Quasi-dynamical Symmetry Research Articles

Partial dynamical symmetry versus quasi dynamical symmetry examination within a quantum chaos analyses of spectral data for even\u2013even nuclei

Statistical analyses of the spectral distributions of rotational bands in 51 deformed prolate even–even nuclei in the 152 ≤ A ≤ 250 mass region R_{{4_{1}^{ + } /2_{1}^{ + } }} ge 3.00 are examined in terms of nearest neighbor spacing distributions. Specifically, the focus is on data for 0+, 2+, and 4+ energy levels of the ground, gamma, and beta bands. The chaotic behavior of the gamma band, especially the position of the 2_{gamma }^{ + } band-head compared to other levels and bands, is clear. The levels are analyzed within the framework of two models, namely, a SU(3)-partial dynamical symmetry Hamiltonian and a SU(3) two-coupled quasi-dynamical symmetry Hamiltonian, with results that are further analyzed using random matrix theory. The partial and quasi dynamics both yield outcomes that are in reasonable agreement with the known experimental results. However, due to the degeneracy of the beta and gamma bands within the simplest SU(3) picture, the theory cannot be used to describe the fluctuation properties of excited bands. By changing relative weights of the different terms in the partial and quasi dynamical Hamiltonians, results are obtained that show more GOE-like statistics in the partial dynamical formalism as the strength of the pairing term is increased. Also, in the quasi-dynamical symmetry limit, more correlations are found because of the stronger couplings.

Unmixing Symmetries.

The low-lying spectra of atomic nuclei display diverse behaviors, for example, rotational bands, which can be described phenomenologically by simple symmetry groups such as spatial SU(3). This leads to the idea of dynamical symmetry, where the Hamiltonian commutes with the Casimir operator(s) of a group, and is block diagonal in subspaces defined by the group's irreducible representations or irreps. Detailed microscopic calculations, however, show these symmetries are in fact often strongly mixed and the wave function fragmented across many irreps. More commonly, the fragmentation across members of a band are similar, which is called a quasidynamical symmetry. In this Letter I explicitly, albeit numerically, construct unitary transformations from a quasidynamical symmetry to a dynamical symmetry, adapting the similarity renormalization group (SRG) in order to transform away the symmetry-mixing parts of the Hamiltonian. The standard SRG produces unsatisfactory results, forcing the induced dynamical symmetry to be dominated by high-weight irreps irrespective of the original decomposition. Using spectral distribution theory to rederive and diagnose standard SRG, I introduce a new form of SRG. The new SRG transforms a quasidynamical symmetry to a dynamical symmetry, that is, unmixes the mixed symmetries, with intuitively more appealing results.

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.

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

Shape isomers of light nuclei from the stability and consistency of the SU(3) symmetry

We investigate the stability and self-consistency of the SU(3) symmetry and quadrupole deformation in light nuclei in terms of the Nilsson model and quasi-dynamical symmetry. It turns out that SU(3) is a remarkably good symmetry for commensurable major axes, similarly to the finding of the simple harmonic oscillator interaction. The method serves as an alternative to the well-known energy-minimum calculation for the determination of the shape isomers. In case of the 16O, 20Ne, and 24Mg nuclei the results of the two different procedures are in good agreement with each other.

Microscopic structure of the low-lying negative-parity states inSm154

The proton-neutron symplectic model with Sp(12,$R$) dynamical algebra is applied to the description of the microscopic structure of the low-lying negative-parity states of the ${K}^{\ensuremath{\pi}}={0}_{1}^{\ensuremath{-}}$ and ${K}^{\ensuremath{\pi}}={1}_{1}^{\ensuremath{-}}$ bands in $^{154}\mathrm{Sm}$ without the introduction of additional degrees of freedom that are inherent to other approaches to odd-parity nuclear states. For this purpose, the model Hamiltonian is diagonalized in a U(6)-coupled basis, restricted to state space spanned by the fully symmetric U(6) irreps of the lowest odd irreducible representation of Sp(12,$R$). In this way, the positive- and negative-parity collective bands are treated on equal footing within the framework of the microscopic symplectic-based shell-model scheme. A good description of the energy levels of the two bands under consideration, as well as the reproduction of some energy splitting quantities which are usually introduced in the literature as a measure of the octupole correlations, is obtained. The microscopic structure of low-lying collective states with negative-parity in $^{154}\mathrm{Sm}$ shows that practically there are no admixtures from the higher shells and hence the presence of a very good U(6) dynamical symmetry. Additionally, the structure of the collective states under consideration shows also the presence of a good SU(3) quasidynamical symmetry.

