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arXiv:cond-mat/0401634 · physica status solidi (b)

Quantum Monte Carlo approach to the Holstein polaron problem

Martin Hohenadler, Hans Gerd Evertz, Wolfgang von der Linden

Published Apr 7, 2005• 8 citations
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Abstract

Abstract A recently developed quantum Monte Carlo approach to the Holstein model with one electron [Phys. Rev. B 69, 024301 (2004)] is extended in several aspects. In addition to a straight‐forward generalization to higher dimensions, a checkerboard breakup for the hopping part of the Hamiltonian is introduced, which allows significantly faster simulations. Moreover, results are extrapolated towards the limit of continuous imaginary time to remove the error due to the Trotter–Suzuki approximation. The performance and accuracy of the current approach are compared to existing quantum Monte Carlo methods. To demonstrate the applicability of our method for almost arbitrary parameters, we study the small‐polaron crossover in one to three dimensions by calculating the electronic kinetic energy. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssb.200440019

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Abstract A recently developed quantum Monte Carlo approach to the Holstein model with one electron [Phys. Rev. B 69, 024301 (2004)] is extended in several aspects. In addition to a straight‐forward generalization to higher dimensions, a checkerboard breakup for the hopping part of the Hamiltonian is introduced, which allows significantly faster simulations. Moreover, results are extrapolated towards the limit of continuous imaginary time to remove the error due to the Trotter–Suzuki approximation. The performance and accuracy of the current approach are compared to existing quantum Monte Carlo methods. To demonstrate the applicability of our method for almost arbitrary parameters, we study the small‐polaron crossover in one to three dimensions by calculating the electronic kinetic energy. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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 .
Abstract Quantum Monte Carlo is an important and complementary alternative to density functional theory when performing computational electronic structure calculations in which high accuracy is required. The method has many attractive features for probing the electronic structure of real atoms, molecules and solids. In particular, it is a genuine many‐body theory with a natural and explicit description of electron correlation which gives consistent, highly‐accurate results while at the same time exhibiting favourable (cubic or better) scaling of computational cost with system size. This article is intended to provide a brief and hopefully accessible review of some relevant aspects of quantum Monte Carlo together with an outline of our implementation of it in the Cambridge computer code ‘CASINO’ [1, 2]. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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.
The Holstein model is a benchmark model of systems with electron-phonon interaction. However, its electrical transport properties are not yet fully understood. In this work, we performed numerically exact calculations of the imaginary-time current-current correlation function of the Holstein Hamiltonian for a broad range of model parameters. These calculations were performed using a path-integral-based quantum Monte Carlo method. We compared these results with the results obtained under the assumption of conventional band transport, small polaron hopping, and polaron band transport. From this comparison, we identified the regions in parameter space where each of these transport regimes is valid. In some cases when an imaginary-time comparison of current-current correlation functions could not give conclusive results, we complemented it with real-time comparisons or a comparison with the literature data on numerically exact dc mobilities. Overall, we found that the parameter space is almost completely covered by the three mentioned transport regimes.
Based on a recently developed variational method, we explore the properties of the Holstein polaron on an infinite lattice in D dimensions, where 1< D< 4.1<~D<~4.1< D< 4. The computational method converges as a power law, so that highly accurate results can be achieved with modest resources. We present the most accurate ground state energy (with no small parameter) to date for polaron problems, 21 digits for the one-dimensional (1D) polaron at intermediate coupling. The dimensionality effects on polaron band dispersion, effective mass, and electron-phonon (el-ph) correlation functions are investigated in all coupling regimes. It is found that the crossover to large effective mass of the higher-dimensional polaron is much sharper than the 1D polaron. The correlation length between the electron and phonons decreases significantly as the dimension increases. Our results compare favorably with those of the quantum Monte Carlo, dynamical mean-field theory, density-matrix renormalization-group, and Toyozawa variational methods. We demonstrate that the Toyozawa wave function is qualitatively correct for the ground-state energy and the two-point electron-phonon correlation functions, but fails for the three-point functions. Based on this finding, we propose an improved Toyozawa variational wave function.