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Physical Review C

Selective bosonization of the many-fermion problem in a model framework

Eugene R. Marshalek, Y. Miyanishi

Published May 22, 2001• 0 citations
Publisher

Abstract

The higher-order terms of a novel boson-fermion expansion formalism previously introduced by one of the authors are discussed within the framework of the soluble SU(2) LMG model. A new formulation is provided to rectify certain undesirable features of the original expansion. In particular, the subsidiary condition that defines the physical subspace is obtained in a much simpler form. With the new formulation, the infinite expansion is summed in closed form. A comparison is made with earlier extended Holstein-Primakoff boson-fermion expansions.

DOI
10.1103/physrevc.63.064314

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The higher-order terms of a novel boson-fermion expansion formalism previously introduced by one of the authors are discussed within the framework of the soluble SU(2) LMG model. A new formulation is provided to rectify certain undesirable features of the original expansion. In particular, the subsidiary condition that defines the physical subspace is obtained in a much simpler form. With the new formulation, the infinite expansion is summed in closed form. A comparison is made with earlier extended Holstein-Primakoff boson-fermion expansions.
In order for a boson-expansion theory to be useful for practical purposes, it must satisfy at least two requirements: It must be in the form of a linked-cluster expansion, and the pure (ideal) boson states must be usable as basis states. Previously, we constructed such a boson theory and used it successfully for many realistic calculations. This construction, however, lacked mathematical rigor. In the present paper, we develop an entirely new approach, which results in the same boson expansions obtained earlier, but now in a mathematically rigorous fashion. The achievement of the new formalism goes beyond this. Its framework is much more general and flexible than was that of the earlier formalism, and it allows us to extend the calculations beyond what had been done in the past.NUCLEAR STRUCTURE Boson expansion theories, commutator method, Marumori-Yamamura-Tokunaga method, fermion system, norm matrix, linked-cluster expansion, ideal-boson state, physical boson state, even-even nucleus, odd-AAA nucleus, nuclear collectivity.
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),
We propose a new boson expansion theory that does not premise the closed-algebra approximation, indispensable for formulation until now, as an extension of the conventional practical boson expansion methods that have tried to elucidate nuclear collective motion, which is a method that allows the closed-algebra approximation not to be used or to be used appropriately, enables us to obtain the boson expansion easier, and reproduces the fermion subspace onto the boson subspace more faithfully than the conventional practical methods. The two-phonon norm matrix composed of all the phonon excitation modes is investigated in detail, which reveals the mechanism, essential for the boson expansion methods, of how we should construct the fermion subspace to be mapped from the whole fermion space. The conventional practical boson expansion methods have applied the closed-algebra approximation improperly to strengthen the effect of the Pauli principle inappropriately, which should be replaced by those that do not use that approximation or use it properly.
In the framework of the Marumori-Yamamura-Tokunaga boson expansion, three forms for the su(2)-spin system are presented. The first is related with the well-known Holstein-Primakoff boson expansion and the second is the expansion reinvestigated in a previous paper by the present authors. The third is, in some sense, the original part of this paper. Corresponding with three forms, c-number counterparts are obtained by using three forms of wave packets defined in the su(2)-spin space. With the help of appropriate quantization rule, they go back to the original boson expansions. Furthermore, it is discussed if the c-number counterparts can be regarded as the classical counterparts. Finally, the relation to the su(1, 1)-algebraic model is discussed.
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.