U(6) quasi-dynamical symmetry in 238U

The proton-neutron symplectic model with Sp(12,R) dynamical algebra is applied for the simultaneous description of the microscopic structure of the low-lying positive-parity states of the lowest ground, β and γ bands in U238. For this purpose, the model Hamiltonian is diagonalized in a U(6)-coupled basis, restricted to the state space spanned by the fully symmetric U(6) irreps. A good description of the energy levels of the three bands under consideration, as well as the enhanced intraband B(E2) transition strengths between the states of the ground band is obtained without the use of an effective charge. The results obtained show small admixtures from the higher major shells and a highly coherent mixing of different U(6) irreps which is manifested by the presence of a good U(6) quasi-dynamical symmetry in the microscopic structure of the collective states under consideration.

Symmetries of the interacting boson model

This contribution reviews the symmetry properties of the interacting boson model of Arima and Iachello. 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.

U(6) dynamical and quasi-dynamical symmetry in strongly deformed heavy nuclei

The low-lying collective states of the ground, β and γ bands in154Sm and238U are investigated within the framework of the microscopic proton-neutron symplectic model (PNSM). For this purpose, the model Hamiltonian is diagonalized in a U(6)-coupled basis, restricted to the symplectic state space spanned by the fully symmetric U(6) vectors. A good description of the energy levels of the three bands under consideration, as well as the intraband B(E2) transition strengths between the states of the ground band is obtained for the two nuclei without the use of an effective charge. The calculations show that when the collective quadrupole dynamics is covered already by the symplectic bandhead structure, as in the case of154Sm, the results show the presence of a very good U(6) dynamical symmetry. In the case of238U, when we have an observed enhancement of the intraband B(E2) transition strengths, then the results show small admixtures from the higher major shells and a highly coherent mixing of different irreps which is manifested by the presence of a good U(6) quasi-dynamical symmetry in the microscopic structure of the collective states under consideration.

Quasidynamical symmetries in the backbending of chromium isotopes

Background: Symmetries are a powerful way to characterize nuclear wave functions. A true dynamical symmetry, where the Hamiltonian is block-diagonal in subspaces defined by the group, is rare. More likely is a quasidynamical symmetry: states with different quantum numbers (i.e. angular momentum) nonetheless sharing similar group-theoretical decompositions. Purpose: We use group-theoretical decomposition to investigate backbending, an abrupt change in the moment of inertia along the yrast line, in $^{48,49,50}$Cr: prior mean-field calculations of these nuclides suggest a change from strongly prolate to more spherical configurations as one crosses the backbending and increases in angular momentum. Methods: We decompose configuration-interaction shell-model wavefunctions using the SU(2) groups $L$ (total orbital angular momentum) and $S$ (total spin), and the groups SU(3) and SU(4). We do not need a special basis but only matrix elements of Casimir operators, applied with a modified Lanczos algorithm. Results: We find quasidynamical symmetries, albeit often of a different character above and below the backbending, for each group. While the strongest evolution was in SU(3), the decompositions did not suggest a decrease in deformation. We point out with a simple example that mean-field and SU(3) configurations may give very different pictures of deformation. Conclusions: Persistent quasidynamical symmetries for several groups allow us to identify the members of a band and to characterize how they evolve with increasing angular momentum, especially before and after backbending.

Simultaneous occurrence of distinct symmetries in nuclei

We show that distinct emergent symmetries, such as partial dynamical symmetry and quasi dynamical symmetry, can occur simultaneously in the same or different eigenstates of the Hamiltonian. Implications for nuclear spectroscopy in the rare-earth region and for first-order quantum phase transitions between spherical and deformed shapes, are considered.

Emergent spectral degeneracies in the quantum phase transitions in the interacting boson model

The spectral degeneracies emerging in the critical region of the interacting boson model are systematically investigated. The results indicate that the level crossings in the transitions from the U(5) to the SU(3) quasidynamical symmetry can induce the approximate spectral degeneracies along the first-order transitional line for large boson number $N$. The remnants of these types of degeneracies in finite-$N$ systems have been preliminarily identified from the low-lying spectra of $^{150}\mathrm{Nd}$. In addition, the similar approximate degeneracies emerging in the phase transition between the U(5) and SO(6) quasidynamical symmetries are parallel emphasized and those in other critical models are also mentioned.

Erratum: Linking partial and quasi dynamical symmetries in rotational nuclei [Phys. Rev. C89, 041302(R) (2014)

Received 14 September 2015DOI:https://doi.org/10.1103/PhysRevC.92.039902©2015 American Physical Society

Partial and quasi dynamical symmetries in quantum many-body systems

We introduce the notions of partial dynamical symmetry (PDS) and quasi dynamical symmetry (QDS) and demonstrate their relevance to nuclear spectroscopy, to quantum phase transitions and to mixed systems with regularity and chaos. The analysis serves to highlight the potential role of PDS and QDS towards understanding the emergent “simplicity out of complexity” exhibited by complex many-body systems.

