We consider a many-fermion model which exhibits a transition from a superconducting to a rotational phase with variation of a parameter in its Hamiltonian. The model has analytical solutions in its two limits due to the presence of dynamical symmetries. However, the symmetries are basically incompatible with one another; no simple solution exists in intermediate situations. Exact (numerical) solutions are possible and enable one to study the behavior of competing but incompatible symmetries and the phase transitions that result in a semirealistic situation. The results are remarkably simple and shed light on the nature of phase transitions.
The quantum-phase-transitional behavior of the spherical shell-model mean field plus the geometric quadrupole-quadrupole and standard pairing model within a single-
j shell is analyzed in detail. Various quantities, such as low-lying energy levels, some typical energy ratios, the overlaps of the excited states with those of the corresponding limiting cases,
B(E2) values and electric quadrupole moments of some low-lying states and their ratios, as functions of the control parameter of the model in a
j=15/2 shell are calculated as an example, in which only a crossover occurs due to the Pauli exclusion. The results show that there are noticeable changes within the crossover region of the rotation-like to the pair-excitation (superconducting) phase transition, especially in the half-filling case. As an application to realistic nuclear systems, a chain of isotones $^{212}\mathrm{Rn}\text{\ensuremath{-}}^{213}\mathrm{Fr}\text{\ensuremath{-}}^{214}\mathrm{Ra}\text{\ensuremath{-}}^{215}\mathrm{Ac}$ is chosen to be described by the model with valence protons in the
1h9/2 shell. As far as the low-lying energy levels, the experimentally observed
B(E2) values, and the electric quadrupole moment within the yrast band are concerned, these nuclei seem fitted reasonably well. The results indicate that these nuclei are all within the rotation-like to the pair-excitation phase transition near the crossover point.
The boson expansion method was developed with the purpose of continuously following even nuclear systems near their ground states, from the region of slightly anharmonic vibrations through the transitional region formed by a nonsharp phase transition to a region of permanent distortion characterized by the occurrence of rotational spectra. However, as the boson quantum was defined in the extreme vibrational limit, it was not possible to obtain quantitatively correct spectra in the transitional or deformed region. Believing that the major cause for the lack of success was the disregard of the self- consistent change of the basic degree of freedom, a simple variational method to be used with the boson expansion technique is presented. Its ability to cover the vibration-to-rotation transition is demonstrated by comparison with an exact solution for a four-level model with pairing force. Further the coverage of a quadrupole phase transition is shown for a system interacting via pairing-Plus- quadrupole forces. A successful calculation of static quadrupole moments of Cd and Te isotopes is presented. Finally the approach in general situations with more complicated forces is discussed. (auth)
Aspects of the phase change of the two-level pairing model are investigated in the semi-classical treatment by using the variational approch with the mixed-mode coherent state. In the classical limit, � → 0, the sharp phase transition appears and the two phases exist in the region where the force strength is larger than a certain critical value. However, it is shown that, in the semi-classical treatment, the abovementioned behavior of the phase change disappears in the analytical and the numerical treatments. As a result, new understanding about aspects of the phase change in this model is given. The phase structure and the phase transitions between various phases in condensed matter systems, such as nuclear and/or quark matter as well as the condensed matter in atoms and molecules, are important concepts to understand the nature of the matter under consideration. In the infinite systems, the clear phase separation is often realized and, in that
The program of the algebraic approach to nuclear collective motion is summarized. In essence, it should permit an accurate solution of the many-body problem within a subspace of the many-particle space if the conditions defining collective motion are present. The formulation, containing both dynamical and kinematical aspects, leads to coupled nonlinear algebraic equations for the matrix elements between collective states of fermion pair operators. An explicit set of equations is given for the problem of
N (even) particles confined to a single-
j shell and described by the conventional pairing plus quadrupole-quadrupole interaction Hamiltonian. This formulation is designed to be nearly exact in the pairing-dominant (seniority) limit and to be accurate enough (based on plausibility arguments) to describe the transition from spherical to deformed shape. That this is so is verified for configurations, the largest being
jN=(221)6, for which exact solutions are available. The present method can be viewed as an angular momentum conserving alternative to the description of the transition based on the Hartree-Bogoliubov-BCS method. Qualitative and detailed comparisons of the two methods are carried out. In particular, it is shown from approximate analytic results that the two theories predict the transition to occur for almost identical values of the parameters. Over-all agreement with exact results is much better for the algebraic method, however, confirming its status as an approximate method of variation after projection.
The existence of more than one solution for s-wave pairing in the extended Hubbard model is not often realized. This possibility was noted by Friedberg et al. [Phys. Rev B50, 10190 (1994)] in the case of two electrons on a lattice, without further analysis. In the present paper, the second solution is found also in the case of superconductivity of extended s-wave symmetry in the extended Hubbard model. The properties of both s-wave solutions are examined by mean-field methods and thresholds for their appearance are given. A possibility of the first-order transition is discovered and modifications of the phase diagram are calculated. Results of this paper in the limit of low electron density are also applicable to the bound, two-electron pairs.