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Journal of Physics A Mathematical and General

Dynamic versus kinematic symmetry breaking in a two-dimensional model of collective motion

J. Cseh

Published Nov 21, 1992• 41 citations
Publisher

Abstract

Investigates the question of whether or not the vibrational spectrum of a two-dimensional interacting boson model can be transformed into the rotational one via quantum deformation of the group structure. It turns out that although the spectrum of the q-deformed model shows one or another of the rotational-like features at different values of the deformation parameter, nevertheless, all the requirements of the rotational behavior cannot be obtained simultaneously. This result indicates that the quantum algebraic treatment cannot give an analytical solution to the eigenvalue problem of the interacting boson models of nuclear and molecular physics in the whole range of the physically relevant breaking of dynamical symmetries.

DOI
10.1088/0305-4470/25/22/001

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Investigates the question of whether or not the vibrational spectrum of a two-dimensional interacting boson model can be transformed into the rotational one via quantum deformation of the group structure. It turns out that although the spectrum of the q-deformed model shows one or another of the rotational-like features at different values of the deformation parameter, nevertheless, all the requirements of the rotational behavior cannot be obtained simultaneously. This result indicates that the quantum algebraic treatment cannot give an analytical solution to the eigenvalue problem of the interacting boson models of nuclear and molecular physics in the whole range of the physically relevant breaking of dynamical symmetries.
We reinvestigate the question of whether or not the dynamical symmetries of the two-dimensional interacting boson model can be transformed into each other by the quantum deformation of the algebraic structure. The deformation of both the vibrational and the rotational spectra is considered, and the breaking of the classical symmetry is measured quantitatively, while a systematic search for the best fit is performed. In no case can the quantum-deformed spectrum reach the other classical limit; only a modest improvement is observed with the phase deformation of the rotational spectrum.
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.
We suggest the quantum Heisenberg group Hq(J){\mathit{H}}_{\mathit{q}(\mathit{J})}Hq(J)​(4) as the dynamical symmetry group of a diatomic-molecule system. The deformation parameter q of the Hq(J){\mathit{H}}_{\mathit{q}(\mathit{J})}Hq(J)​(4) symmetry group is a function of J (the quantum number of the angular momentum of the rigid rotation) and characterizes the interaction between vibration and rotation. The vibrational and rotational structures of diatomic molecules are compactly expressed using the quantum-group-theoretic approach. Taylor expansions of the analytic formulas from this approach reproduce the results of the anharmonic-oscillator and the nonrigid-rotator models. For some randomly selected molecules in particular states, the parameters are computed to fit the phenomenological formulas to high accuracy.
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.
An algebraic approach, within a framework using equations of motion, is applied to the study of vibrational collective motion in exactly soluble shell models with symmetry R(5) and R(5) \ifmmode\times\else\texttimes\fi{} R(5). The latter exhibits two phonon branches, permitting the study of the effect of the noncollective on the collective branch. The results of this method, including the splitting of the two-phonon degeneracy, agree satisfactorily with exact results for a range of parameters well into the transition region between vibrational and rotational motion.[NUCLEAR STRUCTURE Approximate many-body methods applied to a solvable model. Vibrational spectrum. Equations of motion method.]
q-deformation of the U(1) dynamical symmetry of U(2) group is introduced for the first time for vibrations of diatomic molecules. The question of pure real and pure imaginary value of q-deformation parameter, often raised in the literature, is found to be related to the particular choice of the constants of the model. The complex q-deformation parameter is allowed, with pure real and pure imaginary values as its special cases. Compared to the other dynamical symmetry Oq(2) of Uq(2), the result of Uq(1) of Uq(2) remains better even when the vibron number in Oq(2) is varied. The role of the vibron number in the two dynamical symmetries of U(2) is shown to be different. In Uq(1) symmetry, the vibron number is fixed by the number of observed states of the system since Uq(1) represents the q-deformation of a truncated anharmonic oscillator. A possible physical meaning to the q-deformation parameter is suggested in terms of higher order anharmonicities of the truncated anharmonic oscillator.