FIG 3 - uploaded by David J Rowe
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Wave functions for the N = 60 ground state and first excited v = 0 state for α = 1 and 0.75. The dots give the coefficients of the states in a basis of d-boson number n. The continuous lines are those of the harmonic approximation.
Source publication
The apparent persistence of symmetry in the face of strong symmetry-breaking interactions is examined in a many-boson model. The model exhibits a transition between two phases associated with U(5) and O(6) symmetries, respectively, as the value of a control parameter progresses from 0 to 1. The remarkable fact is that, in spite of strong mixing of...
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Citations
... As the competing symmetry increases in strength this distortion becomes more important until it reaches breaking point and the system enters a transition phase from where a quasi-dynamical symmetry of the competing phase may emerge. Over the years David and collaborators investigated several models with competing symmetries [11][12][13] the properties of which can be in terms of quasi-dynamical symmetries. ...
David Rowe was a highly respected theoretical physicist who made seminal contributions that improved our understanding of the atomic nucleus, in particular of the collective behaviour of its constituent nucleons - results he often obtained with the use of sophisticated group-theoretical methods. He will also be remembered as the (co-)author of monographs on nuclear physics, written with the scientific rigour that was characteristic of his research.
... In [31], a shell-model calculation was conducted for 100 Mo and 100 Ru, starting from a realistic nucleon-nucleon potential and deriving the effective shell-model Hamiltonian and decay operators within many-body perturbation theory, with a focus on studying neutrinoless double-β decay. In [32], the multiquasiparticle triaxial projected shell model was used to investigate the band structures of [98][99][100][101][102][103][104][105][106] Ru isotopes, providing a consistent description. In [33], the odd-even and even-even isotopes of [95][96][97][98][99][100][101][102] Ru were studied using the nucleon pair approximation with a phenomenological pairing plus quadrupole interaction, yielding good agreement with experimental data. ...
... The g factors of Ru and Pd nuclei were calculated using the IBM-2 in [39]. The even-even [98][99][100][101][102][103][104][105][106][107][108][109][110] Ru isotopes were studied using the affine SU(1, 1) Lie algebra in [40]. In [41], the A = 100 region was described using the IBM-1 with a single Hamiltonian featuring constant parameters. ...
... In [44], a large set of isotopes were studied to identify good candidates for vibrationallike behavior, i.e., U(5) nuclei. Among others, 100 Mo and [98][99][100][101][102][103][104] Ru were identified as suitable candidates. It is important to note that this work is relatively old, and, with the present experimental knowledge, the conclusions may have evolved. ...
Background: Even-even isotopes of Mo (Z=42) and Ru (Z=44) are nuclei close to the subshell closure at Z=40, where shape coexistence plays a significant role. As a result, their spectroscopic properties are expected to resemble those of Sr (Z=38) and Zr (Z=40). Exploring the evolution of these properties as they move away from the subshell closure is of great interest.
... The way the energy levels are rearranged by SPTs has been clarified in terms of quasidynamical symmetries in a series of papers by Rowe et al. [465][466][467][468]. ...
The last decade has seen a rapid growth in our understanding of the microscopic origins of shape coexistence, assisted by the new data provided by the modern radioactive ion beam facilities built worldwide. Islands of the nuclear chart in which shape coexistence can occur have been identified, and the different microscopic particle–hole excitation mechanisms leading to neutron-induced or proton-induced shape coexistence have been clarified. The relation of shape coexistence to the islands of inversion, appearing in light nuclei, to the new spin-aligned phase appearing in N=Z nuclei, as well as to shape/phase transitions occurring in medium mass and heavy nuclei, has been understood. In the present review, these developments are considered within the shell-model and mean-field approaches, as well as by symmetry methods. In addition, based on systematics of data, as well as on symmetry considerations, quantitative rules are developed, predicting regions in which shape coexistence can appear, as a possible guide for further experimental efforts that can help in improving our understanding of the details of the nucleon–nucleon interaction, as well as of its modifications occurring far from stability.
... The way the energy levels get rearranged by SPTs has been clarified in terms of quasidynamical symmetries in a series of papers by Rowe et al. [654,658,659,756]. ...
