self-energy and improves Feynman's variational estimates for d=2 and d=3. A scaling relation between polaron self-energies for different space dimensions is obtained for this term. The next correction to this estimation is calculated by numerical integration for d=2 and d=3.
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.
We evaluate the Fourier path integral for an optical polaron by the method of partial averaging. Higher Fourier coefficients are taken into account by averaging with Feynman's trial action, whereas the first N coefficients are treated explicitly by numerical integration. This method removes the Coulomb singularity in the effective action, yields lower bounds to the partition function, and systematically improves Feynman's variational results. We present numerical results for the exact ground-state energy for small and intermediate values of the coupling constant.
The main quasi-particle characteristics of the one-dimensional polaron are estimated within and beyond the most general Gaussian approximation at arbitrary electron-phonon coupling. We have derived explicitly the ground-state energy and the effective mass in the weak- and strong-coupling regimes. For arbitrary coupling, the Gaussian leading-order term of the polaron self energy improves the corresponding Feynman estimate and represents the lowest upper bound available. We have calculated the next non-Gaussian corrections. Taking into account systematically higher-order corrections does not perturb considerably the obtained results.