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Functional variational approach to polaron models

E. A. Kochetov, S.P. Kuleshov, Smondyrev

Published May 1, 1982• 2 citations
OpenAlex

Abstract

A general approach to the investigation of polaron systems using path integrals is presented. The basis of the approach is a functional variational method which generalizes Feynman's to the case of a nonvanishing momentum of the system. The characteristics of a free polaron and the behavior of a polaron system in the presence of weak external perturbations are investigated. Questions associated with polaron thermodynamics are considered. The high-temperature behavior of the mean energy and the polaron mass are studied, as is the reaction of a polaron system at finite temperatures to an external perturbation.

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Physical review. B, Solid state•Nov 15, 1970•24 citations
A general approach to the investigation of polaron systems using path integrals is presented. The basis of the approach is a functional variational method which generalizes Feynman's to the case of a nonvanishing momentum of the system. The characteristics of a free polaron and the behavior of a polaron system in the presence of weak external perturbations are investigated. Questions associated with polaron thermodynamics are considered. The high-temperature behavior of the mean energy and the polaron mass are studied, as is the reaction of a polaron system at finite temperatures to an external perturbation.
It is well known that two qualitatively different pictures arise in the description of the motion of an electron in an ionic crystal. In the first case, when the electron interacts weakly with the ionic lattice of the crystal, its motion is the same as the motion of a free band electron whose energy is shifted downward with respect to the bottom of the conduction band, and the effective mass is replaced by the renormalized mass (weak-coupling polaron). In the other limiting case, when the electron interacts strongly with the ionic crystal, an entire series of different self-consistent states of the electron and lattice, each of which has its own effective mass and radius (strong-coupling polaron), arises. Although the motion of an electron in an ionic crystal is described in both limiting cases with the help of the Pekar-Frohlich Hamiltonian, each case requires its own special methods of analysis. Thus, for example, in the strong-coupling theory a special form of adiabatic perturbation theory has been developed in which the trans-lational degeneracy is removed before the perturbation expansion is made. It has been possible to relate both descriptions within the framework of Feynman's varia-tional approach, in which a smooth upper bound was obtained for the energy of the ground state of the polaron for all values of the electron-phonon coupling constant a (see Fig. 1). We call attention in this connection to Ref. 6, where the thermodynamic properties are calculated with the help of a special method for performing the calculations and for finding averages of T-products. The approach developed there is more convenient for performing specific calculations than is the use of path integrals. It seems to us that it has definite advantages in the construction of general proofs concerning the choice of the approximating Hamiltonians (this is illustrated in Ref. 6 for the example of the linear Bogolyubov model), in the investigation of the characteristics of the behavior of a polaron in external fields, etc.
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.
Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that nthn\mathrm{th}nth-order terms in the polaron coupling constant are obtained from nthn\mathrm{th}nth-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a 2nth2n\mathrm{th}2nth-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.
New variational ansatz for the large-radius Fröhlich polaron is considered. The corresponding operator estimation for the energy of polaron proves to be very similar to the result found by Feynman on the basis of the variational principle for the functional integral of the system. It allows us to make clear the problem of ``phase transition'' and to anylize the structure of the localized state of the polaron in the intermediate coupling regime. It is shown that the analogous state exists also for the model of particle-field interaction with the divergent perturbation theory.
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%.