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SU (6) bozonska proširena aproksimacija slučajnih faza: novi pristup mikroskopskom ostvarenju interakcijsko bozonskog modela

Kyrchev, Georgi, Paar, Vladimir

Published Jan 1, 1990• 0 citations
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Abstract

Lie algebraic approach to the microscopic foundation of Interacting Boson Model – 1 is developed, treating on equal footing the SU (6) governed dynamics and the accompanying SU (6) constraints. In this approach the collective RPA phonon operators are used as „preferred pairs“ with subsequent enforcement of the relevant SU (6) algebra. The „decoupled“ phonon subspace is identified as carrier space of the totally symmetric irreducible representation of SU (6) group. From the microscopic Hamiltonian the fragment with the ensuing IBM-1 boson form is extracted.

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Microscopic approach to enforced SU(6) symmetry in random phase approximation

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Explicit construction of Dyson, Holstein-Primakoff, and Schwinger representations for random-phase approximation phonon operators is given within Lie algebraic framework. It is shown that these representations emerge as exact boson realizations of quadrupole collective algebra, but the enforcement of SU(6) symmetry involves important constraints embodied in definite nonlinear conditions imposed on random-phase approximation phonon amplitudes. The constructed Schwinger representation could be employed to provide alternative approach to the interacting boson model parameters, avoiding standard procedures of mapping the shell-model SD subspace into the sd boson space.

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Correlated Nucleon Pair Model and Microscopic Basis of Interacting Boson

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•Sep 24, 1984•0 citations

Dynamical group of microscopic collective states. I. One-dimensional case

J. Deenen, C. Quesne

In the present series of papers it is intended to determine the nature and study various realizations of the dynamical group of microscopic collective states for an A-nucleon system, defined as those A-particle states invariant under the orthogonal group O(n) associated with the n = A−1 Jacobi vectors. The present paper discusses the case of a hypothetical one-dimensional space. Simple invariance considerations show that the dynamical group of collective states is then the group 𝒮𝓅c(2,R), which is the restriction to the collective subspace of the group 𝒮𝓅(2,R) of linear canonical transformations in n dimensions conserving the O(n) symmetry. In addition to the well-known realization of the dynamical group in the Schrödinger representation based upon the Dzublik–Zickendraht transformation, two new realizations are proposed. The first acts in a Barut Hilbert space, which is the subspace of a Bargmann Hilbert space of analytic functions left invariant by O(n). A unitary mapping is established between the ordinary Hilbert space of collective states and the Barut Hilbert space and coherent collective states are defined in the latter. The second is carried out in terms of one boson creation and one boson annihilation operator through a generalized Holstein–Primakoff representation. The generator of a U(1) group, which is the one-dimensional analog of the U(6) group of the interacting boson model (IBM), can then be expressed in terms of the generators of 𝒮𝓅c(2,R). Finally the generalization of the preceding analysis to a d-dimensional space is outlined in the cases where d = 2 or 3. The dynamical group of collective states becomes 𝒮𝓅c(2d,R).

