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Physical Review C

Geometrical representation of interacting boson models: U(3) model

Frederik G. Scholtz, F.J.W. Hahne

Published Nov 1, 1987• 4 citations
Publisher

Abstract

A method is presented by which a geometrical interpretation can be given for boson models without leaving the domain of quantum mechanics. This approach turns out to be particularly useful in symmetry-preserving treatments of deformed boson systems. A U(3) boson model having an SO(2) symmetry, which is to be preserved throughout, is analyzed in detail. For a well-deformed system, rotational and vibrational modes are identified. A procedure is developed by which the parameters of the geometrical description for the deformed case can be calculated in a series expansion controlled by the inverse of the boson number. The method is compared with the corresponding random phase approximation approach.

DOI
10.1103/physrevc.36.2076

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A method is presented by which a geometrical interpretation can be given for boson models without leaving the domain of quantum mechanics. This approach turns out to be particularly useful in symmetry-preserving treatments of deformed boson systems. A U(3) boson model having an SO(2) symmetry, which is to be preserved throughout, is analyzed in detail. For a well-deformed system, rotational and vibrational modes are identified. A procedure is developed by which the parameters of the geometrical description for the deformed case can be calculated in a series expansion controlled by the inverse of the boson number. The method is compared with the corresponding random phase approximation approach.
The usefulness of perturbative expansions in deformed bosons has been limited by difficulties associated with zero-frequency (Goldstone) modes. A method developed earlier by Marshalek and Weneser to overcome these difficulties was restricted to the case in which the broken-symmetry generators commute. The present paper shows how to extend the method to the non-Abelian case of three-dimensional rotation of axially symmetric deformed systems. Application is made to a generic anharmonic Hamiltonian constructed from random-phase-approximation modes.
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)
Quantum-deformation of the vibrational symmetry SU(3) supset SU(2) supset SO(2) of the two-dimensional SU(3) interacting boson model is studied from the point of view of the breaking of the dynamical symmetry. It is shown that, in agreement with the earlier work of Gupta et al. (1992) for the one-dimensional problem, the real and complex (more so the complex) q-deformations take this dynamical symmetry very close to the other (rotational) dynamical symmetry SU(3) supset SO(3) supset SO(2) of the two-dimensional SU(3) group. Also, the intermediate results of mixing these two symmetries are obtained, since q-deformations vary continuously. The purely imaginary q-deformation in SU q (3) supset SU q (2) supset SO q (2) restores the SU(3) symmetry for a parameter value larger than 1. Applications to real nuclei are also made.
A deformed version of the vibron model for diatomic molecules is constructed. Both the O(4) and U(3) dynamical symmetries of the model are rewritten, using the concept of complementary subalgebras, in a more convenient form, which is subsequently deformed. The present model unifies the so far independent successful quantum-algebraic approaches to rotational and to vibrational spectra of diatomic molecules. In addition, the method can be used for the construction of deformed versions of the U(5) and O(6) limits of the interacting boson model of nuclear structure.
We study quasicritical phenomena in transitions between two "quantum phases" of a finite boson system, described by the interacting boson model 1 used in nuclear physics. The model is formulated in the algebraic framework and has a simple geometrical interpretation; the "phases" represented by dynamical symmetries U(5) and SU(3) correspond to spherical and deformed nuclear shapes. The quasicriticality of the U(5)-SU(3) transition is shown to be connected with the following phenomena simultaneously occurring in a narrow parameter region between the symmetries: (a) abrupt structural changes of eigenstates, (b) multiple avoided crossing of levels, (c) peaked density of exceptional points, (d) qualitative changes of the corresponding classical potential. We show that these spectroscopic features influence the dynamics of intersymmetry transitions in the model parameter space if the parameters themselves become dynamical variables.
An exactly soluble three-level model for a system of fermions with Hamiltonian formed from generators of the group SU(3) is studied. A basis for the representation of which the ground state is a member is constructed. It is demonstrated that in this representation, the generators can each be expanded in a series as functions of a pair of ``kinematical'' boson operators; the series are uniquely determined to satisfy the operator algebra and the invariants of the representation by a method of Marumori. It is seen that the lowest anharmonic approximation to the Hamiltonian and other operators yields excellent numerical agreement with exact results for all regimes of interaction strength considered. An alternative description of the system in terms of a dynamically more meaningful boson, called the ``physical'' boson, is shown to be appropriate for relatively weak coupling where one has a near harmonic spectrum.