Monte Carlo evaluation of path integrals for the nuclear shell model

Abstract

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.Comment: 38 pages, RevTeX v3.0, figures available upon request; Caltech Preprint #MAP-15

Spectroscopy of discrete energy levels in ultrasmall metallic grains

“…Since the CSPA is conceptually simple, well-documented [30] and straightforward to implement, it seems to be the method of choice for not too low temperatures. A possible alternative is a quantum Monte Carlo evaluation of the path integral (93) [144,145], but the numerics is much more demanding than for the CPSA, while the convergence at low T is in general rather poor, due to the familiar sign problem of Monte Carlo methods.…”

Nuclear state densities of odd-mass heavy nuclei in the shell model Monte Carlo approach

“…In contrast, SMMC calculations in neighboring even-even samarium and neodymium nuclei, for which there is no sign problem, were carried out up to β = 20 MeV −1 [16]. Systematic errors introduced by the discretization of β [2] are corrected by calculating E(β) for the two time slices of ∆β = 1/32 MeV −1 and ∆β = 1/64 MeV −1 and then performing a linear extrapolation to ∆β = 0. For β 3 MeV −1 , the dependence of E(β) on ∆β is weaker and an average value is taken instead.…”

“…Statistical and collective properties of nuclei can be reliably calculated [6,13] by employing a class of interactions that have a good Monte Carlo sign in the grand-canonical formulation [2,13]. The projection on an even number of particles keeps the good sign of the interaction, allowing accurate calculations for even-even nuclei.…”

“…Correlation effects can be taken into account in the context of the configuration-interaction (CI) shell model approach, but the size of the required model space in heavy nuclei is prohibitively large for direct diagonalization of the CI shell model Hamiltonian. This limitation can be overcome in part by using the shell model Monte Carlo (SMMC) method [1][2][3][4][5]. The SMMC method enables the calculation of statistical nuclear properties, and in particular of level densities, in very large model spaces [6][7][8][9][10].…”

“…Fermionic Monte Carlo methods are often hampered by the so-called sign problem, which leads to large fluctuations of observables at low temperatures [1,2,11,12]. Statistical and collective properties of nuclei can be reliably calculated [6,13] by employing a class of interactions that have a good Monte Carlo sign in the grand-canonical formulation [2,13].…”