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arXiv:cond-mat/9910436 · Physical Review Letters

Band Structure of the Jahn-Teller Polaron from Quantum Monte Carlo

P. E. Kornilovitch

Published Feb 14, 2000• 20 citations
arXivHTMLPDF

Abstract

A path-integral representation for the Jahn-Teller polaron (JTP) is constructed. It leads to a perturbation series that can be summed exactly by the diagrammatic quantum Monte Carlo technique. The ground-state energy, effective mass, spectrum, and density of states of the three-dimensional JTP are calculated with no systematic errors. The band structure of JTP is found to be similar to that of the Holstein polaron. The mass of JTP increases exponentially with the coupling constant. At small phonon frequencies, the spectrum of JTP is flat at large momenta, which leads to a strongly distorted density of states with a massive peak at the top of the band.

DOI
10.1103/physrevlett.84.1551

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A path-integral representation for the Jahn-Teller polaron (JTP) is constructed. It leads to a perturbation series that can be summed exactly by the diagrammatic quantum Monte Carlo technique. The ground-state energy, effective mass, spectrum, and density of states of the three-dimensional JTP are calculated with no systematic errors. The band structure of JTP is found to be similar to that of the Holstein polaron. The mass of JTP increases exponentially with the coupling constant. At small phonon frequencies, the spectrum of JTP is flat at large momenta, which leads to a strongly distorted density of states with a massive peak at the top of the band.
The properties of the polaron and bipolaron are explored in the one-dimensional Jahn-Teller model with dynamical quantum phonons. The ground-state properties of the polaron and bipolaron are computed using a recently developed variational method. Dynamical properties of the ground state of a polaron are investigated by calculating the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega}).$ Our numerical results suggest that the Jahn-Teller and Holstein polarons are similar. However, in the strong-coupling regime qualitative differences in $\ensuremath{\sigma}(\ensuremath{\omega})$ between the two models are found and discussed. The influence of the electron-phonon coupling and the electrostatic repulsion on the bipolaron binding energy, bipolaron masses, and correlation functions is investigated.
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A detailed study of the Fr\"ohlich polaron model is performed on the basis of diagrammatic quantum Monte Carlo method [N. V. Prokof'ev and B. V. Svistunov, Phys. Rev. Lett. 81,81,81, 2514 (1998)]. The method is further developed both quantitatively (performance) and qualitatively (new estimators), and is enhanced by spectral analysis of the polaron Green's function, within an approach developed in the present paper. We present up to date results for the binding energy, and make available precise data for the effective mass, including the region of intermediate and strong couplings. We look at the structure of the polaron cloud and answer such questions as the average number of phonons in the cloud and their number/momentum distribution. The spectral analysis reveals nontrivial structure of the spectral density at intermediate and large coupling: the spectral continuum features pronounced peaks that we attribute to unstable excited states of the polaron.
A path-integral quantum Monte Carlo method is applied to the two-dimensional single-polaron Holstein model. Simulation data are shown to exhibit -scaling with the number of imaginary-time slices M used in the Trotter decomposition. Numerical extrapolation to yields very accurate estimation of polaron energetic characteristics. Kinetic, potential and total energy are calculated as functions of the electron - phonon coupling strength for different phonon frequencies. The small-polaron regime is found in the strong-coupling limit for all of the frequencies studied. The transition to the polaron state is sharp in the adiabatic regime but broadens as the frequency increases. In the adiabatic limit the polaron forms at .
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.
A formula is derived that relates the ground-state dispersion of a many-body system with the end-to-end distribution of paths with open boundary conditions in imaginary time. The formula does not involve the energy estimator. It allows direct measurement of the ground-state dispersion by quantum Monte Carlo methods without analytical continuation or auxiliary fitting. The formula is applied to the lattice polaron problem. The exact polaron spectrum and density of states are calculated for several models in one, two, and three dimensions. In the adiabatic regime of the Holstein model, the polaron density of states deviates spectacularly from the free-particle shape.