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Chinese Physics C

1/N Expansion Technique and the Non-SU(3)Intrinsic States of the sdg Interacting Boson Model

Heng Wu

Published Jan 1, 1991• 0 citations
OpenAlex

Abstract

By utilizing the 1/N expansion technique, the SU(3)-symmetry breaking caused by boson energies is discussed for the sdg Interacting Boson Model, and the analytical expressions of the intrinsic states are given under the approximation in the order of 1/N. It is also shown that the many body theories for a interacting boson system, such as Hartree-Bose method, Tamm- Dancoff approximation and Cranked Hartree approximation, are all special cases of the 1/N expansion technique.

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By utilizing the 1/N expansion technique, the SU(3)-symmetry breaking caused by boson
energies is discussed for the sdg Interacting Boson Model, and the analytical expressions of the
intrinsic states are given under the approximation in the order of 1/N. It is also shown that
the many body theories for a interacting boson system, such as Hartree-Bose method, Tamm-
Dancoff approximation and Cranked Hartree approximation, are all special cases of the 1/N
expansion technique.
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.
I specialize the Hartree-Fock, random-phase, and configuration-interaction approximations to a system of interacting bosons in an external potential. This approach affords a more direct connection to the methods and language of atomic-structure calculations. The derivations and equations parallel those found in atomic structure and are contrasted with the more commonly used Bogoliubov approach for bosons. I present a numerical method for solving the Hartree-Fock and random-phase equations to obtain the low-lying excitation energies for trapped alkali-metal atoms.
This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at T=0.T=0.T=0. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3,\sqrt{{\mathrm{na}}^{3}},na3​, where aaa is the sss-wave scattering length and nnn is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include 1/N1/N1/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.