What can we learn from the Interacting Boson Model in the limit of large boson numbers?
Dennis Bonatsos, E. A. McCutchan, R. F. Casten
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8 pages, 5 postscript figures, LaTeX. To appear in the Proceedings of the 17th Symposium of the Hellenic Nuclear Physics Society (Ioannina, Greece, May 30-31, 2008)
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.
A microscopic interpretation of the s and d bosons of the Interacting Boson Model is being suggested: the s, d bosons can be interpreted as symmetric pairs of harmonic oscillator quanta of the valence nuclear shell. Within this interpretation, the SU(3) limit of the Interacting Boson Model results to the Elliott SU(3) symmetry.
We introduce a class of exactly solvable boson models. We give explicit analytic expressions for energy eigenvalues and eigenvectors for an sd-boson Hamiltonian, which is related to the SO(6) chain of the Interacting Boson Model Hamiltonian.
The exact solution of the boson pairing hamiltonian given by Richardson in the sixties is used to study the phenomena of level crossings and quantum phase transitions in the integrable regions of the sd and sdg interacting boson models.
Interacting boson models provide an elegant and powerful method to describe collective excitations of complex systems by introducing a set of effective degrees of freedom. We review the interacting boson model of nuclear structure and discuss a recent extension to the nucleon and its excited states.
In this contribution the symmetry properties of the interacting boson model of Arima and Iachello are reviewed. While the concept of a dynamical symmetry is by now a familiar one, this is not necessarily so for the extended notions of partial dynamical symmetry and quasi dynamical symmetry, which can be beautifully illustrated in the context of the interacting boson model. The main conclusion of the analysis is that dynamical symmetries are scarce while their partial and quasi extensions are ubiquitous.
We study the phase diagram of the proton--neutron interacting boson model (IBM--2) with special emphasis on the phase transitions leading to triaxial phases. The existence of a new critical point between spherical and triaxial shapes is reported.
We introduce an exactly solvable model for interacting bosons that extend up to high spin and interact through a repulsive pairing force. The model exhibits a phase transition to a state with almost complete $sd$ dominance. The repulsive pairing interaction that underlies the model has a natural microscopic origin in the Pauli exclusion principle between contituent nucleons. As such, repulsive pairing between bosons seems to provide a new mechanism for the enhancement of $sd$ dominance, giving further support for the validity of the $sd$ Interacting Boson Model.
The extended interacting boson model with s-, p-, d-, f- and g-bosons being included (spdfg IBM) are investigated. The algebraic structure including the generators, the Casimir operators of the groups at the SU(5) dynamical symmetry and the branching rules of the irreducible representation reductions along the group chain are obtained. The typical energy spectrum of the Symmetry is given.
We intend to provide a consistent description of the even-even Hg isotopes, 172-200Hg, using the interacting boson model including configuration mixing. We pay special attention to the description of the shape of the nuclei and to its connection with the shape coexistence phenomenon.
The relation of the recently proposed E(5) critical point symmetry with the interacting boson model is investigated. The large-N limit of the interacting boson model at the critical point in the transition from U(5) to O(6) is obtained by solving the Richardson equations. It is shown explicitly that this algebraic calculation leads to the same results as the solution of the Bohr differential equation with a $β^4$ potential.
Exact numerical results of the interacting boson model Hamiltonian along the integrable line from U(5) to O(6) are obtained by diagonalization within boson seniority subspaces. The matrix Hamiltonian reduces to a block tridiagonal form which can be diagonalized for large number of bosons. We present results for the low energy spectrum and the transition probabilities for systems up to 10000 bosons, which confirm that at the critical point the system is equally well described by the Bohr Hamiltonian with a $β^4$ potential.
We introduce the basic concepts of catastrophe theory needed to derive analytically the phase diagram of the proton-neutron interacting boson model (IBM-2). Previous studies [1,2,3] were based on numerical solutions. We here explain the whole IBM-2 phase diagram including the precise order of the phase transitions in terms of the cusp catastrophe.
We discuss the symmetry of the parameter space of the interacting boson model (IBM). It is shown that for any set of the IBM Hamiltonian parameters (with the only exception of the U(5) dynamical symmetry limit) one can always find another set that generates the equivalent spectrum. We discuss the origin of the symmetry and its relevance for physical applications.
