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Physical review. B, Condensed matter

Functional-integral approach to the polaron mass

B. Gerlach, Hartmut Löwen, H. Schliffke

Published Oct 15, 1987• 20 citations
Publisher

Abstract

Making use of functional-integration techniques, we calculate the formal free energy of a polaron in the subspace of fixed total momentum Q. Utilizing this result, we prove an analytical expression for the polaron mass. We include numerical results of variational and Monte Carlo types. As a by-product, we comment on the position of the continuum edge of the polaronic energy spectrum for fixed Q.

DOI
10.1103/physrevb.36.6320

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Making use of functional-integration techniques, we calculate the formal free energy of a polaron in the subspace of fixed total momentum Q. Utilizing this result, we prove an analytical expression for the polaron mass. We include numerical results of variational and Monte Carlo types. As a by-product, we comment on the position of the continuum edge of the polaronic energy spectrum for fixed Q.
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.
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.
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.
We consider a (band) electron---LO-phonon system described by Fr\"ohlich's Hamiltonian. Making use of functional-integration methods, one can eliminate the phonon terms exactly leading to an effective one-particle system with the same spectral properties as the original one. The (formal) free energy of this effective system is calculated numerically by means of the Monte Carlo technique. We give a detailed description and first-error estimates of our numerical procedure. Furthermore, we draw a comparison with previous work. Our results compare favorably with all earlier results of variational type of which we are aware.
The basic principles behind a path integral approach to the lattice polaron are reviewed. Analytical integration of phonons reduces the problem to one self-interacting imaginary-time path, which is then simulated by Metropolis Monte Carlo. Projection operators separate states of different symmetry, which provides access to various excited states. Shifted boundary conditions in imaginary time enable calculation of the polaron mass, spectrum and density of states. Other polaron characteristics accessible by the method include the polaron energy, number of excited phonons and isotope exponent on mass. Monte Carlo updates are formulated in continuous imaginary time on infinite lattices and as such provide statistically unbiased results without finite-size and finite time-step errors. Numerical data are presented for models with short-range and long-range electron?phonon interactions.