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Physical review. B, Condensed matter

The Feynman effective mass of the polaron

Virulh Sa‐Yakanit

Published Feb 15, 1979• 17 citations
Publisher

Abstract

The polaron effective mass of the Fr\"ohlich polaron model is considered. It is shown that the ad hoc effective-mass expression given by Feynman can be derived from the off-diagonal density matrix in the path-integral representation while the effective mass given by Krivoglaz and Pekar is obtained from the study of the diagonal density matrix.

DOI
10.1103/physrevb.19.2377

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Alternative derivation of the effective mass of the polaron

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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.

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The effective mass of the Fröhlich polaron model in d spatial dimensions is obtained within the Gaussian-equivalent representation, that is, within a suitably rearranged perturbative path-integral approach, whose leading order already takes care of all Gaussian fluctuations. In doing so, the authors extend their earlier investigations ( Phys. Rev. B 50 3733 (1994)) of the polaron ground-state energy. Analytical expressions for the effective mass are derived in the weak- and strong-coupling limit. In the intermediate-coupling regime, the effective mass is calculated numerically for the cases of two and three dimensions. The new results are compared with previously known data.

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Polaron density matrix and effective mass at finite temperature

Ashok Sethia, Fumio Hirata, Yoshitaka Tanimura +1 more

We calculate the density matrix elements of the polaron using Feynman's variational method. The density matrix is diagonalized, and the eigenvalues and eigenfunctions are derived. These results and the calculated density matrix are used to evaluate the root-mean-square (rms) displacements ${R}_{\ensuremath{\beta}}$ and the effective mass (EM) ${m}^{*}$ for various values of coupling strengths between electron and medium and of the temperature. We find that EM is related to the rms displacement. The temperature dependence of the EM is in qualitative agreement with experiment.

Physical review. B, Condensed matter•Sep 1, 1999•6 citations
The polaron effective mass of the Fr"ohlich polaron model is considered. It is shown that the ad hoc effective-mass expression given by Feynman can be derived from the off-diagonal density matrix in the path-integral representation while the effective mass given by Krivoglaz and Pekar is obtained from the study of the diagonal density matrix.
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.
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.
In the expansion of the ground-state energy of a polaron in a weak magnetic field, the zeroth-order term is the polaron self-energy, while the first-order term is inversely proportional to the polaron effective mass. The effective mass so obtained is exactly equivalent to the freepolaron effective mass as defined by Fr\"ohlich. This equivalence principle is used to approximate the polaron effective mass by employing an approximate expression for the ground-state energy of a polaron in a weak magnetic field obtained by applying Feynman's path-integral variational method. The resultant polaron effective mass is found to be higher than Feynman's result by less than 1%.
The effective mass of the Fröhlich polaron model in d spatial dimensions is obtained within the Gaussian-equivalent representation, that is, within a suitably rearranged perturbative path-integral approach, whose leading order already takes care of all Gaussian fluctuations. In doing so, the authors extend their earlier investigations ( Phys. Rev. B 50 3733 (1994)) of the polaron ground-state energy. Analytical expressions for the effective mass are derived in the weak- and strong-coupling limit. In the intermediate-coupling regime, the effective mass is calculated numerically for the cases of two and three dimensions. The new results are compared with previously known data.
We calculate the density matrix elements of the polaron using Feynman's variational method. The density matrix is diagonalized, and the eigenvalues and eigenfunctions are derived. These results and the calculated density matrix are used to evaluate the root-mean-square (rms) displacements ${R}_{\ensuremath{\beta}}$ and the effective mass (EM) m∗{m}^{*}m∗ for various values of coupling strengths between electron and medium and of the temperature. We find that EM is related to the rms displacement. The temperature dependence of the EM is in qualitative agreement with experiment.