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Physical Review C

Canonical number projection in a simple nuclear model

DAVID M. VAN DER WALT, R. M. Quick

Published Apr 1, 1994• 3 citations
Publisher

Abstract

The heat capacity for a system of nucleons as a function of temperature is determined from the canonical partition function, which is projected from an approximation to the grand canonical partition function. This approximation is obtained from a functional integral representation of the partition function, including both thermodynamic fluctuations and first order quantum corrections. The number projection methods include a simple contour integration of ``Wick rotation'', and a Chebyschev polynomial expansion, both of which are formally exact. The above procedure is tested in a simple ${\mathit{i}}_{13/2}$ pairing model for which the exact canonical and grand canonical results can be obtained. It is found that both number projection methods fail to predict the position or the height of the peak in the heat capacity due to errors in the approximation to the grand partition function, even though the path integral approximation is very good.

DOI
10.1103/physrevc.49.1955

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The heat capacity for a system of nucleons as a function of temperature is determined from the canonical partition function, which is projected from an approximation to the grand canonical partition function. This approximation is obtained from a functional integral representation of the partition function, including both thermodynamic fluctuations and first order quantum corrections. The number projection methods include a simple contour integration of ``Wick rotation'', and a Chebyschev polynomial expansion, both of which are formally exact. The above procedure is tested in a simple i13/2{\mathit{i}}_{13/2}i13/2​ pairing model for which the exact canonical and grand canonical results can be obtained. It is found that both number projection methods fail to predict the position or the height of the peak in the heat capacity due to errors in the approximation to the grand partition function, even though the path integral approximation is very good.
While the thermodynamics of a system of non-interacting fermions can be straightforwardly determined in the grand canonical ensemble, results for a specific number of N particles are more difficult to obtain. In this talk we will present a general recursion scheme for the canonical partition function of free fermions with a quadratic energy level spacing as might be present for ultracold fermionic atoms confined inside a box trap. Exact results for the entropy, specific heat and one and two particle occupation probabilities are numerically obtained to arbitrary precision and compared with their corresponding grand canonical values. The numerical stability of the recursion relation allows us to quantify deviations from Wicks theorem in the canonical ensemble for a variety of temperatures and densities.
In order to compare the approach of the mixed state representation of the pairing model with the exact evaluation of the grand partition function, we introduce the fermion space of a grand canonical ensemble. For the definition of the grand partition function, we adopt a quasi-spin space and evaluate its multiplicity. then we show how to derive the mixed state representation, which yields results that are approximately the same as those of the thermal Hartree-Fock-Bogoliubov theory for the thermal equilibrium state, from the grand partition function under the mean field approximation. We compare numerical results of the mean field approximation with those obtained from the exact evaluation of the grand partition function. We find that the mean field approximation gives good results. However, we find a distinctive difference between the two sets of results in the condensed phase.
A systematic comparison is conducted for pairing properties of finite systems at nonzero temperature as predicted by the exact solutions of the pairing problem embedded in three principal statistical ensembles, as well as the unprojected (FTBCS1+++SCQRPA) and Lipkin+++Nogami projected (FTLN1+++SCQRPA) theories that include the quasiparticle number fluctuation and coupling to pair vibrations within the self-consistent quasiparticle random-phase approximation. The numerical calculations are performed for the pairing gap, total energy, heat capacity, entropy, and microcanonical temperature within the doubly folded equidistant multilevel pairing model. The FTLN1+++SCQRPA predictions agree best with the exact grand-canonical results. In general, all approaches clearly show that the superfluid-normal phase transition is smoothed out in finite systems. A novel formula is suggested for extracting the empirical pairing gap in reasonable agreement with the exact canonical results.
We define a mean field approximation of the modified grand partition function of the pairing model, which has been defined on the modified grand canonical ensemble with even or odd particle number. Using numerical evaluations, it is shown that this approximation reproduces the even-odd effect exhibited in the case of the modified grand partition function at low temperature very well in a qualitative sense. The expectation values of one-body operators can be expressedsimply in terms of those obtainedwith the conventional mean field approximation when a Fermi-type distribution is replaced by a renormalized distribution. Although Wick’s theorm does not hold in this case, the dominant parts of the expectation values of two-body operators are also given using those obtained by the conventional mean fieldapproximation with the replacement mentionedabove.
We exploit the grassmannian nature of the variables involved in the path integral expression of the grand canonical partition function for self--interacting fermionic models to show, in one-space dimension, a general relation among the terms of it expansion in the high temperature limit and a combination of co-factors of a suitable matrix with commuting entries. As an application, we apply this framework to calculate the exact coefficients, up to order β^3, of the expansion of the grand canonical partition function for the Hubbard model in d=(1+1) in the high temperature limit. The results are valid for any set of parameters that characterize the model.
Methods which include the effect of mean-field fluctuations on the nuclear partition function and level density of hot nuclei are tested by applying them to a many-body system for which exact solutions are available, the Lipkin model. It is found that for finite systems fluctuation corrections to the mean field partition function and level density can be substantial, and especially so near a shape phase transition where these fluctuations smooth out discontinuities in the mean-field quantities. The model calculations suggest that only time-independent fluctuations are important at high temperatures or at high excitation energies. Moreover, even when fluctuations are restricted to a small set of constraints, a significant improvement over the mean-field calculations can result. This is thus a practical approximation since it requires only knowledge of the deformation free energy surfaces at finite temperature.