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Journal of Computational Electronics

A path-sum Monte Carlo approach for many-electron systems within a tight-binding basis

John K. David, Leonard Frank Register, Sanjay Kumar Banerjee

Published Feb 14, 2012• 0 citations
Publisher
DOI
10.1007/s10825-012-0393-z

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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 rigorous expression for the partition function of N identical fermions with spin 1/2 is obtained in a form suitable for a path-integral Monte Carlo (PIMC) simulation of a many-electron system at a finite temperature. A sum over permutations is reduced to a sum over classes with readily determinable factors. A distribution over projections and values of the total spin is also obtained, enabling PIMC calculations of spin-dependent quantities. An MC algorithm is proposed which accounts simultaneously for all types of trajectory linkages. A test calculation, four electrons in a spherical cavity with a radius of 1 nm at a temperature of 300--2000 K, is performed.
Thermal equilibrium properties are usually obtained from the imaginary-time path integral representation of the Boltzmann operator in combination with Monte Carlo integration methods. In some situations (identical fermions or frustrated Hamiltonians), the Boltzmann matrix leads to terms of alternating sign, which leads to a sign problem that severely impacts convergence. In this paper, we develop a robust and efficient quadrature-based method suitable for computing the Boltzmann matrix for discrete systems coupled to common or local harmonic baths. By expressing the discretized path integral with the influence functional in terms of a sum of matrix products, we develop a small matrix path integral (SMatPI) decomposition that allows iterative propagation in imaginary time while circumventing the storage of tensors employed in earlier work. The method is illustrated with several examples that involve two- and three-level systems coupled to common or local baths. We show that cyclic tight-binding Hamiltonians with positive coupling parameters give rise to Boltzmann matrix elements with alternating signs, presenting a severe sign problem to Monte Carlo approaches, while the SMatPI algorithm is stable and efficient.