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Lecture notes in physics. M, Monographs

Numerical Techniques

Published Jan 1, 1993• 0 citations
Publisher
DOI
10.1007/978-3-540-47575-0_9

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Over the last two years, we have used several different numerical techniques to study a variety of intrinsically correlated electron systems. The three main types of numerical techniques we have used are: (1) finite- temperature quantum Monte Carlo (QMC); (2) exact diagonalization; and (3) numerical renormalization groups. We have made substantial progress both on developing and improving the numerical algorithms and in using these methods to study interesting systems. We have developed techniques for analytically continuing imaginary-time QMC data to obtain dynamical properties, and have used these techniques to study pseudogap and gap formation in the repulsive and attractive Hubbard models. We have developed a program to calculate two particle interaction vertices with full frequency and momentum dependence using QMC, and have studied possible magnetic and superconducting instabilities in the Hubbard model.
The reweighted random series techniques provide finite-dimensional approximations to the quantum density matrix of a physical system that have fast asymptotic convergence. We study two special reweighted techniques that are based upon the Lévy–Ciesielski and Wiener–Fourier series, respectively. In agreement with the theoretical predictions, we demonstrate by numerical examples that the asymptotic convergence of the two reweighted methods is cubic for smooth enough potentials. For each reweighted technique, we propose some minimalist quadrature techniques for the computation of the path averages. These quadrature techniques are designed to preserve the asymptotic convergence of the original methods.