First-order quantum phase transitions: Test ground for emergent chaoticity, regularity and persisting symmetries

We present a comprehensive analysis of the emerging order and chaos and enduring symmetries, accompanying a generic (high-barrier) first-order quantum phase transition (QPT). The interacting boson model Hamiltonian employed, describes a QPT between spherical and deformed shapes, associated with its U(5) and SU(3) dynamical symmetry limits. A classical analysis of the intrinsic dynamics reveals a rich but simply-divided phase space structure with a Hénon–Heiles type of chaotic dynamics ascribed to the spherical minimum and a robustly regular dynamics ascribed to the deformed minimum. The simple pattern of mixed but well-separated dynamics persists in the coexistence region and traces the crossing of the two minima in the Landau potential. A quantum analysis discloses a number of regular low-energy U(5)-like multiplets in the spherical region, and regular SU(3)-like rotational bands extending to high energies and angular momenta, in the deformed region. These two kinds of regular subsets of states retain their identity amidst a complicated environment of other states and both occur in the coexistence region. A symmetry analysis of their wave functions shows that they are associated with partial U(5) dynamical symmetry (PDS) and SU(3) quasi-dynamical symmetry (QDS), respectively. The pattern of mixed but well-separated dynamics and the PDS or QDS characterization of the remaining regularity, appear to be robust throughout the QPT. Effects of kinetic collective rotational terms, which may disrupt this simple pattern, are considered.

Partial and Quasi Dynamical Symmetries in Nuclei

One of the interesting aspects in the study of atomic nuclei is the strikingly regular behavior many display in spite of being complex quantum-mechanical systems, prompting the universal question of how regularity emerges out of complexity. It is often conjectured that symmetries play a pivotal role in our understanding of this emerging simplicity. But most symmetries are likely to be broken, partial or both. Under such more realistic conditions, does the concept of symmetry still provide a basis for our understanding of regularity? I suggest that this requires the enlarged concepts of partial and quasi dynamical symmetry.

Linking partial and quasi dynamical symmetries in rotational nuclei

Background: Quasi dynamical symmetries (QDS) and partial dynamical symmetries (PDS) play an important role in the understanding of complex systems. Up to now these symmetry concepts have been considered to be unrelated. Purpose: Establish a link between PDS and QDS and find an emperical manifestation. Methods: Quantum number fluctuations and the intrinsic state formalism are used within the framework of the interacting boson model of nuclei. Results: A previously unrecognized region of the parameter space of the interacting boson model that has both O(6) PDS (purity) and SU(3) QDS (coherence) in the ground band is established. Many rare-earth nuclei approximately satisfying both symmetry requirements are identified. Conclusions: PDS are more abundant than previously recognized and can lead to a QDS of an incompatible symmetry.

SU(3) Quasidynamical Symmetry Underlying the Alhassid-Whelan Arc of Regularity

The first example of an empirically manifested quasidynamical symmetry trajectory in the interior of the symmetry triangle of the interacting boson approximation model is identified for large boson numbers. Along this curve, extending from SU(3) to near the critical line of the first order phase transition, spectra exhibit nearly the same degeneracies that characterize the low energy levels of SU(3). This trajectory also lies close to the Alhassid-Whelan arc of regularity, the unique interior region of regular behavior connecting the SU(3) and U(5) vertices, thus offering a possible symmetry-based interpretation of that narrow zone of regularity amidst regions of more chaotic spectra.

Phases of clusterized nuclei

After reviewing some basic features of the temperature-governed phase-transitions in macroscopic systems and in atomic nuclei we consider non-thermal phase-transitions of nuclear structure in the example of cluster states. The vibron model, as the algebraic model of the relative motion, is considered from this aspect. The extension by the coupling to the internal degrees of freedom of the clusters is discussed. Phenomenological and semimicroscopical algebraic cluster models with identical interactions are applied to binary cluster systems of closed and non-closed shell clusters. A phase-diagram for the binary cluster systems is proposed, as well as its relation to that of the shell-model. The role of the quasi-dynamical symmetry, as a possible signature of the phase of a finite quantum system, is supported.

Transition fromγ-rigid toγ-soft dynamics in the interacting boson model: Quasicriticality and quasidynamical symmetry

We study the transition from the {gamma}-rigid to {gamma}-soft collective nuclear dynamics across the Casten triangle of the interacting boson model using mean-field techniques and confront the description with the exact diagonalization. We inspect the domain of validity of the SU(3) quasidynamical symmetry inside the Casten triangle and reveal critical behavior within the low-lying excited spectrum due to a degeneracy of {beta} and {gamma} vibrations.