Classical and Quantal Descriptions of Small Amplitude Fluctuations Around Equilibriums in the Two-Level Pairing Model

Various classical counterparts for the two-level pairing model in a many-fermion system are presented in the Schwinger boson representation. It is shown that one of the key ingredients giving the classical descriptions for quantal system is the use of the various trial states besides the $su(2)\otimes su(2)$-coherent state, which may be natural selection for the two-level pairing model governed by the $su(2)\otimes su(2)$-algebra. It is pointed out that the fictitious behavior like the sharp phase transition can be avoided by using the other states such as the $su(2)\otimes su(1,1)$- and the $su(1,1)\otimes su(1,1)$-coherent states, while the sharp phase transition appears in the usual Hartree-Fock-Bogoliubov and the quasi-particle random phase approximations in the original fermion system.