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arXiv:cond-mat/9808155 · Physical Review Letters

Continuous-Time Quantum Monte Carlo Algorithm for the Lattice Polaron

P. E. Kornilovitch

Published Dec 14, 1998• 139 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.81.5382

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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 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.
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.
Abstract The polaron problem is a very old problem in condensed matter physics that dates back to the thirties, but still remains largely unsolved today, especially when electron–electron interaction is taken into consideration. The presence of both electron–electron and electron–phonon interactions in the problem invalidates most existing numerical methods, which are either computationally too expensive or simply intractable. The continuous-time quantum Monte Carlo (CTQMC) methods could tackle this problem, but they are only effective on the imaginary-time axis. In this work, we present a method based on tensor networks and the path integral formalism to solve polaron impurity problems. As both the electron and phonon baths can be integrated out via the Feynman–Vernon influence functional in the path integral formalism, our method is free of bath discretization error. It can also flexibly work on imaginary time, Keldysh contour, and L-shaped Kadanoff contour. In addition, our method can naturally resolve several long-existing challenges: (i) non-diagonal hybridization function; (ii) measuring multi-time correlations beyond single-particle Green’s functions. We demonstrate the effectiveness and accuracy of our method with extensive numerical examples against analytic solutions, exact diagonalization, and CTQMC. We also perform full-fledged real-time calculations that have never been done before to our knowledge, which could serve as a benchmarking baseline for future method developments.
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.