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Physical Review

Intermediate-Coupling Polaron Effective Mass

David M. Larsen

Published Oct 15, 1968• 34 citations
Publisher

Abstract

We have calculated the polaron effective mass ${m}^{*}$ by extending the variational ansatz of a previous paper, where, in the Lee-Low-Pines representation, pair correlations between wave vectors of virtually emitted phonons are taken into account. Our results suggest that (1) effective masses calculated by fourth-order perturbation theory lie below the true values but are quite accurate for $\ensuremath{\alpha}<~3$ and (2) the Feynman theory gives effective masses which are somewhat too high in the intermediate-coupling region. For $\ensuremath{\alpha}\stackrel{\ensuremath{\sim}}{<}4.5$ we propose the estimate $\frac{{m}^{*}}{m}=1+\frac{\ensuremath{\alpha}}{6}+0.02363{\ensuremath{\alpha}}^{2}+0.0014{\ensuremath{\alpha}}^{3}$.

DOI
10.1103/physrev.174.1046

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Physical review. B, Condensed matter•Apr 15, 1983•23 citations
We have calculated the polaron effective mass m∗{m}^{*}m∗ by extending the variational ansatz of a previous paper, where, in the Lee-Low-Pines representation, pair correlations between wave vectors of virtually emitted phonons are taken into account. Our results suggest that (1) effective masses calculated by fourth-order perturbation theory lie below the true values but are quite accurate for \ensuremathα< 3\ensuremath{\alpha}<~3\ensuremathα< 3 and (2) the Feynman theory gives effective masses which are somewhat too high in the intermediate-coupling region. For \ensuremathα<\ensuremath∼4.5\ensuremath{\alpha}\stackrel{\ensuremath{\sim}}{<}4.5\ensuremathα<\ensuremath∼​4.5 we propose the estimate m∗m=1+\ensuremathα6+0.02363\ensuremathα2+0.0014\ensuremathα3\frac{{m}^{*}}{m}=1+\frac{\ensuremath{\alpha}}{6}+0.02363{\ensuremath{\alpha}}^{2}+0.0014{\ensuremath{\alpha}}^{3}mm∗​=1+6\ensuremathα​+0.02363\ensuremathα2+0.0014\ensuremathα3.
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.
A refined approach where the wave function is gradually improved is developed in the one-dimensional (1D) large-Frohlich-polaron problem. The results for the ground-state energy, the effective mass, and the average number of virtual phonons are obtained for a wide range of the coupling constant It, and are in agreement with those of the Feynman path-integral formalism. In addition, in the weak-coupling limit, the expansions of these observables in powers of the coupling constant are exactly calculated up to the alpha 3 -term. They are E=- alpha -0.06066 alpha 2 -0.00844 alpha 3 , m*=1+0.5 alpha +0.19194 alpha 2 +0.06912 alpha 3 , and N=0.5 alpha +0.12132 alpha 2 +0.029540 alpha 3 , which agree with the known results of the fourth-order perturbation theory.
A general method is presented for obtaining an accurate intermediate-coupling theory from weak- and strong-coupling perturbation theory. The method uses two-point Pad\'e approximants to extrapolate (low-order) expansions about the weak- and the strong-coupling limits into the intermediate-coupling regime. The method is used to evaluate the ground-state energy and effective mass of the polaron with gratifying success. In addition, the weak-coupling perturbation theory of the polaron dispersion relation, ground-state energy, and effective mass are extended to fourth, sixth, and fourth order, respectively. A scheme based on two-point Pad\'e approximants is used to obtain an optimal polaron dispersion relation.
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.