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Progress of Theoretical Physics

Pairing Model and Mixed State Representation. I: Thermal Equilibrium State

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

Published May 1, 2001• 2 citations
Publisher

Abstract

We apply the mixed state representation to the description of the thermal properties of the pairing model. We derive an extended form of the boson realization of the quasi-spin operators in a classical image. We introduce a density operator and entropy combined with the mixed state representation. Then we define a thermal equilibrium state with use of the variation principle. In this representation, the seniority number is specified by a distribution of the Fermi type as a function of temperature. We study the pairing collective modes that induce no change in the entropy.

DOI
10.1143/ptp.105.729

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We apply the mixed state representation to the description of the thermal properties of the pairing model. We derive an extended form of the boson realization of the quasi-spin operators in a classical image. We introduce a density operator and entropy combined with the mixed state representation. Then we define a thermal equilibrium state with use of the variation principle. In this representation, the seniority number is specified by a distribution of the Fermi type as a function of temperature. We study the pairing collective modes that induce no change in the entropy.
A pure squeezed coherent state is amalgamated with a mixed state for the purpose of describing bose condensation in an imperfect spinless boson system at non-zero temperature. Three types of variables are introduced: One describes the bose condensation, one describes the quantum fluctuations of a condensed pair induced by the interaction, and one describes the thermal properties. We define an entropy in association with the amalgamated state. Then, the thermal equilibrium state is defined through a variational principle with a fixed boson number and entropy.
We study the thermal effect with the use of Lipkin model. We define the density and entropy operator associated with the mixed state representation of Lipkin model, which has been developed with the aid of auxiliary fermion field. We investigate the thermal equilibrium state and its phase transition. In super phase, the thermal effect breaks the particle-hole pairs with coupled angular momentum 0 and does not lift up nucleons from the lower level to upper one, contrary to the case of normal phase.
An imperfect spinless bose system is studied with use of the mixed state representation, which was originally developed to study highly excited nuclear states. With this representation, the degrees of freedom are classified into two types: One describes the thermal behaviour of the non-condensed boson and the other mainly describes the dynamical aspects of the condensed boson. The entropy of the system is specified by only the former type of degrees of freedom. Comparison is made with the conventional results for a dilute bose gas with an interaction of the hard sphere type.
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 classical microscopic theory of mixed states has recently been formulated in order to deal with the theoretical description of nuclear phenomena observed in highly excited states. In this theory, which is here briefly reviewed, the degrees of freedom of real fermions are doubled by fictitious fermions, in the context of the formalism of thermofield dynamics. The requirement of static stability of the thermal state of equilibrium plays, in the theory, a relevant role. The possibility of constraining the dynamics of the fictitious degrees of freedom, in such a way as to eliminate spurious thermal excitations, is discussed. The concept of canonical collective variables for mixed states is analyzed. Boson expansions appropriate to describe collective excitations of hot quantal systems are introduced and their properties investigated. It is shown that thermal boson expansions based on the formalism of thermofield dynamics are free from formal inconsistencies which may be present in boson expansions derived by an ad hoc quantization of conventional thermal mean field theories. The formalism is applied to the Lipkin model, and the correlation energy of a hot system is discussed.
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.