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Low-lying states of $^{94}$Zr were investigated via low-energy multi-step Coulomb excitation. From the measured $\gamma$-ray yields, 13 reduced transition probabilities between low-spin states were determined, together with the spectroscopic quadrupole moments of the $2_{1,2}^+$ states. Based on this information, for the first time in the Zr isotopic chain, the shapes of the $0_{1,2}^+$ states including their deformation softness were inferred in a model-independent way using the quadrupole sum rules approach. The ground state of $^{94}$Zr possesses a rather diffuse shape associated with a spherical configuration, while the $0_2^+$ state is oblate and more strongly deformed. The observed features of shape coexistence in $^{94}$Zr are in agreement with Monte-Carlo shell-model predictions, and the present results are vital to refine the IBM-CM description of the Zr isotopes around $A\approx 100$ in terms of an intertwined quantum phase transition., Comment: 6 pages, 3 figures

Nuclei in the $A\approx100$ region exhibit intricate shape-evolution and configuration crossing signatures. Exploring both even-even and their adjacent odd-mass nuclei gives further insight on the emergence of deformation and shape-phase transitions. We employ the algebraic frameworks of the interacting boson model with configuration mixing and the new interacting boson-fermion model with configuration mixing in order to investigate the even-even zirconium with neutron number 52-70 ($^{40}$Zr) and odd-mass niobium ($_{41}$Nb) isotopes with 52-62. We compare between the evolution in energy levels, configuration and symmetry content of the wave functions, two neutron separation energies and $E2$ transition rates. The comparisons between the two chains of isotopes denote the occurrence of intertwined quantum phase transitions (IQPTs) in both chains. Such a situation occurs when two configurations, normal and intruder, cross through the critical point of a Type II quantum phase transition (QPT), and the intruder configuration undergoes on its own a Type I shape-evolution QPT from a spherical shape (weak coupling scenario) to axially deformed rotor (strong coupling scenario) in the Zr (Nb) isotopes., Comment: 13 pages, 6 figures

The empirical spectra and $E2$ decay rates in $^{110,112,114,116}$Cd are shown to be consistent with a vibrational interpretation for low-lying normal states, coexisting with a single deformed $\gamma$-soft band of intruder states. The observed deviations from this paradigm show up in particular non-yrast states, which are properly described by a Hamiltonian with U(5) partial dynamical symmetry. The latter is characterized by a good (broken) symmetry in most (in selected) normal states, weakly coupled to intruder states., Comment: 5 pages, 3 figures, 2 tables, accepted for publication as a Letter in Phys. Rev. C

Nuclei in the $Z\!\approx\!40,N\!\approx\!60$ region have one of the most complicated structural evolution across the nuclear chart, with coexisting shapes arising from different mixed configurations. In such a region, it is difficult to investigate odd-mass nuclei. In this paper a new algebraic framework is introduced, the interacting boson-fermion model with configuration mixing. Using this framework, with a boson core and a proton in the $1f_{5/2},2p_{3/2},2p_{1/2},1g_{9/2}$ orbits, a calculation is carried out to understand the structural evolution of the odd-mass niobium isotopes $(Z=41)$ with neutron number 52-62. The calculated results are compared to energy levels, two neutron separation energies, $E2$ and $M1$ transition rates, and to quadrupole and magnetic moments. The detailed analysis discloses the effects of an abrupt crossing of states between normal and intruder configurations (Type II QPT), which is accompanied by a gradual evolution from spherical- to deformed-core shapes within the intruder configuration (Type I QPT), where both types of QPTs occur around the critical point of neutron number 60. The identification of both types of QPTs in the same chain of isotopes provides an empirical manifestation of intertwined quantum phase transitions (IQPTs) in odd-mass nuclei and the relevance of IQPTs to the niobium chain., Comment: 20 pages, 24 figures, 2 tables. arXiv admin note: text overlap with arXiv:2210.00441

The algebraic framework of the interacting boson model with configuration mixing is employed to demonstrate the occurrence of intertwined quantum phase transitions (IQPTs) in the $_{40}$Zr isotopes with neutron number 52-70. The detailed quantum and classical analyses reveal a QPT of crossing normal and intruder configurations superimposed on a QPT of the intruder configuration from U(5) to SU(3) and a crossover from SU(3) to SO(6) dynamical symmetries., Comment: 12 pages, 4 figures. Accepted for publication in SciPost Physics Proceedings, 34th International Colloquium on Group Theoretical Methods in Physics

We use self-consistent mean-field methods in combination with the interacting boson model (IBM) of nuclei, to establish a linkage between universal energy density functionals (EDFs) and partial dynamical symmetry (PDS). An application to $^{168}$Er shows that IBM Hamiltonians derived microscopically from known nonrelativistic and relativistic EDFs in this region, conform with SU(3)-PDS., Comment: 12 pages, 4 figures, 1 table, Proceedings International Workshop "Shapes and Dynamics of Atomic Nuclei: Contemporary Aspects" (SDANCA-21), September 16-18, 2021, Sofia, Bulgaria

