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arXiv:cond-mat/0404121

On Feynman's calculation of the Froehlich polaron mass

P. E. Kornilovitch

Published Apr 6, 2004• 0 citations
arXivHTMLPDF

Abstract

Feynman's formula for the effective mass of the Froehlich polaron is rederived from the formalism of projected partition functions. The mass is calculated as inverse of the diffusion coefficient of the polaron trajectory in imaginary time. It is shown that correlation between the electron and phonon boundary conditions in imaginary time is necessary for consistent derivation of the Feynman result.

DOI
10.48550/arxiv.cond-mat/0404121

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Feynman's formula for the effective mass of the Froehlich polaron is rederived from the formalism of projected partition functions. The mass is calculated as inverse of the diffusion coefficient of the polaron trajectory in imaginary time. It is shown that correlation between the electron and phonon boundary conditions in imaginary time is necessary for consistent derivation of the Feynman result.
The polaron effective mass using the path integral approach was first introduced by Feynman. He has shown that from the partition function, he obtained the ground state energy and the effective mass mF of the polaron. By using the partition function, Krivoglaz and Pekar proposed another definition of the effective mass mKP. Sa-yakanit has shown that using the density matrix Feynmam's path integral representation, it is possible to obtain both Feynman's and Krivoglaz and Pekar's results. In this thesis we propose a consistent definition of effective mass that mF and mKP should be equal. We have shown that the wave functions obtained from the density matrix should be normalized. We have also shown that the dynamic mass of the polaron can be obtained from the density matrix. The comparison of our results with recent work by Wang et.al.are given.
A Monte Carlo algorithm for calculating polaron effective mass is proposed. It is based on the path-integral representation of a partial partition function with fixed total quasimomentum. Phonon degrees of freedom are integrated out analytically resulting in a single-electron system with retarded self-interaction and open boundary conditions in imaginary time. The effective mass is inversely proportional to the covariance of total energy calculated on an electron trajectory and the square distance between ends of the trajectory. The method has no limitations on values of model parameters and on the size and dimensionality of the system although large statistics are required for stable numerical results. The method is tested on the one-dimensional Holstein model for which simulation results are presented.
Using the exactly soluble Feynman polaron as a point of departure, I discuss and compare the properties of three projected, or momentum dependent, partition functions of a particle coupled to an environment and reconsider the anomalies that appear in some definitions of the effective mass. Although the momentum-dependent free energy associated with the projected partition function at fixed total momentum has exponentially suppressed finite-temperature corrections and yields the correct zero-temperature effective mass, the same is not necessarily true of the free energies associated with the simpler momentum-dependent partition functions derived from either the reduced density matrix of the particle or the density matrixlike quantity obtained by replacing in the particle's nonlocal effective action the phonon Green's function with its zero-temperature form.
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.
The approximated density matrix of the polaron system obtained in our previous paper for deriving the Feynman effective mass mF{m}_{F}mF​ and the Krivoglaz-Pekar mass mKP{m}_{\mathrm{KP}}mKP​ is reconsidered. It is shown that to obtain a consistent definition of the polaron effective mass one must impose the condition mF=mKP.{m}_{F}{=m}_{\mathrm{KP}}.mF​=mKP​. This consistent definition is based on the fact that the wave functions derived from the density matrix corresponding to each excited state must be normalized and orthogonal. This condition leads to a consistent definition of the effective mass and a different variation principle for treating excited states. Numerical results are presented.