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arXiv:cond-mat/0407250 · Physical Review B

Effect of electron-phonon interaction range on lattice polaron dynamics: A continuous-time quantum Monte Carlo study

Paul Spencer, John H. Samson, P. E. Kornilovitch, A. S. Alexandrov

Published May 31, 2005• 49 citations
arXivHTMLPDF

Abstract

We present the numerically exact ground-state energy, effective mass, and isotope exponents of a one-dimensional lattice polaron, valid for any range of electron-phonon interaction, applying a continuous-time quantum Monte Carlo (QMC) technique in a wide range of coupling strength and adiabatic ratio. The QMC method is free from any systematic finite-size and finite-time-step errors. We compare our numerically exact results with analytical weak-coupling theory and with the strong-coupling $1∕\ensuremath{\lambda}$ expansion. We show that the exact results agree well with the canonical Fr\"ohlich and Holstein-Lang-Firsov theories in the weak and strong coupling limits, respectively, for any range of interaction. We find a strong dependence of the polaron dynamics on the range of interaction. An increased range of interaction has a similar effect to an increased (less adiabatic) phonon frequency: specifically, a reduction in the effective mass.

DOI
10.1103/physrevb.71.184310

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We present the numerically exact ground-state energy, effective mass, and isotope exponents of a one-dimensional lattice polaron, valid for any range of electron-phonon interaction, applying a continuous-time quantum Monte Carlo (QMC) technique in a wide range of coupling strength and adiabatic ratio. The QMC method is free from any systematic finite-size and finite-time-step errors. We compare our numerically exact results with analytical weak-coupling theory and with the strong-coupling 1∕\ensuremathλ1∕\ensuremath{\lambda}1∕\ensuremathλ expansion. We show that the exact results agree well with the canonical Fr"ohlich and Holstein-Lang-Firsov theories in the weak and strong coupling limits, respectively, for any range of interaction. We find a strong dependence of the polaron dynamics on the range of interaction. An increased range of interaction has a similar effect to an increased (less adiabatic) phonon frequency: specifically, a reduction in the effective mass.
We study the effects of lattice type on polaron dynamics using a continuous-time quantum Monte Carlo approach. Holstein and screened Fr\"ohlich polarons are simulated on a number of different Bravais lattices. The effective mass, isotope coefficients, ground-state energy and energy spectra, phonon numbers, and density of states are calculated. In addition, the results are compared with weak- and strong-coupling perturbation theory. For the Holstein polaron, it is found that the crossover between weak- and strong-coupling results becomes sharper as the coordination number is increased. In higher dimensions, polarons are much less mobile at strong coupling, with more phonons contributing to the polaron. The total energy decreases monotonically with coupling. Spectral properties of the polaron depend on the lattice type considered, with the dimensionality contributing to the shape and the coordination number to the bandwidth. As the range of the electron-phonon interaction is increased, the coordination number becomes less important, with the dimensionality taking the leading role.
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 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 describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.
We explore the quasiparticle properties of lattice polarons on the basis of a quite general electron-phonon Hamiltonian with a long-range displacement-type interaction. To treat the dynamical quantum phonons without significant loss of accuracy we adapt an exact Lanczos diagonalization method and compute various static and dynamical quantities, such as the electron-lattice correlation function, the polaron band dispersion, the effective polaron mass, the kinetic energy, the single-particle spectral function, and the optical conductivity, on finite one-dimensional lattices for a wide range of model parameters. We compare the results with those obtained for the standard Holstein model with short-range electron-phonon interaction only.
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 .