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Physical Review C

Large-amplitude collective motion in the Lipkin model

Eugene R. Marshalek

Published Jul 1, 1976• 2 citations
Publisher

Abstract

It is shown how a simple reinterpretation of the "transition-operator boson method" eliminates the difficulty noted by Okubo. The revised method is applied to the problem of anharmonic motion in the Lipkin model. The occurrence of a non-positive-definite mass parameter, as obtained by Holzwarth and by Giraud and Grammaticos, is then exposed as the easily avoidable consequence of an inauspicious choice of collective coordinate.NUCLEAR STRUCTURE Approximate many-body method applied to a solvable model.

DOI
10.1103/physrevc.14.342

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It is shown how a simple reinterpretation of the "transition-operator boson method" eliminates the difficulty noted by Okubo. The revised method is applied to the problem of anharmonic motion in the Lipkin model. The occurrence of a non-positive-definite mass parameter, as obtained by Holzwarth and by Giraud and Grammaticos, is then exposed as the easily avoidable consequence of an inauspicious choice of collective coordinate.NUCLEAR STRUCTURE Approximate many-body method applied to a solvable model.
In a previous paper the concept of an ideal collective coordinate was introduced as the basis for a phenomenological theory of nuclear collective motion. The application of this idea to the microscopic domain is illustrated by the study of a simple two-level model with monopole-monopole interactions, previously considered by Lipkin and collaborators. The complete scheme of calculation which is presented is seen as an alternative to the boson expansion methods currently in fashion for the study of anharmonic vibrations. The new scheme should, however, be applicable to a more extended range of problems such as the adiabatic limit for vibrations and the general problem of rotation-vibration interactions.
Large amplitude collective motions in a many-Fermion system are discussed from a standpoint of the variational method. In our method, the variational space itself is also determined by a variational procedure which is an extension of the Peierls-Yoccoz method. The Schrödinger equation for large amplitude collective motions is obtained under the additional approximations: ( i ) The variational space forms a Lie group and ( ii ) it is separated adiabatically from the remaining space. Our method may be regarded as the Born-Oppenheimer approximation in the unclear physics. Finally it is applied to the Lipkin model.
We illustrate some of the concepts introduced in a recent paper devoted to the formulation of a self-consistent theory of large amplitude collective motion at finite excitation energy. We apply these concepts to a Hamiltonian with the symmetry of $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ (a doubled Lipkin model), in which the latter, exemplifying the slow, collective degrees of freedom, interacts with a set of harmonic oscillators representing the fast, noncollective variables. We pass to a mean-field approximation, which implies a collisionless regime, but show as well how the formalism may be extended to include the limit of collisions that establish instantaneous local equilibrium. In the representation of natural orbitals, the variables split into classical canonical sets and occupation numbers of the orbitals. Careful attention is paid to the problem of eliminating the intrinsic canonical coordinates from the equations of motion in such a way (white noise approximation) as to yield dissipative equations of motion for the collective variables. For the case of local equilibrium, the relaxation problem requires in addition an equation relating the slow change in temperature to the slow change of the collective variables. The two limiting relaxation scenarios are studied numerically.
In phenomenological descriptions of nuclear vibrations a close connection is implied or expressed between the most-collective transition operator and the harmonic-oscillator (boson) variables which are introduced to describe the collective motion. With the help of a widely used example - the monopole vibration model of Meshkov, Lipkin, and Glick - it is shown how this relationship suggests, for the microscopic case, a new and flexible class of boson representations. As shown, these are capable of yielding accurate descriptions of the system not only in the extremes of weak and strong coupling, but in the usually more inaccessible intermediate regions as well.
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.]