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arXiv:cond-mat/0011332 · Physics Letters A

On the best quadratic approximation in Feynman's path integral treatment of the polaron

R. Rosenfelder, ANDREAS W. SCHREIBER

Published Jun 1, 2001• 19 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(01)00287-0

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Abstract The dielectric polaron is treated using the Gross unitary transform and the Matz‐Green's function techniques. A two‐particle Feynman‐like model is introduced to compare these methods with the best Feynman path integral formulation. It is found that the introduction of a second particle in the model problem does not improve on the usual result for a one‐particle model. However, a comparison of these different approaches leads to the conclusion that no vertex contributions are contained in the Feynman approximation. It is also concluded that it is the very special treatment of the kinetic energy of the electron in Feynman's theory that makes his results lower in energy.
In order to calculate the effective mass and free energy of a polaron at finite temperatures, Feynman's path-integral formalism is generalized in the following two points: (a) the most general quadratic form is used to fully optimize the partition function, and (b) the new definition of the effective mass is introduced such that the effective mass is determined from the acceleration rate against the fictitious driving force which is incorporated in the original Lagrangian. The use of (a) leads to a non-linear integral equation which determines the best trial form. This non-linear integral equation is solved analytically for the Fröhlich polaron model to obtain the explicit expressions for the effective mass and free energy at finite temperatures.