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arXiv:cond-mat/9903011 · Physical review. B, Condensed matter

Ground-state dispersion and density of states from path-integral Monte Carlo: Application to the lattice polaron

P. E. Kornilovitch

Published Aug 1, 1999• 43 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.60.3237

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Journal of Physics Condensed Matter•Dec 1, 1997•9 citations
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.
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.
We developed a path-integral quantum Monte Carlo-based methodology for calculation of polaron mobility in systems with electron-phonon interaction. Within the method, the current-current correlation function in both the imaginary and real time is calculated in a numerically exact way. The choice of basis for representation of the path integral enabled us to reduce the sign problem and perform real-time calculations for longer times. The DC polaron mobility was extracted by performing analytic continuation that makes use of both the real and imaginary-time data. The method was applied to the Holstein polaron model in one dimension. We obtained reliable results for the temperature dependence of the Holstein polaron mobility for interactions ranging from weak to strong and temperatures that are not too low.
An efficient continuous-time path-integral quantum Monte Carlo algorithm for the lattice polaron is presented. It is based on Feynman's integration of phonons and subsequent simulation of the resulting single-particle self-interacting system. The method is free from the finite-size and finite-time-step errors and works in any dimensionality and for any range of electron-phonon interaction. The ground-state energy and effective mass of the polaron are calculated for several models. The polaron spectrum can be measured directly by Monte Carlo, which is of general interest.
The path-integral approach to the tight-binding polaron is extended to multiple optical phonon modes of arbitrary dispersion and polarization. The nonlinear lattice effects are neglected. Only one electron band is considered. The electron-phonon interaction is of the density-displacement type, but can be of arbitrary spatial range and shape. Feynman's analytical integration of ion trajectories is performed by transforming the electron-ion forces to the basis in which the phonon dynamical matrix is diagonal. The resulting polaron action is derived for the periodic and shifted boundary conditions in imaginary time. The former can be used for calculating polaron thermodynamics while the latter for the polaron mass and spectrum. The developed formalism is the analytical basis for numerical analysis of such models by path-integral Monte Carlo methods.
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 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 .