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Progress of Theoretical Physics

Instability of Thermal Equilibrium State of the Lipkin Model

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

Published May 1, 1999• 0 citations
Publisher

Abstract

The Lipkin model is the simplest algebraic model in which we can investigate a ground state transition in a fermion system. This model is also applied to the description of the thermal equilibrium state. In this paper, we investigate its stability with respect to changes in the particle number and the volume. In order to enable us to take the thermal limit, an appropriate number dependence is introduced into the force strength. We show with the use of a mean field approximation that the thermal equilibrium state in the condensed phase of particle-hole pairs becomes unstable.

DOI
10.1143/ptp.101.1001

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The Lipkin model is the simplest algebraic model in which we can investigate a ground state transition in a fermion system. This model is also applied to the description of the thermal equilibrium state. In this paper, we investigate its stability with respect to changes in the particle number and the volume. In order to enable us to take the thermal limit, an appropriate number dependence is introduced into the force strength. We show with the use of a mean field approximation that the thermal equilibrium state in the condensed phase of particle-hole pairs becomes unstable.
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.
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 study the thermodynamics of quantum particles with long-range interactions at T=0. Specifically, we generalize the Hamiltonian Mean Field (HMF) model to the case of fermions and bosons. In the case of fermions, we consider the Thomas-Fermi approximation that becomes exact in a proper thermodynamic limit. The equilibrium configurations, described by the Fermi (or waterbag) distribution, are equivalent to polytropes with index n=1/2. In the case of bosons, we consider the Hartree approximation that becomes exact in a proper thermodynamic limit. The equilibrium configurations are solutions of the mean field Schrödinger equation with a cosine interaction. We show that the homogeneous phase, that is unstable in the classical regime, becomes stable in the quantum regime. This takes place through a first order phase transition for fermions and through a second order phase transition for bosons where the control parameter is the normalized Planck constant. In the case of fermions, the homogeneous phase is stabilized by the Pauli exclusion principle while for bosons the stabilization is due to the Heisenberg uncertainty principle. As a result, the thermodynamic limit is different for fermions and bosons. We point out analogies between the quantum HMF model and the concepts of fermion and boson stars in astrophysics. Finally, as a by-product of our analysis, we obtain new results concerning the Vlasov dynamical stability of the waterbag distribution.
We study the thermodynamics of quantum particles with long-range interactions at T = 0. Specifically, we generalize the Hamiltonian mean-field (HMF) model to the case of fermions. We consider the Thomas–Fermi approximation that becomes exact in a proper thermodynamic limit with a coupling constant k ~ N. The equilibrium configurations, described by the mean-field Fermi (or waterbag) distribution, are equivalent to polytropes of index n = 1/2. We show that the homogeneous phase, which is unstable in the classical regime, becomes stable in the quantum regime. The homogeneous phase is stabilized by the Pauli exclusion principle. This takes place through a first-order phase transition where the control parameter is the normalized Planck constant. The homogeneous phase is unstable for , metastable for and stable for . The inhomogeneous phase is stable for , metastable for and disappears for (for , there exists an unstable inhomogeneous phase with magnetization ). We point out analogies between the fermionic HMF model and the concept of fermion stars in astrophysics. Finally, as a by-product of our analysis, we obtain new results concerning the Vlasov dynamical stability of the waterbag distribution which is the ground state of the Lynden-Bell distribution in the theory of violent relaxation of the classical HMF model. We show that spatially homogeneous waterbag distributions are Vlasov-stable iff ≥ c = 1/3 and spatially inhomogeneous waterbag distributions are Vlasov-stable iff ≤ * = 0.379 and b ≥ b* = 0.37, where and b are the normalized energy and magnetization. The magnetization curve displays a first-order phase transition at t = 0.352 and the domain of metastability ranges from c to *.
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.