Path-Following Method to Determine the Field of Values of a Matrix with High Accuracy

We describe a novel and efficient algorithm for calculating the field of values boundary, $\partial\textrm{W}(\cdot)$, of an arbitrary complex square matrix: the boundary is described by a system of ordinary differential equations which are solved using Runge--Kutta (Dormand--Prince) numerical integration to obtain control points with derivatives then finally Hermite interpolation is applied to produce a dense output. The algorithm computes $\partial\textrm{W}(\cdot)$ both efficiently and with low error. Formal error bounds are proven for specific classes of matrix. Furthermore, we summarise the existing state of the art and make comparisons with the new algorithm. Finally, numerical experiments are performed to quantify the cost-error trade-off between the new algorithm and existing algorithms

Finite-Temperature Auxiliary-Field Quantum Monte Carlo for Bose-Fermi Mixtures

We present a quantum Monte Carlo (QMC) technique for calculating the exact finite-temperature properties of Bose-Fermi mixtures. The Bose-Fermi Auxiliary-Field Quantum Monte Carlo (BF-AFQMC) algorithm combines two methods, a finite-temperature AFQMC algorithm for bosons and a variant of the standard AFQMC algorithm for fermions, into one algorithm for mixtures. We demonstrate the accuracy of our method by comparing its results for the Bose-Hubbard and Bose-Fermi-Hubbard models against those produced using exact diagonalization for small systems. Comparisons are also made with mean-field theory and the worm algorithm for larger systems. As is the case with most fermion Hamiltonians, a sign or phase problem is present in BF-AFQMC. We discuss the nature of these problems in this framework and describe how they can be controlled with well-studied approximations to expand BF-AFQMC's reach. The new algorithm can serve as an essential tool for answering many unresolved questions about many-body physics in mixed Bose-Fermi systems.Comment: 19 pages, 6 figure

Correlation energies by the generator coordinate method: computational aspects for quadrupolar deformations

We investigate truncation schemes to reduce the computational cost of calculating correlations by the generator coordinate method based on mean-field wave functions. As our test nuclei, we take examples for which accurate calculations are available. These include a strongly deformed nucleus, 156Sm, a nucleus with strong pairing, 120Sn, the krypton isotope chain which contains examples of soft deformations, and the lead isotope chain which includes the doubly magic 208Pb. We find that the Gaussian overlap approximation for angular momentum projection is effective and reduces the computational cost by an order of magnitude. Cost savings in the deformation degrees of freedom are harder to realize. A straightforward Gaussian overlap approximation can be applied rather reliably to angular-momentum projected states based on configuration sets having the same sign deformation (prolate or oblate), but matrix elements between prolate and oblate deformations must be treated with more care. We propose a two-dimensional GOA using a triangulation procedure to treat the general case with both kinds of deformation. With the computational gains from these approximations, it should be feasible to carry out a systematic calculation of correlation energies for the nuclear mass table.Comment: 11 pages revtex, 9 eps figure