We describe a class of exactly-solvable models of interacting bosons based on the algebra SO(3,2). Each copy of the algebra represents a system of neutron and proton bosons in a given bosonic level interacting via a pairing interaction. The model that includes s and d bosons is a specific realization of the IBM2, restricted to the transition regime between vibrational and gamma-soft nuclei. By including additional copies of the algebra, we can generate proton-neutron boson models involving other boson degrees of freedom, while still maintaining exact solvability. In each of these models, we ca…
The interacting boson model (IBM) of Arima and Iachello is applied to calculate nuclear charge radii and electric monopole transitions of even-even nuclei in the rare-earth region. Consistent operators are used for the two observables. A relation between summed M1 strength and $ρ({\rm E0})$ values is pointed out.
Double-$γ$ vibrations in deformed nuclei are studied in the context of the interacting boson model with special reference to their anharmonic character. It is shown that large anharmonicities can be obtained with interactions that are (at least) of three-body nature between the bosons. As an example the $γ$ vibrations of the nucleus $^{166}_{\phantom{0}68}$Er$^{\phantom{00}}_{98}$ are studied in detail.
The interacting boson model is extended to describe the spectroscopy of superdeformed bands. Microscopic structure of the model in the second minimum is discussed and superdeformed bosons are introduced as the new building blocks. Solutions of a quadrupole Hamiltonian are implemented through the $1/N$ expansion method. Effects of the quadrupole parameters on dynamic moment of inertia and electric quadrupole transition rates are discussed and the results are used in a description of superdeformed bands in the Hg-Pb and Gd-Dy regions.
The boson-conserving one-nucleon transfer operator in the interacting boson model (IBA) is reanalyzed. Extra terms are added to the usual form used for that operator. These new terms change generalized seniority by one unit, as the ones considered up to now. The results obtained using the new form for the transfer operator are compared with those obtained with the traditional form in a simple case involving the pseudo-spin Bose-Fermi symmetry $U^{B}(6) \otimes U^F(12)$ in its $U^{BF}(5) \otimes U^F(2)$ limit. Sizeable differences are found. These results are of relevance in the study of transf…
The self-consistent cranking method is tested by comparing the cranking calculations in the interacting boson model with the exact results obtained from the SU(3) and O(6) dynamical symmetries and from numerical diagonalization. The method is used to study the spin dependence of shape variables in the $sd$ and $sdg$ boson models. When realistic sets of parameters are used, both models lead to similar results: axial shape is retained with increasing cranking frequency while fluctuations in the shape variable $γ$ are slightly reduced.
The signature splitting of the $γ$-vibrational band of several Ru, Pd, Xe, Ba, Os and Pt isotopes is analyzed in the framework of the interacting boson model (IBM). The nuclei studied are close to the $γ$-unstable SO(6) limit of the IBM and have well-known $γ$ bands. It is shown that in most nuclei the signature splitting is better reproduced by the inclusion of a three-body interaction between the $d$ bosons. In none of the nuclei evidence for a stable, triaxial ground-state shape is found.
The case of U(5)--$\hat{Q}(χ)\cdot\hat{Q}(χ)$ mixing in the configuration-mixed Interacting Boson Model is studied in its mean-field approximation. Phase diagrams with analytical and numerical solutions are constructed and discussed. Indications for first-order and second-order shape phase transitions can be obtained from binding energies and from critical exponents, respectively.
This work introduces into the Interacting Boson Model, which was created in 1974 by F. Iachello and A. Arima and then extended by numerous papers. Many-body configurations with s- and d-boson states are described and creation- and annihilation-operators for bosons are introduced. States with defined total angular momentum are dealt with and the Hamilton operator of the IBM1-model is expressed in terms of Casimir operators. Level energies and electromagnetic transition probabilities are compared with measured data. A short introduction to Lie algebras and their application to the IBM1-model are…
Analysis of the recently obtained experimental data for collective states of ^{160}Dy is presented. The Interacting Vector Boson Model (IVBM) was applied for the classification of low lying states with positive parity 0^{+},2^{+},4^{+},6^{+}and for description of rotational ground, S, γ and two octupole bands. The energies of the bands are reproduced with high accuracy using only one set of the model parameters for all bands.
