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Progress of Theoretical Physics

Perturbation Expa]nsion in Dynamical Nuclear Field Theory and Its Relation with Boson Expansion Theory

Teruo Kishimoto, Tetsuo Kammuri

Published Sep 1, 1990• 1 citation
Publisher

Abstract

With the Dynamical Nuclear Field Theory (DNFT) in the Tamm-Dancoff representation we examine higher order corrections in the vibrational mode of a spherical nuclear system. Due to the effects of bubble diagrams, the perturbation expansion in terms of the unrenormalized coupling strength and boson energy fails at full self-consistency. On the other hand, it becomes applicable in the form of linked-cluster expansion when we use these constants renormalized by the effect of bubble diagrams, in the schematic model. However, in comparison with the boson expansion theory, serious defects are found in higher order DNFT.

DOI
10.1143/ptp.84.470

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With the Dynamical Nuclear Field Theory (DNFT) in the Tamm-Dancoff representation we examine higher order corrections in the vibrational mode of a spherical nuclear system. Due to the effects of bubble diagrams, the perturbation expansion in terms of the unrenormalized coupling strength and boson energy fails at full self-consistency. On the other hand, it becomes applicable in the form of linked-cluster expansion when we use these constants renormalized by the effect of bubble diagrams, in the schematic model. However, in comparison with the boson expansion theory, serious defects are found in higher order DNFT.
In the treatment of the vibrational mode of even-even nuclei, Dynamical Nuclear Field Theory (DNFT) in the Tamm-Dancoff (TD) representation exhibited the defect to include unlinked cluster terms. We aim to remove this deficiency. In lowest order of the repetitive actions of the particle-boson couplings, we generate |0)B⊗|nTD fermion pairs> states out of |0>F⊗|nbosons) states. We impose the lowest order self-consistency. The modified DNFT gives the normal-ordered linked-cluster expansion, in complete equivalence with the boson expansion theory (BET), of Kishimoto, Tamura and Sakamoto.
Dynamical Nuclear Field Theory which was modified in the previous paper so as to bring complete equivalence with Boson Expansion Theory (BET) is now extended to derive a BET under the influence of two-body correlations. For this purpose, the driving Hamiltonian of the vibrating Hartree field approximation is supplemented by the residual two-body interaction. Effective forms of Tamm-Dancoff fermion-pair and scattering operators are expanded perturbatively. Effects of three kinds of bubble diagrams can be incorporated by the renormalization procedure, but the contributions from unlinked diagrams remain uncancelled. Comparisons with Nuclear Field Theory are discussed.
A many•body technique called the dynamical nuclear field theory is developed by taking the example of pair rotation.The time-dependent one-body Hamiltonian contains the collective coordinates; the magnitude and phase aIigleof the energy gap.Effective forms of the original Hamiltonian and other operators are derived for a direct-product space of the collective states and a restricted set of the particle states defined by the driving Hamiltonian in the intrinsic frame.Pair rotations of even-and odd-A systems are treated in the lowest order approximation.The particle-rotor picture and the cranking model are shown to be equivalent.*) Brief report on DNFT in the Tamm-Dancoff approximation can be found in Ref. 13).
We derive an approximate analytical solution of the self-consistency equations of the bosonic dynamical mean-field theory (B-DMFT) in the strong-coupling limit. The approach is based on a linked-cluster expansion in the hybridization function of normal bosons around the atomic limit. The solution is used to compute the phase diagram of the bosonic Hubbard model for different lattices. We compare our results with numerical solutions of the B-DMFT equations and numerically exact methods, respectively. The very good agreement with those numerical results demonstrates that our approach captures the essential physics of correlated bosons both in the Mott insulator and in the superfluid phase. Close to the transition into the superfluid phase the momentum distribution function at zero momentum is found to be strongly enhanced already in the normal phase. The linked-cluster expansion also allows us to compute dynamical properties such as the spectral function of bosons. The evolution of the spectral function across the transition from the normal to the superfluid phase is seen to be characteristically different for the interaction driven and density driven transition, respectively.
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),
The Goldstone expansion is rederived by elementary time-independent methods, starting from Brillouin-Wigner (BW) perturbation theory. Interaction energy terms $\ensuremath{\Delta}E$ are expanded out of the BW energy denominators, and the series is then rearranged to obtain the linked-cluster result. Similar algebraic methods lead to the linked expansions for the total wave function (Hugenholtz) and the expectation value of a general operator (Thouless). Starting again with a degenerate version of BW perturbation theory, these methods are used to obtain the Bloch-Horowitz energy expansion, as well as the corresponding wave function, expectation-value, and transition-amplitude expansions. A "reduced" form of the Block-Horowitz expansion is described, and also a "completely linked" version. The latter is suggested as a tool for investigating superfiuid phenomena in nuclear matter, and for establishing contact with the Landau theory of Fermi liquids. The physical interpretation of these expansions is carefully studied, especially with regard to nuclear applications, to determine how they handle such "physical" features as antisymmetry, self-energy effects, wave-function renormalizations, and the distinction between "true" and "model" single-particle occupation probabilities. The problem of a correct theoretical definition for the shell-model potential is carefully examined, and a specific theory is presented. These expansions are seen to form a convenient and very powerful set of tools for studying the structure of actual nuclei.