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Physical Review C

Quonic expansion and its random-phase approximation counterpart

Sidney S. Avancini, F. Cruz, José Ricardo Marinelli, Débora Peres Menezes

Published Jul 19, 2000• 3 citations
Publisher

Abstract

The usefulness of the quon expansion to treat Pauli principle violations and many-body correlations in many-fermion systems is investigated within two models, namely, the Lipkin and the two level pairing models. The q deformation parameter of the quon algebra is fixed through a dynamical criterion. The spectra obtained are compared with the random-phase approximation (RPA), usual boson expansion, and exact results. The quonic expansion is shown to take into account correlations which are not included either in the second order boson expansion or in the RPA approach.

DOI
10.1103/physrevc.62.024312

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The usefulness of the quon expansion to treat Pauli principle violations and many-body correlations in many-fermion systems is investigated within two models, namely, the Lipkin and the two level pairing models. The q deformation parameter of the quon algebra is fixed through a dynamical criterion. The spectra obtained are compared with the random-phase approximation (RPA), usual boson expansion, and exact results. The quonic expansion is shown to take into account correlations which are not included either in the second order boson expansion or in the RPA approach.
The random-phase approximation (RPA) is examined in a boson formalism and special attention is focused on the problem of the violation of the Pauli principle affecting this theory. A mapping technique is discussed and the boson image of a two-body fermion Hamiltonian is constructed. Within the boson space so defined, an extension of RPA is proposed where structure and energy of the ground state, as well as those of the one-phonon states, are analyzed with regard to the role of the spurious components associated with a violation of the Pauli principle. A multistep minimization procedure for the determination of a better ground state is also examined. Numerical tests are performed within the Lipkin-Meshkov-Glick model.
The quon algebra, which interpolates between the Bose and Fermi algebras and depends on a free parameter q , is used to generate deformed Marumori boson expansions for the pair and quadrupole operators. Then we obtain a pairing plus quadrupole-quadrupole Hamiltonian, for a single j -shell, in terms of these deformed bosonic operators. The Hamiltonian is diagonalized and its eigenvalues are compared with the ones obtained from the counterpart fermionic Hamiltonian. The deformation parameter helps in achieving the correct energy levels, which cannot be encountered in practice with the usual non-deformed boson expansion.
The quon algebra describes particles, ``quons,'' that are neither fermions nor bosons. The parameter qqq attached to a quon labels a smooth interpolation between bosons ( q=+1q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+1q=+1) and fermions ( $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}1$). Wigner and Ehrenfest and Oppenheimer showed that a composite system of identical bosons and fermions is a fermion if it contains an odd number of fermions and is a boson otherwise. We generalize this and show that qcomposite=qconstituentn2{q}_{\mathrm{composite}}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}q}_{\mathrm{constituent}}^{{n}^{2}}qcomposite​=qconstituentn2​ for a system of nnn identical quons. Using this generalization, we find bounds on possible violations of the Pauli exclusion principle for nucleons and quarks based on such bounds for nuclei.
I discuss the statistics of "quons" (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the qqq-deformed commutation relation (the quon algebra or qqq-mutator) which interpolates between fermions and bosons. Topics discussed include representations of the quon algebra, proof of the TCP\mathrm{TCP}TCP theorem and clustering, violation of the usual locality properties, and experimental constraints on violations of the Pauli exclusion principle (i.e., Fermi statistics) and of Bose statistics.
The exactly solvable Luttinger model for a system of one-dimensional, massless, two-component fermions interacting via general two-body forces is used as a test case for a systematically generated sequence of decoupling approximations commonly referred to as random-phase approximations (RPA) of which the simple RPA is the lowest. It is found, irrespective of both the nature and strength of the interaction, that the exact (collective) spectrum, as well as the exact correlation energy, can already be obtained in the simple RPA, and furthermore, that the next higher RPA gives a vanishing correction to these quantities. It is also shown, quite in contradistinction to the exact result, that the free-gas momentum distribution n‾k{\overline{n}}_{k}nk​ persists in the second RPA as well as in the first, thereby eliminating the higher RPA as a possible approach to the calculation of n‾k{\overline{n}}_{k}nk​. Thus, in the context of this particular model, although the first few orders of RPA are highly effective for calculating the excitation spectrum and correlation energy, they are sensibly powerless in their ability to predict any variation in structure beyond that of the free-gas distribution, a result which most probably holds as well in the realistic three-dimensional case.