An intrinsic-state formalism for IBM-4 is presented. A basis of deformed bosons is introduced which allows the construction of a general trial wave function which has Wigner's spin-isospin SU(4) symmetry as a particular limit. Intrinsic-state calculations are compared with exact ones showing good agreement.
A method of deriving the Hamiltonian of the interacting boson model, that is based on the microscopic framework of the nuclear energy density functional, is presented. The constrained self-consistent mean-field calculation with a given energy density functional provides potential energy surface within the relevant collective coordinates, which is subsequently mapped onto the expectation value of the interacting-boson Hamiltonian in the boson condensate state. This procedure completely determines the strength parameters of the IBM, and the diagonalization of the mapped Hamiltonian yields excita…
We introduce a method for isospin restoration in the calculation of nuclear matrix elements (NME) for $0νββ$ and $2νββ$ decay within the framework of interacting boson model (IBM-2). With this method, we calculate NME for all processes of interest in $0νβ^-β^-$, $2νβ^-β^-$, and in $0νβ^+β^+$, $0νβ^+ EC^+$, $R0νECEC$, $2νβ^+β^+$, $2νβ^+EC$, and $2νECEC$. With this method, the Fermi (F) matrix elements for $2νββ$ vanish, and those for $0νββ$ are considerably reduced.
The Interacting Boson Model with broken-pairs has been extended to include mixed proton-neutron configurations in the fermion model space. The extended version of the model has been used to describe high-spin bands in the transitional nucleus $^{136}$Nd. Model calculations reproduce ten bands of positive and negative parity states, including the two dipole high-spin structures based on the $(πh_{11/2})^2$ $(νh_{11/2})^2$ configuration.
It is shown that the proton-neutron interacting boson model (pnIBM) admits new symmetry limits with O(12) algebra which break F-spin but preserves the quantum number M_F. The generators of O(12) are derived and the quantum number `v' of O(12) for a given boson number N is determined by identifying the corresponding quasi-spin algebra. The O(12) algebra generates two symmetry schemes and for both of them, complete classification of the basis states and typical spectra are given. With the O(12) algebra identified, complete classification of pnIBM symmetry limits with good M_F is established.
In terms of the Interacting Boson Model, shape invariants for the ground state, formed by quadrupole moments up to sixth order, are studied in the dynamical symmetry limits and, for the first time, over the whole structural range of the IBM-1. The results are related to the effective deformation parameters and their fluctuations in the geometrical model. New signatures that can distinguish vibrator and gamma-soft rotor structures, and one that is related to shape coexistence, are identified.
We exploit the rich algebraic structure of the interacting boson model to explain the notion of partial dynamical symmetry (PDS), and present a procedure for constructing Hamiltonians with this property. We demonstrate the relevance of PDS to various topics in nuclear spectroscopy, including K-band splitting, odd-even staggering in the gamma-band and anharmonicity of excited vibrational bands. Special emphasis in this construction is paid to the role of higher-order terms.
In the framework of the interacting boson model the three transitional regions (rotational-vibrational, rotational-$γ$-unstable and, vibrational-$γ$-unstable transitions) are reanalyzed. A new kind of plot is presented for studying phase transitions in finite systems such as atomic nuclei. The importance of analyzing binding energies and not only energy spectra and electromagnetic transitions, describing transitional regions is emphasized. We finally discuss a number of realistic examples.
A general study of excitations up to two-phonon states is carried out using the intrinsic-state formalism of the Interacting Boson Model (IBM). Spectra and transitions for the different dynamical symmetries are analyzed and the correspondence with states in the laboratory frame is established. The influence of multi-phonon states is discussed. The approach is useful in problems where the complexity of the IBM spectrum renders the analysis in the laboratory frame difficult.
The particle-hole symmetry (equivalence) of the full shell-model Hilbert space is straightforward and routinely used in practical calculations. In this work we show that this symmetry is preserved in the subspace truncated at a certain generalized seniority, and give the explicit transformation between the states in the two types (particle and hole) of representations. Based on the results, we study the particle-hole symmetry in popular theories that could be regarded as further truncations on top of the generalized seniority, including the microscopic interacting boson (fermion) model, the nu…
Spectrum generating algebras are used in various fields of physics as models to determine quantum structure, including energy levels and transition strengths. The advantage of such models is that their group structure allows an extensive understanding of the system being studied. In addition they possess a simple classical limit, at least for bosonic systems. In this paper we discuss an algebraic model of nuclear structure, the Interacting Boson Model (IBM), and show that in one limit its group structure is particularly simple. For zero angular momentum there is an effective lower dimensional …
We investigate the evolution of quantal spectra and the corresponding wave functions along the [O(6)-U(5)]$\supset$O(5) transition of the interacting boson model. The model is integrable in this regime and its ground state passes through a second-order structural phase transition. We show that the whole spectrum as a function of the Hamiltonian control parameter, as well as structures of all excited states, exhibit rather organized and correlated behaviors, that provide deeper insight into the nature of this transitional path.
