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Physical Review Letters

Fourier path integrals, partial averaging, and the polaron problem

Constantia Alexandrou, Wolfgang Fleischer, R. Rosenfelder

Published Nov 19, 1990• 28 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.65.2615

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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.
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.
This chapter presents a brief introduction to the path-integral method before applying it to the polaron problem. The coupling theory is based on the path-integral formulation of quantum mechanics developed by Feynman in 1955. The path-integral formalism proves to be a useful tool for many field-theoretic problems. Feynman's path-integral method as applied by Feynman to determine the polaron effective mass has an intuitive rather than formal basis. The comparison revealed unequivocally the superiority of the Feynman model over the entire range of the coupling constant. Schultz also simulated the Feynman approximation by replacing the whole lattice with a second, fictitious particle harmonically bound to the electron by a spring of constant k. Thus, the Feynman's strong-coupling energy is better than the Landau-Pekar energy with the Gaussian trial function. The numerical results reported by Marshall and Mills show that the second-order correction to the Feynman result for the ground state energy is less than 2% for the entire coupling rang.
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.
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.