The last decade has seen a rapid growth of our understanding of the microscopic origins of shape coexistence, assisted by the new data provided by the modern radioactive ion beam facilities built worldwide. Islands of the nuclear chart in which shape coexistence can occur have been identified, and the different microscopic particle-hole excitation mechanisms leading to neutron-induced or proton-induced shape coexistence have been clarified. The relation of shape coexistence to the islands of inversion, appearing in light nuclei, to the new spin-aligned phase appearing in N=Z nuclei, as well as to shape/phase transitions occurring in medium mass and heavy nuclei, has been understood. In the present review, these developments are considered within the shell model and mean field approaches, as well as by symmetry methods. In addition, based on systematics of data, as well as on symmetry considerations, quantitative rules are developed, predicting regions in which shape coexistence can appear, as a possible guide for further experimental efforts, which can help in improving our understanding of the details of the nucleon-nucleon interaction, as well as of its modifications occurring far from stability.
... Apart from the exact DSs, approximate DSs are suggested to exist in many-body systems too and yield some important symmetry-based concepts. For instance, the partial dynamical symmetries [4][5][6] and quasidynamical symmetries [7][8][9] have been found to occur in the IBM and other algebraic models. Approximate DSs are usually hidden behind a complicate parameter relation of the Hamiltonian of a given system. ...
Based on the boson realization of the Euclidean algebras, it is found that the E(n) dynamical symmetry (DS) may emerge at the critical point of the U(n)-SO(n+1) quantum phase transition. To justify this finding, we provide a detailed analysis of the transitional Hamiltonian in the U(n+1) vibron model in both quantal and classical ways. It is further shown that the low-lying structure of 82Kr can serve as an excellent empirical realization of the E(5) DS, which provides a specific example of the Euclidean DS in experiments.
... To display how the entanglement entropy changes as a 449 function of the control parameter in case j f = 3/2, we used 450 Eq. (55) or Eq. (57) which leads to Fig. 6. ...
... In other words, 465 we can see in Fig. 6 that the modification induced by the 466 presence of a fermion shift the entanglement entropy value 467 by S(m j f , L). 468 In the following, the possible occurrence of transitional 469 characteristics intermediate between spherical and γ -unstable 470 shapes in 102-110 Pd, 122-134 Xe, and 123-133 Xe isotopic chains 471 is investigated by using entanglement entropy. The theo-472 retical and experimental studies of energy spectra done in 473Refs.[49][50][51][52][53][54][55] show Pd and Xe isotopes have U(5) ↔ O(6) 474 transitional characteristics. ...
Background: One of the fundamental problems of quantum information science is quantum entanglement. Recently, quantum phase transition in nuclear systems is studied in connection with quantum entanglement. Purpose: In this paper, the use of entanglement entropy as a suitable signal for the study of quantum phase transition in the even-even and odd-A nuclei is investigated. The effect of the coupling of a single fermion to a boson core on entanglement entropy is studied. Method: By use of the affine SU (1, 1) Lie algebra and through the Schmidt decomposition in the framework of the interacting boson model (IBM) and interacting boson-fermion model (IBFM), entanglement entropy in the even-even and odd-A are obtained. The entanglement entropy is used for tracking and studying the shape phase transition in these nuclei. Results: The entanglement entropy in the IBM and the IBFM is calculated. The entanglement entropy values of the low-lying states of 122-134 Xe, 102-110 Pd, and 123-133 Xe were calculated and analyzed. It is found that entanglement entropy is a suitable order parameter to detect shape phase transition in nuclear systems. Conclusions: The obtained results indicate that the entropy of entanglement is sensitive to the shape-phase transition between spherical and γ-unstable regions in nuclei. The results show that entanglement entropy is a powerful tool for identifying shape phase transitions in nuclei. It is found that the coupling of the single fermion with angular momentum j to the even-even system does not change the geometry imposed by the boson core performing the transition and only the entanglement entropy values have been shifted by the addition of the odd particle with respect to the even case.
... 2, the participation ratio does not allow in all cases for an unambiguous assignment of a linear or bent character to a given state. The large mixing that occurs once the system is far enough from the dynamical symmetry limits hinders this assignment, a fact that can be explained using the quasidynamical symmetry concept [97]. ...
