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Progress of Theoretical Physics

Thermal Effect in Lipkin Model. II: Grand Partition Function and Mean Field Approximation

A. Kuriyama, João da Providência, Constança Providência, Y. Tsue, M. Yamamura

Published Feb 1, 1996• 4 citations
Publisher

Abstract

In a previous paper, referred to as I, we developed a new approach for the description of thermal effects in the Lipkin model. In order to clarify the physical ingredient of this approach, we introduce the grand partition function of the present system. We compare numerically the results from these two approaches and show that they nearly coincide, except for the region near the transition temperature. Further, we show that our approach is equivalent to that based on the grand partition function under the mean field approximation.

DOI
10.1143/ptp.95.339

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In a previous paper, referred to as I, we developed a new approach for the description of thermal effects in the Lipkin model. In order to clarify the physical ingredient of this approach, we introduce the grand partition function of the present system. We compare numerically the results from these two approaches and show that they nearly coincide, except for the region near the transition temperature. Further, we show that our approach is equivalent to that based on the grand partition function under the mean field approximation.
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.
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.
The problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.
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.