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Physical Review C

Mean field description of the ground state of many boson systems relevant to nuclei

Fermin Aldabe, G.G. Dussel, Hugo M. Sofia

Published Sep 1, 1994• 2 citations
Publisher

Abstract

In the present paper we give the explicit expressions for the ground state of a many boson system in different mean field approximations, such as Hartree-Bose, Bogoliubov, the particle-hole random phase approximation (RPA), and its coupling with the particle-particle (RPA). The ground states obtained satisfy the requirement that the annihilation operators of the ``elementary excitations'' annihilates them. In all cases the ground state wave functions can be understood as a condensate of pairs of bosons.

DOI
10.1103/physrevc.50.1518

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In the present paper we give the explicit expressions for the ground state of a many boson system in different mean field approximations, such as Hartree-Bose, Bogoliubov, the particle-hole random phase approximation (RPA), and its coupling with the particle-particle (RPA). The ground states obtained satisfy the requirement that the annihilation operators of the ``elementary excitations'' annihilates them. In all cases the ground state wave functions can be understood as a condensate of pairs of bosons.
A general theory of long range correlations in the ground state of a closed shell nucleus is outlined, using a number of ideas introduced or adumbrated, but not always successfully developed, in previous work of the authors and others. These include the following: (i) Any operator can be formally expressed as a series in particle-hole (p-h) excitation and destruction operators which should converge rapidly for a suitable choice of single-particle basis at the closed shells. By means of these relations and with the help of sum rules all observables can be computed without the explicit use of wave functions. (ii) For a fixed choice of single-particle basis, the ground state energy can be computed from a variational principle in which in lowest nontrivial approximation, in consequence of (i), the parameters are amplitudes of the same type as occur in the random phase approximation (RPA). The precise relation to RPA is fully detailed. (iii) The single particle basis chosen is the natural orbital basis of Lowdin, which presents itself as the obvious choice both for formulation of the dynamics and for the algorithm of solution. (iv) Though the theory deals with pairs of fermion operators, particular attention is paid that all Pauli principle restrictions are satisfied. (v) It is verified that in the perturbation limit, known results are reproduced. The method is generally applicable to any system for which the correlations can be viewed as particle-hole excitations with respect to a reference Slater determinant.
We reconsider the long-standing problem that the ground-state correlations predicted by the standard quasiboson (random phase) approximation are too large by a factor of two in the limit of weak residual interaction, and are thus inconsistent with the Pauli exclusion principle. We solve this problem by noting that for the derivation of the equations of the random phase approximation (RPA), which determine a set of matrix elements and excitation energies, it is unnecessary to assume that the ground state is the vacuum of a set of boson excitations, whereas in the faculty calculation of the ground-state correlation energy, this assumption is made. For the study of excitation energies and the restoration of symmetries broken by an initial mean-field solution, we apply a symmetry-preserving form of the equations of motion for fermion particle-hole excitation operators rather than for bosons. We illustrate this for the case of translations. We then describe a separate calculation of the ground-state energy that is consistent with the Pauli principle, but can nevertheless be evaluated in terms of the solutions of the RPA equations of motion. In contrast to some earlier work, we treat direct and exchange terms on an equal footing throughout.
Identical Particles Second Quantization Independent-Particle Model for Fermions in Finite Systems Two-Particle States and Interactions Noninteracting Bosons and Fermions Propagators in One-Particle Quantum Mechanics Single-Particle Propagator in Many-Particle Systems Perturbation Expansion of the Single-Particle Propagator Dyson Equation and Self-consistent Green's Functions Mean-Field or Hartree-Fock Approximation Beyond the Mean-Field Approximation Interacting Boson Systems Excited States in Finite Systems Excited States in Infinite Systems Excited States in N 2 Systems and In-Medium Scattering Dynamical Treatment of the Self-energy in Infinite Systems Dynamical Treatment of the Self-energy in Finite Systems Bogoliubov Perturbation Expansion for the Bose Gas Boson Perturbation Theory Applied to Physical Systems In-Medium Interaction and Scattering of Dressed Particles Conserving Approximations and Excited States Pairing Phenomena Propagator Framework for Phenomenological Description of Many-Body Systems Finite Temperature Formalism and Applications.
We verify Bogoliubov's approximation for translation invariant Bose gases in the mean field regime, i.e. we prove that the ground state energy ENE_NEN​ is given by EN=NeH+inf⁡σ(H)+oN→∞(1)E_N=Ne_\mathrm{H}+\inf σ\left(\mathbb{H}\right)+o_{N\rightarrow \infty}(1)EN​=NeH​+infσ(H)+oN→∞​(1), where NNN is the number of particles, eHe_\mathrm{H}eH​ is the minimal Hartree energy and H\mathbb{H}H is the Bogoliubov Hamiltonian. As an intermediate result we show the existence of approximate ground states ΨNΨ_NΨN​, i.e. states satisfying ⟨HN⟩ΨN=EN+oN→∞(1)\langle H_N\rangle_{Ψ_N}=E_N+o_{N\rightarrow \infty}(1)⟨HN​⟩ΨN​​=EN​+oN→∞​(1), exhibiting complete Bose--Einstein condensation with respect to one of the Hartree minimizers.
We discuss a variational approach to collective excitations in a boson formalism based on quasiparticles. Bosons are defined in correspondence with pairs of quasiparticles and boson images of fermion operators are constructed by means of a mapping procedure of Marumori-type. Phonons of the type used within the random phase approximation (RPA) are introduced as Bogoliubov transformations of these bosons. The variables entering into the definition of these phonons as well as of the quasiparticle operators are fixed simultaneously by minimizing the expectation value of the boson Hamiltonian in the vacuum of the phonons. The approach is tested within an exactly solvable two-level model which is characterized by a pairing Hamiltonian. A quite good agreement is found for the energies of the ground state and of the first 0+{0}^{+}0+ excited state. The comparison with the Bardeen-Cooper-Schrieffer method and the quasiparticle RPA as well as with some recent self-consistent RPA-type approaches is discussed.
Working within a schematic model and in a boson formalism, we provide a descrip tion of the ground and first-excited states of a nuclear system in terms of two independent phonon operators. The description reveals itself quite effective in reproducing the energies of the states. A comparison between these two phonon operators allows us to localize a region where a correct description of these states is compatible with the Random Phase Approximation (RPA) requirement of a single phonon. This region is found to coincide approximately with that spanned by the standard RPA. Outside this region, the two phonons start differing rapidly therefore making the application of the RPA scheme impossible. This allows a better understanding of recently performed higher-order RPA calculations.