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Anharmonic collective excitation in a solvable model

G. F. Bertsch, P. F. Bortignon, K. Hagino

Published Jan 1, 1999• 9 citations
OpenAlex

Abstract

We apply the time-dependent variational principle, the nuclear field theory, and the boson expansion method to the Lipkin model to discuss anharmonicities of collective vibrational excitations. It is shown that all of these approaches lead to the same anharmonicity to leading order in the number of particles. Comparison with the exact solution of the Lipkin model shows that these theories reproduce it quite well. I.

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Physical Review C•May 1, 1992•4 citations
We apply the time-dependent variational principle, the nuclear field theory, and the boson expansion method to the Lipkin model to discuss anharmonicities of collective vibrational excitations. It is shown that all of these approaches lead to the same anharmonicity to leading order in the number of particles. Comparison with the exact solution of the Lipkin model shows that these theories reproduce it quite well. I.
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.
A general hierarchy of approximations to the many-body Schr\"odinger equation is presented which reduces to the time-dependent mean-field approximation in lowest order and provides systematic corrections in subsequent orders. The theory is applied to two interacting systems described by the Lipkin model Hamiltonian. Comparison of the results of the lowest two orders of approximation with the exact solution demonstrates the practicality of the method and its potential for generalizing nuclear dynamics beyond the mean field theory.NUCLEAR STRUCTURE Time-dependent approximation to nuclear dynamics. Corrections to time-dependent mean-field theory. Solution of Lipkin model for two interacting systems.
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.
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)
For the last decades Boson expansion theory has played an effective role in the development of approximate solutions to the many-body problem in nuclear physics [1]. These techniques also provide promising tools to address solutions for quantum field theories [2]. In a recent work, we have succeeded in combining an extended form of the Holstein-Primakoff mapping (EHPM) [3] with thermofield-dynamics [4] to generate a consistent thermal boson expansion for quantum mechanical systems such as the bosonic O(N) anharmonic oscillator as well as the fermionic Lipkin model [7]. The analogue of the first of these two models in four space-time dimensions can be used to address important questions such as the chiral transition for two flavors QCD or the scattering of pions at zero and finite temperature. It is also found that the EHPM removes number of ambiguities when computing anharmonicities in the next-to-leading order of the 1/N expansion [7], which are very much relevant to the ππ scattering problem, for instance. Furthermore, the EHPM allows to extend the use of the boson expansion approach to quantum systems with internal SU(N) symmetry where one is confronted with theories having tensor couplings LI = fijklφiφjφkφl + gijkφiφjφk +..., (1) and which calls for a simultaneous bosonization of both pair and single modes. This opens, indeed, the possibility to ultimately address the question of the chiral transition in SU(3) flavor QCD in four space-time dimensions. This point, although crucial for the understanding of the QCD phase diagram, has not been adequately addressed before due to the fact that the 1/Nf expansion cannot be correctly sorted out with known techniques [6]. The approach highlighted above and which fundamentally relies on the bosonization of the generators of the Sp2⊗N1 semidirect product group, (aa)I = √ 2+4(n + m)A, (a + a)I =2n+ m, (a)I = √ 2+4(n + m)Γ(m)α +2α + A Γ(m),
A classical microscopic theory of mixed states has recently been formulated in order to deal with the theoretical description of nuclear phenomena observed in highly excited states. In this theory, which is here briefly reviewed, the degrees of freedom of real fermions are doubled by fictitious fermions, in the context of the formalism of thermofield dynamics. The requirement of static stability of the thermal state of equilibrium plays, in the theory, a relevant role. The possibility of constraining the dynamics of the fictitious degrees of freedom, in such a way as to eliminate spurious thermal excitations, is discussed. The concept of canonical collective variables for mixed states is analyzed. Boson expansions appropriate to describe collective excitations of hot quantal systems are introduced and their properties investigated. It is shown that thermal boson expansions based on the formalism of thermofield dynamics are free from formal inconsistencies which may be present in boson expansions derived by an ad hoc quantization of conventional thermal mean field theories. The formalism is applied to the Lipkin model, and the correlation energy of a hot system is discussed.