Large Boson Number Limit Research Articles

What can we learn from the Interacting Boson Model in the limit of large boson numbers?

Over the years, studies of collective properties of medium and heavy mass nuclei in the framework of the Interacting Boson Approximation (IBA) model have focused on finite boson numbers, corresponding to valence nucleon pairs in specific nuclei. Attention to large boson numbers has been motivated by the study of shape/phase transitions from one limiting symmetry of IBA to another, which become sharper in the large boson number limit, revealing in parallel regularities previously unnoticed, although they survive to a large extent for finite boson numbers as well. Several of these regularities will be discussed. It will be shown that in all of the three limiting symmetries of the IBA [U(5), SU(3), and O(6)], energies of 0+ states grow linearly with their ordinal number. Furthermore, it will be proved that the narrow transition region separating the symmetry triangle of the IBA into a spherical and a deformed region is described quite well by the degeneracies E(0^+_2 ) = E(6^+_1 ), E(0^+_3 ) = E(10^+_1 ), E(0^+_4 ) = E(14^+_1 ), the energy ratio E(6^+_1 )/E(0^+_2 ) turning out to be a simple, empirical, easy-to-measure effective order parameter, distinguishing between first- and second-order transitions. The energies of 0+ states near the point of the first order shape/phase transition between U(5) and SU(3) will be shown to grow as n(n+3), where n is their ordinal number, in agreement with the rule dictated by the relevant critical point symmetries studied in the framework of special solutions of the Bohr Hamiltonian. The underlying dynamical and quasi-dynamical symmetries are also discussed.

Line of approximate SU(3) symmetry inside the symmetry triangle of the Interacting Boson Model

The U(5), SU(3), and O(6) symmetries of the Interacting Boson Model (IBM) have been traditionally placed at the vertices of the symmetry triangle, while an O(5) symmetry is known to hold along the U(5)–O(6) side of the triangle. We construct [1] for the first time a symmetry line in the interior of the triangle, along which the SU(3) symmetry is preserved. This is achieved by using the contraction of the SU(3) algebra to the algebra of the rigid rotator in the large boson number limit of the IBM. The line extends from the SU(3) vertex to near the critical line of the first order shape/phase transition separating the spherical and prolate deformed phases. It lies within the Alhassid–Whelan arc of regularity, the unique valley of regularity connecting the SU(3) and U(5) vertices amidst chaotic regions, thus providing an explanation for its existence.

Approximate symmetries in the Interacting Boson Model

Dynamical symmetries have played a central role for many years in the study of nuclear structure. Recently, the concepts of Partial Dynamical Symmetry (PDS) and Quasi-Dynamical Symmetry (QDS) have been introduced. We shall discuss examples of PDS and QDS appearing in the large boson number limit of the Interacting Boson Mod

Majorana Representation and Mean Field Approach for Interacting-Boson System☆☆Supported by the National Natural Science Foundation of China under Grant Nos. 11405008, 11175044, and the Plan for Scientific and Technological Development of Jilin Province under Grant No. 20160520173JH

The Majorana representation, which represents a quantum state by stars on the Bloch sphere, provides us an intuitive tool to study the quantum evolution in high dimensional Hilbert space. In this work, we investigate the second quantized model and the mean-field model for the interacting-boson system in the Majorana representation. It is shown that the motions of states in the two models are same in the linear case. Furthermore, the contribution of the nonlinear interaction to the star motions in the second quantized model can be expressed by a single star part which is equal to the nonlinear part of the equation for the star in mean-field model under large boson number limit and an extra part caused by the correlation between stars. These differences and relations can not only be reflected by the population differences between the two boson modes in the two models, but also lie with the differences between the continuous changes of the second quantized evolution with the nonlinear interacting strength and the critical behavior of the mean-field evolution which related to the self-trapping effect. The reason of the difference between the two models is also discussed by an effective Hamiltonian.

