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Progress of Theoretical Physics

On the Applicability of Generalized Schwinger Representation of the Fermion Pairs to the Anharmonicity. I

Y. Mizobuchi, S. Nishiyama, M. Yamamura

Published May 1, 1978• 1 citation
Publisher

Abstract

The generalized Schwinger representation of the fermion pairs is reformulated with the aim of investigating the applicability to the anharmonicity. Through this reformulation, the seniority quantum number and quasi-boson operator as the generator of higher seniority states are introduced into the Schwinger boson space. An approximate diagonalization of pairing Hamiltonian gives the same result as that of the BCS-Bogoliubov theory.

DOI
10.1143/ptp.59.1551

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The generalized Schwinger representation of the fermion pairs is reformulated with the aim of investigating the applicability to the anharmonicity. Through this reformulation, the seniority quantum number and quasi-boson operator as the generator of higher seniority states are introduced into the Schwinger boson space. An approximate diagonalization of pairing Hamiltonian gives the same result as that of the BCS-Bogoliubov theory.
A new method for describing the two-level pairing model in a many-fermion system is presented. The basic idea comes from the Schwinger boson representation for the su(2) ⊗ su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the su(2) ⊗ su(1,1)-coherent state is mainly discussed in this paper.
The lowest seniority-zero eigenstates of an exactly solvable multilevel pairing Hamiltonian for a finite Fermi system are examined at different pairing regimes. After briefly reviewing the form of the eigenstates in the Richardson formalism, we discuss a different representation of these states in terms of the collective pairs resulting from the diagonalization of the Hamiltonian in a space of two degenerate time-reversed fermions. We perform a two-fold analysis by working both in the fermionic space of these collective pairs and in a space of corresponding elementary bosons. On the fermionic side, we monitor the variations which occur, with increasing the pairing strength, in the structure of both these collective pairs and the lowest eigenstates. On the bosonic side, after reviewing a fermion-boson mapping procedure, we construct exact images of the fermion eigenstates and study their wave function. The analysis allows a close examination of the phenomenon of pair condensation in a finite Fermi system and gives new insights into the evolution of the lowest (seniority-zero) excited states of a pairing Hamiltonian from the unperturbed regime up to a strongly interacting one.
A derivation of the random phase approximation is given, which is based on the generalized Schwinger representation of the fermion-pairs. The method is made of two stages : First, the truncation of the boson space is done, and secondly comes the separation of boson operators into static and fluctuating parts. The pairing vibration in the super phase and the so-called quadrupole oscillation are treated by the method, which leads to the same results as those of RPA. Then the relations of these bosons to ones appearing in RPA are clarified.
A new method for describing the two-level pairing model in a many-fermion system is presented. Basic idea comes from the Schwinger boson representation for the su(2) cross su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the mixed-mode coherent state is mainly discussed in this paper.
The two-level pairing model was investigated in the framework of the Schwinger representation. It has been shown that zero-point fluctuations, which are important in the crossover region, are well described by certain coherent states which conserve the constants of motion. These so-called conserving approximations provide excellent descriptions of the ground-state energy and of the first excitation energy over the whole range of coupling constant values. The description of the ground-state energy provided by the Glauber coherent state is also reasonable, but not so good. Although the Glauber coherent state fails to provide the description of the excitation energy around the critical point, its performance in the strong coupling limit is very good both for the description of the ground-state energy and of the first excitation energy. The present results also show that already at the mean-field level, the bosonic description in the independent particle approximation incorporates important correlation corrections to the conventional mean-field description based on the fermionic realization of the su (2) algebra (BCS theory).
A nonlinear generalization of the quasiparticle random phase approximation is constructed with the aim of describing anharmonic effects in the low-lying spectra of vibrational nuclei. Angular momentum zero pairing correlations of the ground state are described by a generalized BCS calculation based on a variational method. The dynamics of the vibrational states are described by the Heisenberg equations of motion for fermion pair operators; matrix elements of products of pair operators are treated by sum rule methods. Kinematical constraints are expressed by the algebra of the pair operators and by boson-like polynomial expansions for the quasiparticle conserving multipole operators in terms of the quasiparticle pair changing operators, these expansions enforcing the content of the Pauli principle. The physical basis for the choice of approximations is described. The final algorithm is organized so as to involve little more than repeated BCS-like and RPA-like calculations, converging to a self-consistent solution of the full nonlinear scheme. Detailed discussion is carried out for a modified form of the pairing plus quadrupole interaction Hamiltonian, but the method is not confined to this model.NUCLEAR STRUCTURE Approximate many-body methods applied to modified pairing plus quadrupole model. Description of vibrational spectrum of single closed shell nuclei. Equations of motion and algebraic methods.