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arXiv:nucl-th/9307018 · Physical Review C

Demonstration of the auxiliary-field Monte Carlo approach forsd-shell nuclei

W. E. Ormand, David J. Dean, C. W. Johnson, G. H. Lang, S. E. Koonin

Published Mar 1, 1994• 73 citations
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Abstract

We apply the auxiliary-field Monte Carlo approach to the nuclear shell model in the 1s-0d configuration space. The Hamiltonian was chosen to have isovector pairing and isoscalar multipole-multipole interactions, and the calculations were performed within the fixed-particle, canonical ensemble. The results demonstrate the feasibility of the method for N\ensuremath{\ne}Z even-even and odd-odd N=Z nuclei. In particular, static observables for even-even Ne isotopes and $^{22}\mathrm{Na}$ compare well with results obtained from exact diagonalization of the Hamiltonian. Response functions are presented for $^{22}\mathrm{Ne}$ and compared with exact results, and the viability of cranked calculations for N\ensuremath{\ne}Z even-even nuclei is addressed. We present methods for computing observables in the canonical ensemble using Fourier extraction, and for determining the nucelar shape.

DOI
10.1103/physrevc.49.1422

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We apply the auxiliary-field Monte Carlo approach to the nuclear shell model in the 1s-0d configuration space. The Hamiltonian was chosen to have isovector pairing and isoscalar multipole-multipole interactions, and the calculations were performed within the fixed-particle, canonical ensemble. The results demonstrate the feasibility of the method for N\ensuremath{\ne}Z even-even and odd-odd N=Z nuclei. In particular, static observables for even-even Ne isotopes and 22Na^{22}\mathrm{Na}22Na compare well with results obtained from exact diagonalization of the Hamiltonian. Response functions are presented for 22Ne^{22}\mathrm{Ne}22Ne and compared with exact results, and the viability of cranked calculations for N\ensuremath{\ne}Z even-even nuclei is addressed. We present methods for computing observables in the canonical ensemble using Fourier extraction, and for determining the nucelar shape.
This thesis presents the path-integral formulation of the nuclear shell model using the Hubbard-Stratonovich transformation, which linearizes the two-body interaction by auxiliary fields. The path-integral was evaluated via Monte Carlo. The method scales favorably with valence-nucleon number and shell-model basis: full-basis calculations can be done up to the rare-earth region, which cannot be treated by other methods. Observables are calculated for the ground state and in a thermal ensemble. Dynamical correlations are obtained, from which strength functions are extracted through the Maximum Entropy method. Examples in the s-d shell, where exact diagonalization can be carried out, compare well with exact results. The "sign problem", which is generic to fermion Monte Carlo calculations, is proved to be absent in a wide class of interactions including the attractive pairing-plus-multipole interactions. The formulation is general for interacting fermion systems and is well suited for parallel computation. The method has been implemented on the Intel Touchstone Delta System, achieving better than 99% parallelization.
We present in detail a formulation of the shell model as a path integral and Monte Carlo techniques for its evaluation. The formulation, which linearizes the two-body interaction by an auxiliary field, is quite general, both in the form of the effective ``one-body'' Hamiltonian and in the choice of ensemble. In particular, we derive formulas for the use of general (beyond monopole) pairing operators, as well as a novel extraction of the canonical (fixed-particle-number) ensemble via an activity expansion. We discuss the advantages and disadvantages of the various formulations and ensembles and give several illustrative examples. We also discuss and illustrate calculation of the imaginary-time response function and the extraction, by maximum entropy methods, of the corresponding strength function. Finally, we discuss the ``sign problem'' generic to fermion Monte Carlo calculations, and prove that a wide class of interactions are free of this limitation.
The formalism for the auxiliary-field Monte Carlo approach to the nuclear shell model is presented.The method is based on a linearization of the two-body part of the Hamiltonian in an imaginary-time propagator using the Hubbard-Stratonovich transformation.The foundation of the method, as applied to the nuclear many-body problem, is discussed.Topics presented in detail include: (1) the density-density formulation of the method, (2) computation of the overlaps, (3) the sign of the Monte Carlo weight function, (4) techniques for performing Monte Carlo sampling, and ( 5) the reconstruction of response functions from an imaginary-time auto-correlation function using MaxEnt techniques.Results obtained using schematic interactions, which have no sign problem, are presented to demonstrate the feasibility of the method, while an extrapolation method for realistic Hamiltonians is presented.In addition, applications at finite temperature are outlined.
The formalism for the auxiliary-field Monte Carlo approach to the nuclear shell model is presented. The method is based on a linearization of the two-body part of the Hamiltonian in an imaginary-time propagator using the Hubbard-Stratonovich transformation. The foundation of the method, as applied to the nuclear many-body problem, is discussed. Topics presented in detail include: (1) the density-density formulation of the method, (2) computation of the overlaps, (3) the sign of the Monte Carlo weight function, (4) techniques for performing Monte Carlo sampling, and (5) the reconstruction of response functions from an imaginary-time auto-correlation function using MaxEnt techniques. Results obtained using schematic interactions, which have no sign problem, are presented to demonstrate the feasibility of the method, while an extrapolation method for realistic Hamiltonians is presented. In addition, applications at finite temperature are outlined.
In this thesis, I discuss the use of the Auxiliary Field Diffusion Monte Carlo method to compute the ground state of nuclear Hamiltonians, and I show several applications to interesting problems both in nuclear physics and in nuclear astrophysics. In particular, the AFDMC algorithm is applied to the study of several nuclear systems, finite, and infinite matter. Results about the ground state of nuclei (4^44He, 8^88He, 16^{16}16O and 40^{40}40Ca), neutron drops (with 8 and 20 neutrons) and neutron rich-nuclei (isotopes of oxygen and calcium) are discussed, and the equation of state of nuclear and neutron matter are calculated and compared with other many-body calculations. The 1S0^1S_01S0​ superfluid phase of neutron matter in the low-density regime was also studied.
The auxiliary-field Monte Carlo (AFMC) method, also known as the shell model Monte Carlo (SMMC), enables us to calculate microscopically statistical and collective properties of nuclei in the presence of strong correlations. These calculations can be carried out in shell model spaces that are many orders of magnitude larger than spaces that can be treated with conventional diagonalization methods. A recent major development has been the extension of the AFMC approach to heavy nuclei. Such applications to heavy nuclei have been a major challenge. On the conceptual level, a crucial question is whether a truncated spherical shell model Hamiltonian can describe the proper collectivity observed in heavy nuclei and, in particular, the rotational character of strongly deformed nuclei. On the technical level, the low excitation energies make it necessary to perform calculations down to much lower temperatures. At such low temperatures the propagator becomes ill-conditioned and requires the introduction of stabilization methods.