Group contractions and conformal maps in nuclear structure

Group contraction is a procedure in which a symmetry group is reduced into a group of lower symmetry in a certain limiting case. Examples are provided in the large boson number limit of the Interacting Boson Approximation (IBA) model by a) the contraction of the SU(3) algebra into the [R5]SO(3) algebra of the rigid rotator, consisting of the angular momentum operators forming SO(3), plus 5 mutually commuting quantities, the quadrupole operators, b) the contraction of the 0(6) algebra into the [R5]SO(5) algebra of the γ-unstable rotator. We show how contractions can be used for constructing symmetry lines in the interior of the symmetry triangle of the IBA model.In mathematics, a conformal map is a function which preserves angles. We show how this procedure can be used in the framework of the Bohr Hamiltonian, leading to a Hamiltonian in a curved space, in which the mass depends on the nuclear deformation β, while it remains independent of the collective variable 7 and the three Euler angles. This Hamiltonian is proved to be equivalent to that obtained using techniques of Supersymmetric Quantum Mechanics.

Analytic derivation of the Alhassid–Whelan arc of regularity

The Alhassid–Whelan arc of regularity is a unique valley of regularity connecting, amidst chaotic regions, the SU(3) and U(5) vertices of the symmetry triangle of the Interacting Boson Model (IBM). Using the contraction of the SU(3) algebra to the algebra of the rigid rotator in the large boson number limit of the IBM, we construct for the first time a symmetry line in the interior of the triangle, along which the SU(3) symmetry is preserved. The line extends from the SU(3) vertex to near the critical line of the first order shape/phase transition separating the spherical and prolate deformed phases. It lies within the Alhassid–Whelan arc of regularity, thus providing an explanation for its existence.

Analytic derivation of an approximate SU(3) symmetry inside the symmetry triangle of the interacting boson approximation model

Using the contraction of the SU(3) algebra to the algebra of the rigid rotator in the large boson number limit of the Interacting Boson Approximation (IBA) model, a line is found inside the symmetry triangle of the IBA, along which the SU(3) symmetry is preserved. The line extends from the SU(3) vertex to near the critical line of the first order shape/phase transition separating the spherical and prolate deformed phases, and lies within the Alhassid--Whelan arc of regularity, the unique valley of regularity connecting the SU(3) and U(5) vertices amidst chaotic regions. In addition to providing an explanation for the existence of the arc of regularity, the present line represents the first example of an analytically determined approximate symmetry in the interior of the symmetry triangle of the IBA. The method is applicable to algebraic models possessing subalgebras amenable to contraction. This condition is equivalent to algebras in which the equilibrium ground state (and its rotational band) become energetically isolated from intrinsic excitations, as typified by deformed solutions to the IBA for large numbers of valence nucleons.

Relation between the E(5) symmetry and the interacting boson model beyond the mean-field approximation

In a unified algebraic scheme, we investigate the relation between the E(5) symmetry and the interacting boson model beyond the mean-field level. The results indicate that the E(5) symmetry is actually in between the critical point of the U(5)-O(6) transition and the O(6) limit but it is fairly close to the former based on the phase diagram of the interacting boson model at the large boson number limit. In addition, an algebraic Hamiltonian of the E(5)-β 2n model is proposed.

Regularities and symmetries of subsets of collective0+states

The energies of subsets of excited ${0}^{+}$ states in geometric collective models are investigated and found to exhibit intriguing regularities. In models with an infinite square well potential, it is found that a single formula, dependent on only the number of dimensions, describes a subset of ${0}^{+}$ states. The same behavior of a subset of ${0}^{+}$ states is seen in the large boson number limit of the interacting boson approximation (IBA) model near the critical point of a first-order phase transition, in contrast to the fact that these ${0}^{+}$ state energies exhibit a harmonic behavior in all three limiting symmetries of the IBA. Finally, the observed regularities in ${0}^{+}$ energies are analyzed in terms of the underlying group theoretical framework of the different models.

Correspondence between the SU(3) ⊗ U(2) limit of IBF 2M and two quasi-particle Nilsson configurations

In the SU(3) ⊗ U(2) limit of the interacting boson two-fermion model (IBF 2M), appropriate for the description of rotational bands built on two quasi-particle excitations in heavy even-even deformed nuclei, the group chain and the basis states are given. In the large boson number limit, exact correspondence between the Bose-Fermi SU(3) representations in this limit and the two quasi-particle Nilsson configurations is established using SU(3)⊃SU(2) ⊗U(1) reduced Wigner coefficients.