... This is specially relevant for systems with a low barrier to linearity, and for states that lie far from both limiting physical cases, the U(2) and SO(3) dynamical symmetries. This is in good accordance with the quasidynamical symmetry concept [97], that explains the high degree of mixing expected as one gets further from the dynamical symmetries, even for states retaining most of the characteristic features of a dynamical symmetry. Therefore, in such cases, the direct comparison of the PR values for the U(2) and SO(3) bases does not allow an unambiguous assignment of the eigenstate to a linear or bent character. ...
We characterize excited state quantum phase transitions in the two dimensional limit of the vibron model with the quantum fidelity susceptibility, comparing the obtained results with the information provided by the participation ratio. As an application, we locate the eigenstate closest to the barrier to linearity and determine the linear or bent character of the different overtones for particular bending modes of six molecular species. We perform a fit and use the optimized eigenvalues and eigenstates in three cases and make use of recently published results for the other three cases.
... 2, the participation ratio does not allow in all cases for an unambiguous assignment of a linear or bent character to a given state. The large mixing that occurs once the system is far enough from the dynamical symmetry limits hinders this assignment, a fact that can be explained using the quasidynamical symmetry concept [96]. ...
... This is specially relevant for systems with a low barrier to linearity, and for states that lie far from both limiting physical cases, the U (2) and SO(3) dynamical symmetries. This is in good accordance with the quasidynamical symmetry concept [96], that explains the high degree of mixing expected as one gets further from the dynamical symmetries, even for states retaining most of the characteristic features of a dynamical symmetry. Therefore, in such cases, the direct comparison of the PR values for the U (2) and SO(3) bases does not allow an unambiguous assignment of the eigenstate to a linear or bent character. ...
We characterize excited state quantum phase transitions in the two dimensional limit of the vibron model with the quantum fidelity susceptibility, comparing the obtained results with the information provided by the participation ratio. As an application, we perform fits using a four-body algebraic Hamiltonian to bending vibrational data for several molecular species and, using the optimized eigenvalues and eigenstates, we locate the eigenstate closest to the barrier to linearity and determine the linear or bent character of the different overtones.
... Each state is characterized by the SU BF (3) quantum numbers (λ, µ), which are strictly valid only at x = 1, and the O BF (3) quantum number, L. As seen in this figure, the general behavior of the energy levels is rather smooth close to the U BF (5) and SU BF (3) dynamical symmetries and changes rapidly in the neighborhood of the critical point, indicated by a dashed vertical line. This flat behavior close to the dynamical symmetries has been observed before, also in even systems, and called quasi-dynamical symmetry by Rowe [44]. ...
The quantum phase transition studies we have done during the last few years for odd-even systems are reviewed. The focus is on the quantum shape phase transition in Bose-Fermi systems. They are studied within the Interacting Boson-Fermion Model (IBFM). The geometry is included in this model by using the intrinsic frame formalism based on the concept of coherent states. First, the critical point symmetries E(5/4) and E(5/12) are summarized. E(5/4) describes the case of a single j=3/2 particle coupled to a bosonic core that undergoes a transition from spherical to γ-unstable. E(5/12) is an extension of E(5/4) that describes the multi-j case (j=1/2,3/2,5/2) along the same transitional path. Both, E(5/4) and E(5/12), are formulated in a geometrical context using the Bohr Hamiltonian. Similar situations can be studied within the IBFM considering the transitional path from UBF(5) to OBF(6). Such studies are also presented. No critical points have been proposed for other paths in odd-even systems as, for instance, the transition from spherical to axially deformed shapes. However, the study of such shape phase transition can be done easily within the IBFM considering the path from UBF(5) (spherical) to SUBF(3) (axial deformed). Thus, in a second part, this study is presented for the multi-j case. Energy levels and potential energy surfaces obtained within the intrinsic frame formalism of the IBFM Hamiltonian are discussed. Finally, our recent works within the IBFM for a single-j fermion coupled to a bosonic core that performs different shape phase transitional paths are reviewed. All significant paths in the model space are studied: from spherical to γ-unstable shape, from spherical to axially deformed (prolate and oblate) shapes, and from prolate to oblate shape passing through the γ-unstable shape. The aim of these applications is to understand the effect of the coupled fermion on the core when moving along a given transitional path and how the coupled fermion modifies the bosonic core around the critical points.
... The intrinsic coherent state allows the evaluation of potential energy surface (PES) given as a function of the intrinsic shape variables (the deformation parameters β and γ ) [20,21]. The IBM reveals rich features of their shape phase transitions [22][23][24][25]. ...