The zirconium isotopes with $A=$ 92$-$110 have one of the most complicated evolution of structure in the nuclear chart. In order to understand the structural evolution of these isotopes, we carry a detailed calculation in a definite symmetry-based framework, the interacting boson model with configuration mixing (IBM-CM). We compare our calculation to a large range of experimental data, such as energy levels, two neutron separation energies, $E2$ and $E0$ transition rates, isotope shifts and magnetic moments. The structural evolution of the low lying spectra of these isotopes is explained using the notion of intertwined quantum phase transitions (IQPTs), for which a QPT involving a crossing of two configurations (Type II) is accompanied by a QPT involving a shape evolution of each configuration separately (Type I). In our study, we find the occurrence of Type I QPT within the intruder configuration, changing from weakly deformed to prolate deformed and finally to $\gamma$-unstable, associated with the U(5), SU(3) and SO(6) dynamical symmetry limits of the IBM, respectively. Alongside the Type I QPT, we also find the occurrence of Type II QPT between the normal and intruder configurations, where both Types I and II have a critical-point near $A\approx100$. The good agreement of our calculation with the vast empirical data along the chain of isotopes demonstrates the relevance of IQPTs to the zirconium isotopes, and can serve as a case study to set path for new investigations of IQPTs in other nuclei and other physical systems., Comment: 25 pages, 20 figures, 6 tables

We show that the notion of partial dynamical symmetry is robust and founded on a microscopic many-body theory of nuclei. Based on the universal energy density functional framework, a general quantal boson Hamiltonian is derived and shown to have essentially the same spectroscopic character as that predicted by the partial SU(3) symmetry. The principal conclusion holds in two representative classes of energy density functionals: nonrelativistic and relativistic. The analysis is illustrated in application to the axially-deformed nucleus $^{168}$Er., Comment: 7 pages, 3 figures, 1 table

Lifetimes of low-spin excited states in $^{98}$Zr were measured using the recoil-distance Doppler-shift technique and the Doppler-shift attenuation method. The nucleus of interest was populated in a $^{96}$Zr($^{18}$O,$^{16}$O)$^{98}$Zr two-neutron transfer reaction at the Cologne FN Tandem accelerator. Lifetimes of six low-spin excited states, of which four are unknown, were measured. The deduced $B(E2)$ values were compared with Monte Carlo shell model and interacting boson model with configuration mixing calculations. Both approaches reproduce well most of the data but leave challenging questions regarding the structure of some low lying states., Comment: 13 pages, 12 figures, 3 tables, accepted for publication in Physical Review C

We study the properties of quantum cusp and butterfly catastrophes from an algebraic viewpoint. The analysis employs an interacting boson model Hamiltonian describing quantum phase transitions between specific quadrupole shapes by interpolating between two incompatible dynamical symmetry limits. The classical properties are determined by using coherent states to construct the complete phase diagrams associated with Landau potentials exhibiting such catastrophes.The quantum properties are determined by analyzing the spectra, transition rates and symmetry character of the eigenstates of critical Hamiltonians., Comment: 14 pages, 11 figures, talk at Symposium "Symmetries in Science XVIII", August 4-9, 2019, Bregenz, Austria

We present a detailed analysis of spectra and other observables for the entire chain of Zr isotopes, from neutron number 52 to 70, in the framework of the interacting boson model with configuration mixing. The results suggest a remarkable interplay of multiple quantum phase transitions (QPTs). One type of QPT involves an abrupt crossing of normal and intruder configurations, superimposed on a second type of QPT involving gradual shape-changes within each configuration., Comment: 13 pages, 8 figures, Proceedings International Workshop "Shapes and Dynamics of Atomic Nuclei: Contemporary Aspects" (SDANCA-19), October 3-5, 2019, Sofia, Bulgaria. arXiv admin note: substantial text overlap with arXiv:1909.08305, arXiv:1908.06677, arXiv:1904.09919

We investigate the evolution of structure in the zirconium isotopes where one of the most complex situations encountered in nuclear physics occurs. We demonstrate the role of two concurrent types of quantum phase transitions, sharing a common critical point. The first type, involves an abrupt crossing of coexisting normal and intruder configurations. The second type, involves a gradual shape-phase transition within the intruder configuration, changing from weakly-deformed to prolate-deformed and finally to gamma-unstable. Evidence for this scenario is provided by a detailed comparison with experimental data, using a definite algebraic framework., Comment: 16 pages, 9 figures, Final version, Physica Scripta Focus Issue on Nuclear Shapes and Symmetries: From Experiment to Theory. arXiv admin note: substantial text overlap with arXiv:1904.09919