The negative and positive parity low-spin states of the even-odd Tungsten isotopes, 183,185,187W are studied in the frame work of the Interacting Boson-Fermion Approximation (IBFA) model. The fermion that is coupled to the system of bosons is taken to be in the negative parity 2f_7|2, 2f_5\2, 3p_3\2, 3p_1\2 and in the positive parity 1i_13\2 single-particle orbits. The calculated energies of low-spin energy levels of the odd isotopes are found to agree well with the experimental data. Also B(E2) values and spectroscopic factors for single-neutron transfer are calculated and found to be in good…
Bosonic degrees of freedom and their emergence as a part of complex quantum many-body dynamics, symmetries, collective behavior, clustering and phase transitions play an important role in modern studies of quantum systems. In this work we present a systematic study of many-boson systems governed by random interactions. Our findings show that ground states of randomly interacting bosons are not random, being dominated by a few collective configurations containing condensates of clusters.
The phase diagram of two-level boson Hamiltonians, including the Interacting Boson Model (IBM), is studied beyond the standard mean field approximation using the Holstein-Primakoff mapping. The limitations of the usual intrinsic state (mean field) formalism concerning finite-size effects are pointed out. The analytic results are compared to numerics obtained from exact diagonalizations. Excitation energies and occupation numbers are studied in different model space regions (Casten triangle for IBM) and especially at the critical points.
Random interactions are used to investigate to what extent the low-lying behavior of even-even nuclei depend on particular nucleon-nucleon interactions. The surprising results that were obtained for the interacting boson model, i.e. the dominance of ground states with L=0 and the occurrence of both vibrational and rotational structure, are interpreted and explained in terms of a mean-field analysis.
We investigate the origin of the regular features observed in numerical studies of the interacting boson model with random interactions, in particular the dominance of L=0 ground states and the occurrence of vibrational and rotational band structures. It is shown that all of these properties can be interpreted and explained in terms of a Hartree-Bose mean-field analysis, in which different regions of the parameter space are associated with geometric shapes. The same conclusions hold for the vibron model.
The evolution of the total energy surface and the nuclear shape in the isotopic chain $^{172-194}$Pt are studied in the framework of the interacting boson model, including configuration mixing. The results are compared with a self-consistent Hartree-Fock-Bogoliubov calculation using the Gogny-D1S interaction and a good agreement between both approaches shows up. The evolution of the deformation parameters points towards the presence of two different coexisting configurations in the region 176 $\leq$ A $\leq$ 186.
We investigate the finite-size scaling exponents for the critical point at the shape phase transition from U(5) (spherical) to O(6) (deformed $γ$-unstable) dynamical symmetries of the Interacting Boson Model, making use of the Holstein-Primakoff boson expansion and the continuous unitary transformation technique. We compute exactly the leading order correction to the ground state energy, the gap, the expectation value of the $d$-boson number in the ground state and the $E2$ transition probability from the ground state to the first excited state, and determine the corresponding finite-size scal…
We study the quantum phase transition mechanisms that arise in the Interacting Boson Model. We show that the second-order nature of the phase transition from U(5) to O(6) may be attributed to quantum integrability, whereas all the first-order phase transitions of the model are due to level repulsion with one singular point of level crossing. We propose a model Hamiltonian with a true first-order phase transition for finite systems due to level crossings.
A general time-dependent projection technique is applied to the study of the dynamics of quantum correlations in a system consisting of interacting fermionic and bosonic subsystems, described by the Jaynes-Cummings Hamiltonian. The amplitude modulation of the Rabi oscillations which occur for a strong, coherent initial bosonic field is obtained from the spin intrinsic depolarization resulting from collisional corrections to the mean-field approximation.