Journal of Mathematical Physics•May 1, 1982•55 citations
Lie algebraic approach to the microscopic foundation of Interacting Boson Model – 1 is developed, treating on equal footing the SU (6) governed dynamics and the accompanying SU (6) constraints. In this approach the collective RPA phonon operators are used as „preferred pairs“ with subsequent enforcement of the relevant SU (6) algebra. The „decoupled“ phonon subspace is identified as carrier space of the totally symmetric irreducible representation of SU (6) group. From the microscopic Hamiltonian the fragment with the ensuing IBM-1 boson form is extracted.
Explicit construction of Dyson, Holstein-Primakoff, and Schwinger representations for random-phase approximation phonon operators is given within Lie algebraic framework. It is shown that these representations emerge as exact boson realizations of quadrupole collective algebra, but the enforcement of SU(6) symmetry involves important constraints embodied in definite nonlinear conditions imposed on random-phase approximation phonon amplitudes. The constructed Schwinger representation could be employed to provide alternative approach to the interacting boson model parameters, avoiding standard procedures of mapping the shell-model SD subspace into the sd boson space.
Considering the present situation of Lie-algebraic studies of high-temperature superconductivity, a Schwinger-type boson realization of the su(4) algebra is investigated. The basic framework comes from a possible form of the su(n) and the su(n,1) algebras in the Schwinger representation, which was recently proposed by two of the present authors (M. Y. & A. K.), together with J. da Providência. The form investigated in this paper is, in some sense, a natural extension of the Schwinger boson representation for the so(5), so(4) and su(2)⊗su(2) algebras, which was also proposed by the present authors. The orthogonal set for the irreducible representation of the su(4) algebra is completely analogous to that expressed in terms of fermion operators for pairing correlation.
An earlier work dealing with low-lying collective states in nuclei is revised and improved. Correlated and independent nucleon and pairs properly modified so that they behave approximately like bosons are constructed in terms of which a model space for low-lying collective states is defined, and a microscopic basis for IBM is thus set up. All dynamical variables can also be "bosonized", i.e., expressed in terms of these "bosons" but with state-dependent coefficients or operators to incorporate their state dependence in model space.
Algebraic mean field theory constructs a group theoretical model of quantum systems that have a weak dynamical symmetry but may break dynamical symmetry. The strong defining condition for dynamical symmetry is that states belong to one irreducible representation space. Weak dynamical symmetry demands that the densities corresponding to the states have a constant value for each Casimir. Quantum phase transitions and other complex systems exhibit weak dynamical symmetry. Furthermore mean field theory often yields analytic formulae for expectations and energy spectra that are not feasible in representation theory. This paper develops mean field theory on any coadjoint orbit of su (4) densities. The simple Lie algebra su (4) ≃ so (6) is a 15-dimensional algebra that contains the subalgebra usp (4) ≃ so (5) and the angular momentum algebra su (2). The su (4) dual space consists of density matrices which are defined by the expectations of the su (4) generators. A coadjoint orbit is a common level surface in the dual space of the three su (4) Casimirs. A Lax pair determines the dynamics of these densities on each coadjoint orbit. Analytic solutions are reported for rotating su (4) densities in equilibrium for a particular energy function.
The microscopic basis of the Interacting Boson Model for deformed nuclei is discussed. The IBM Hamiltonian is constructed microscopically in the following two steps. In the first step, the collective nucleon pairs of J = 0/sup +/ (S), 2/sup +/ (D), etc. are mapped onto the corresponding bosons. Nucleon-nucleon interactions are also mapped onto boson-boson interactions. This mapping method for deformed nuclei was proposed recently, and it turned out that this method is consistent with the Hartree-Fock-Bogoliubov + angular momentum projection calculation. Low-lying collective states primarily consist of S and D pairs. Consequently, the corresponding boson states mainly consist of s and d bosons, while there are some admixture of g-bosons. In the second step, effects of these g-bosons are included within the s-d boson space by a unitary transformation which transforms a combination of d and g bosons into a new d-boson. By minimizing the coupling between new d and g bosons with an appropriate mixing angle, one can neglect the coupling and obtain the IBM Hamiltonian with s and d bosons. It is demonstrated that the s-d Hamiltonian thus derived indeed reproduces spectra of the original s-d-g Hamiltonian.
In the present series of papers it is intended to determine the nature and study various realizations of the dynamical group of microscopic collective states for an A-nucleon system, defined as those A-particle states invariant under the orthogonal group O(n) associated with the n = A−1 Jacobi vectors. The present paper discusses the case of a hypothetical one-dimensional space. Simple invariance considerations show that the dynamical group of collective states is then the group 𝒮𝓅c(2,R), which is the restriction to the collective subspace of the group 𝒮𝓅(2,R) of linear canonical transformations in n dimensions conserving the O(n) symmetry. In addition to the well-known realization of the dynamical group in the Schrödinger representation based upon the Dzublik–Zickendraht transformation, two new realizations are proposed. The first acts in a Barut Hilbert space, which is the subspace of a Bargmann Hilbert space of analytic functions left invariant by O(n). A unitary mapping is established between the ordinary Hilbert space of collective states and the Barut Hilbert space and coherent collective states are defined in the latter. The second is carried out in terms of one boson creation and one boson annihilation operator through a generalized Holstein–Primakoff representation. The generator of a U(1) group, which is the one-dimensional analog of the U(6) group of the interacting boson model (IBM), can then be expressed in terms of the generators of 𝒮𝓅c(2,R). Finally the generalization of the preceding analysis to a d-dimensional space is outlined in the cases where d = 2 or 3. The dynamical group of collective states becomes 𝒮𝓅c(2d,R).