We introduce the notion of intertwined quantum phase transitions (IQPTs), for which a crossing of two configurations coexists with a pronounced shape-evolution of each configuration. A detailed analysis in the framework of the interacting boson model with configuration mixing, provides evidence for this scenario in the Zr isotopes. The latter exhibit a normal configuration which remains spherical along the chain, but exchanges roles with an intruder configuration, which undergoes first a spherical to prolate-deformed [U(5)$\to$SU(3)] QPT and then a crossover to $\gamma$-unstable [SU(3)$\to$SO(6)]., Comment: 4 pages, 5 figures, Proceedings of the IV International Conference on Nuclear Structure and Dynamics NSD2019, May 13-17, 2019, Venice, Italy

We explore the situation of intertwined quantum phase transitions (IQPTs), for which a QPT involving a crossing of two configurations is accompanied by a shape evolution of each configuration with its own separate QPT. We demonstrate the relevance of IQPTs to the Zr isotopes, with such coexisting Type I and Type II QPTs, and ground state shapes changing from spherical to prolate axially deformed and finally to gamma-unstable. Evidence for this scenario is provided by a detailed comparison with experimental data, using a definite symmetry-based conceptual framework., Comment: 6 pages, 4 figures, 1 table, final version, accepted for publication in Physical Review C

The phonon structure and spectral properties of states in $^{110}$Cd are addressed by including proton excitations in the phonon basis and exploiting a partial dynamical symmetry that mixes only certain classes of states and maintains the vibrational character in the majority of normal states., Comment: 6 pages, 2 figures, 2 Tables, Proceedings of the Zakopane Conference on Nuclear Physics "Extremes of the Nuclear Landscape", Zakopane, Poland, August 26 - September 2, 2018. arXiv admin note: text overlap with arXiv:1810.05917

A key question concerning the spherical-vibrator attributes of states in Cadmium isotopes is addressed by means of a boson Hamiltonian encompassing U(5) partial dynamical symmetry. The U(5) symmetry is preserved in a segment of the spectrum and is broken in particular non-yrast states, and the resulting mixing with the intruder states is small. The vibrational character is thus maintained in the majority of low-lying normal states, as observed in $^{110}$Cd., Comment: 6 pages, 1 figure, 2 tables, Physical Review C (Rapid Communication). arXiv admin note: text overlap with arXiv:1810.05917

A symmetry-based approach for describing shape-coexistence, is presented in the framework of the interacting boson model of nuclei. It involves a construction of a number-conserving Hamiltonian which preserves the dynamical symmetry of selected bands associated with each shape, while breaking the symmetries in other states. The resulting structure embodies multiple partial dynamical symmetries. The procedure is applied to prolate-oblate and spherical-prolate-oblate coexistence, at and slightly away from the critical points., Comment: 12 pages, 5 figures, Final version, Proc. Int. Workshop on "Shapes and Dynamics of Atomic Nuclei: Contemporary Aspects" (SDANCA-17), October 5-7, 2017, Sofia, Bulgaria. arXiv admin note: substantial text overlap with arXiv:1708.06321, arXiv:1803.03982, arXiv:1605.00782

The recently reported deviations of selected non-yrast states in $^{110}$Cd from the expected spherical-vibrator behaviour, is addressed by means of an Hamiltonian with U(5) partial dynamical symmetry. The latter preserves the U(5) symmetry in a segment of the spectrum and breaks it in other states. The effect of intruder states is treated in the framework of the interacting boson model with configuration mixing., Comment: 3 pages, 2 figures, talk at 16th Int. Symposium on Capture Gamma-Ray Spectroscopy and Related Topics (CGS16), September 18-22, 2017, Shanghai Jiao Tong University, China

We present an algebraic procedure for constructing Hamiltonians with several distinct partial dynamical symmetries (PDSs), of relevance to shape-coexistence phenomena. The procedure relies on a spectrum generating algebra encompassing several dynamical symmetry (DS) chains and a coherent state which assigns a particular shape to each chain. The PDS Hamiltonian maintains the DS solvability and quantum numbers in selected bands, associated with each shape, and mixes other states. The procedure is demonstrated for a variety of multiple quadrupole shapes in the framework of the interacting boson model of nuclei., Comment: 18 pages, 11 figures, talk at Symposium "Symmetries in Science XVII", July 30 - August 4, 2017, Bregenz, Austria

We present a symmetry-based approach for shape coexistence in nuclei, founded on the concept of partial dynamical symmetry (PDS). The latter corresponds to a situation when only selected states (or bands of states) of the coexisting configurations preserve the symmetry while other states are mixed. We construct explicitly critical-point Hamiltonians with two or three PDSs of the type U(5), SU(3), ${\overline{\rm SU(3)}}$ and SO(6), appropriate to double or triple coexistence of spherical, prolate, oblate and $\gamma$-soft deformed shapes, respectively. In each case, we analyze the topology of the energy surface with multiple minima and corresponding normal modes. Characteristic features and symmetry attributes of the quantum spectra and wave functions are discussed. Analytic expressions for quadrupole moments and $E2$ rates involving the remaining solvable states are derived and isomeric states are identified by means of selection rules., Comment: 25 pages, 14 figures, accepted for publication as a Special Issue on "Shapes and Symmetries in Nuclei: From Experiment to Theory" in Physica Scripta