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Journal of the Physical Society of Japan

Theory of a Polaron at Finite Temperatures

Motohiko Saitoh

Published Sep 1, 1980• 70 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.49.878

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Physical Review B•Mar 15, 2005•2 citations
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.
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%.
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.
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.
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.