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Progress of Theoretical Physics

Description of Pairing Correlation Based on a New Dyson-Like Representation

M. Hasegawa, Nobuo Kanesaki

Published Nov 1, 1977• 6 citations
Publisher

Abstract

By introduction of the specific operators describing the pairing collective degrees of freedom, a new Dyson-like representation of fermion space is given. The separation of pairing “collective” and “intrinsic” degrees of freedom can be done in the original fermion space. The representation leads to a finite but hermitian Hamiltonian, for example, the pairing Hamiltonian containing at most fourth-order operators. Thus one gets a powerful instrument to deal with the kinematical and dynamical effects of the pairing degrees of freedom. To see the usefulness, the pairing correlation is studied within the framework of this formulation. It is demonstrated that unified description of pairing correlation in both the normal and super-phases is possible.

DOI
10.1143/ptp.58.1405

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By introduction of the specific operators describing the pairing collective degrees of freedom, a new Dyson-like representation of fermion space is given. The separation of pairing “collective” and “intrinsic” degrees of freedom can be done in the original fermion space. The representation leads to a finite but hermitian Hamiltonian, for example, the pairing Hamiltonian containing at most fourth-order operators. Thus one gets a powerful instrument to deal with the kinematical and dynamical effects of the pairing degrees of freedom. To see the usefulness, the pairing correlation is studied within the framework of this formulation. It is demonstrated that unified description of pairing correlation in both the normal and super-phases is possible.
The nonunitary character of the Dyson boson mapping is explored in a model case with the monopole pairing interaction. It is shown that because of the very special form of this interaction, the calculations in the intrinsic state formalism with the particle-particle representation of the Hamiltonian reproduce the exact results of the original fermion space. In contrast, we show that with the particle-hole representation of the Hamiltonian the intrinsic state approach produces erroneous results. Of course, upon employing the physical boson states we recover the exact results in the particle-hole represenation as well.
This Brief Report deals with the solution of the shell-model Hamiltonian in the boson representation. In the present approach, the fermion Hamiltonian is exactly mapped to the boson space using the Dyson boson mapping. The problem of mapping the basis states is resolved by employing the intrinsic boson basis. The method has been applied to a model Hamiltonian consisting of a monopole pairing and a deformation force. It is observed that only s and d bosons are dominant in the intrinsic wave function for the model case studied. This is synonymous to the interacting boson model approach.
The present series of papers deals with various realizations of the dynamical group 𝒮𝓅c(2d,R) of microscopic collective states for an A nucleon system in d dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=A−1 Jacobi vectors. In the present paper, we derive two boson representations of 𝒮𝓅c(2d,R), namely the Dyson representation and the Holstein–Primakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert space ℱc of analytic functions in ν =(1/2)d(d+1) complex variables. Basis functions in ℱc, classified according to the chain 𝒮𝓅c(2d,R)⊇𝒰c(d), can be put into one-to-one correspondence with basis functions, classified according to the chain 𝔘(ν)⊇𝔘(d), in a Bargmann Hilbert space ℬ of analytic functions in ν complex variables representing ν-dimensional boson states. By equating the complex variables of ℱc and their conjugate momenta with those of ℬ, we get the non-Hermitian Dyson representation of 𝒮𝓅c(d,R). We then go from the latter to the Hermitian HP representation by means of a canonical transformation that restores the Hermiticity properties of the variables and conjugate momenta. The inverse of the HP representation gives the unitary representation in quantum mechanics of the classical canonical transformation relating the oscillator Hamiltonians of the microscopic collective model and the boson macroscopic one. From the ν boson creation and annihilation operators, it is possible to build the generators of a 𝔘(ν) group, which in the physical three-dimensional case reduces to 𝔘(6). The latter is finally compared with the U(6) group appearing in the interacting boson model.
A canonical theory with constraints is proposed, in the classical framework, for describing pairing rotation and intrinsic degrees of freedom. The basic idea is a direct translation of a theory developed by the present authors for the case of particle-hole pair vibration. The total particle-number and the phase angle for the rotation and the Grassmann numbers for the intrinsic degrees of freedom are used as the canonical variables, the number of which exceeds apparently the original one. Certain constraints play the role of avoiding the double counting of the degrees of freedom concerning the extra variables. Hamiltonian and other physical quantities can be expressed in terms of the canonical variables and their forms can be well understood in the language of two-dimensional rotor model.
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 present paper attempts to give independent descriptions of the collective degrees of freedom of nuclei such as the monopole and quadrupole ones in order to avoid the seniority mixing problem. This is done by the operatorized Bogoliubov transformation (OBT) which separates the degree of freedom from the rest described by the seniority particle operator which is not coupling to the pairs, and in terms of which the independent pair is composed. The OBT is derived in such a way that it can be generalized easily to include a new kind of “seniority particle operator” neither coupling to the nor to the pairs. In terms of this new particle operator the independent pair or the independent odd particle in odd nuclei is constructed.