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Alternative derivation of the effective mass of the polaron

K. Tayanasanti, Virulh Sa-Yakanit

Published Jan 1, 1996• 0 citations

Abstract

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.

DOI: 10.58837/chula.the.1996.2337

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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.

The approximated density matrix of the polaron system obtained in our previous paper for deriving the Feynman effective mass

{m}_{F}

and the Krivoglaz-Pekar mass

{m}_{\mathrm{KP}}

is reconsidered. It is shown that to obtain a consistent definition of the polaron effective mass one must impose the condition

{m}_{F}{=m}_{\mathrm{KP}}.

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%.

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.

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.