Evolution of order and chaos across a first-order quantum phase transition

We study the evolution of the dynamics across a generic first-order quantum phase transition in an interacting boson model of nuclei. The dynamics inside the phase coexistence region exhibits a very simple pattern. A classical analysis reveals a robustly regular dynamics confined to the deformed region and well separated from a chaotic dynamics ascribed to the spherical region. A quantum analysis discloses regular bands of states in the deformed region, which persist to energies well above the phase-separating barrier, in the face of a complicated environment. The impact of kinetic collective rotational terms on this intricate interplay of order and chaos is investigated.     

Well-Known Distinctive Signatures of Quantum Phase Transition in Shape Coexistence Configuration of Nuclei

It is interesting that a change of nuclear shape may be described in terms of a phase transition. This paper studies the quantum phase transition of the U(5) to SO(6) in the interacting boson model (IBM) on the finite number N of bosons. This paper explores the well-known distinctive signatures of transition from spherical vibrational to γ-soft shape phase in the IBM with the variation of a control parameter. Quantum phase transitions occur as a result of properties of ground and excited states levels. We apply an affine \(\widehat {SU(1,1)}\) approach to numerically solve non-linear Bethe Ansatz equation and point out what observables are particularly sensitive to the transition. The main aim of this work is to describe the most prominent observables of QPT by using IBM in shape coexistence configuration. We calculate energies of excited states and signatures of QPT as energy surface, energy ratio, energy differences, quadrupole electric transition rates and expectation values of boson number operators and show their behavior in QPT. These observables are calculated and examined for ^{98 ? 102}Mo isotopes.     

Partial dynamical symmetries

This overview focuses on the notion of partial dynamical symmetry (PDS), for which a prescribed symmetry is obeyed by a subset of solvable eigenstates, but is not shared by the Hamiltonian. General algorithms are presented to identify interactions, of a given order, with such intermediate symmetry structure. Explicit bosonic and fermionic Hamiltonians with PDS are constructed in the framework of models based on spectrum generating algebras. PDSs of various types are shown to be relevant to nuclear spectroscopy, quantum phase transitions and systems with mixed chaotic and regular dynamics.     

Structure of theN=59 nucleus97Sr: coexistence of spherical and deformed states

A band with a rotational pattern based on a state at 585.1 keV has been identified in theN=59 neutron-rich nucleus⁹⁷Sr. Its properties lead to the [422] 3/2 Nilsson-orbital assignment for the band head. There is evidence for a second band with the head at 644.7 keV and the configuration [541] 3/2. Since the ground state and the lowest excited levels are spherical, shape coexistence is established for⁹⁷Sr. A deformed nature of several levels at 500–600 keV results also from QRPA-model calculations. The structure of the low-lying spherical levels has been studied in the frame of the IBF model. The results of the present investigations lead to a better understanding of theN=59 isotones which constitute the link between the spherical and deformed nuclei atA～100 as a species with shape coexistence but without any indications of particular softness.     

Energy dependence of the rotational enhancement factor in the level density

The one-shell SU(3) energies are given and the corresponding level density is calculated approximately by use of a distribution function for the SU(3) quantum numbers. The calculation is extended to include many shells by a renormalization procedure and an effective one-shell interaction. The traditional level density is then obtained from the related mean-field hamiltonian which corresponds to a deformed harmonic oscillator potential. Various rotational enhancement factors are considered. Numerical results, are obtained and comparisons between the SU(3) and the traditional level densities allow the first computation of the energy dependence of the rotational enhancement factor. A transition from axial to spherical level density is found. A simple parametrization is suggested in terms of a deformation-dependent half-value energy and a transition width.     

Rotational bands from shell-model Hamiltonians

Shell-model Hamiltonians with eigenstates forming rotational bands are considered. Such states have eigenvalues proportional to J(J+ 1) and can be projected from a Slater determinant of deformed orbitals. The latter are linear combinations of single-nucleon wave functions of j-orbits in a major shell. Conditions on matrix elements and single-nucleon energies are obtained in terms of the deformation parameters. An actual effective interaction is constructed yielding exact ground-state rotational bands for ²⁰Ne and ²⁴Mg which gives reasonable agreement with energies of other sd shell nuclei. Unlike the case of SU(3) symmetry, spin-orbit interaction and different single-nucleon energies can be accommodated and the procedure is not confined to oscillator major shells. Other welcome departures of our effective interaction from the SU(3) picture are the absence of rotational spectra in oxygen isotopes and that the ²⁴Mg ground-state band is projected from a Nilsson-type deformed state with axial symmetry.     

q-Deformed SU(1,1) ⊗ SO(6) Spectra of Interacting s,d Bosons in Nuclei

The quantum groups SU_q(1,1) and SU_q(2) are used to describe the interacting s, d bosons system which hae the q-deformed SU(1,1) ⊗ SO(6) dynamic symmetry and a Hamiltonian similar to that of O(6)-limit in the interacting boson model is constructed. Three regions of q-number are taken to analyze the influences of deformation on the deformed SU(1,1) ⊗ SO(6) spectra. It is found that the deformation parameter has important influences on the higher excited states within a quasi- rotational band. Yt isotopes with O(6) characteristics are investigated.     

U(5)-SU(3) nuclear shape transition within the interacting boson model applied to dysprosium isotopes

In the framework of the interacting boson model (IBM) with intrinsic coherent state, the shape Hamiltonian from spherical vibrator U(5) to axially symmetric prolate deformed rotator SU(3) are examined. The Hamiltonian used is composed of a single boson energy term and quadrupole term. The potential energy surfaces (PES’ s) corresponding to the U(5)-SU(3) transition are calculated with variation of a scaling and control parameters. The model is applied to ^150–162Dy chain of isotopes. In this chain a change from spherical to well deformed nuclei is observed when moving from the lighter to heavier isotopes. ¹⁵⁶Dy is a good candidate for the critical point symmetry X(5). The parameters of the model are determined by using a computer simulated search program in order to minimize the deviation between our calculated and some selected experimental energy levels, B(E2) transition rates and the two neutron separation energies S_2n. We have also studied the energy ratios and the B(E2) values for the yrast state of the critical nucleus. The nucleon pair transfer intensities between ground-ground and ground-beta states are examined within IBM and boson intrinsic coherent framework.     

Coulomb reorientation in near-barrier fusion of deformed+spherical systems in classical dynamical approach

A classical rigid-body dynamics model which takes into account all the translational and the rotational degrees of freedom is developed to study Coulomb reorientation of deformed nuclei in heavy-ion collisions. Various aspects of the collision dynamics in the case of near-barrier fusion of ²⁴Mg + ²⁰⁸Pb system due to the Coulomb reorientation are studied; the dependence of the extent of reorientation of the symmetry axis of the deformed nucleus, isotropy of the initial orientations, barrier parameters, and rotational excitation energy are discussed in detail. It is found that the barrier parameters not only depend on the initial orientations of the deformed nucleus but also on the collision energy; with maximum reorientation effect at near- and below-barrier energies. Even small amount of the rotational excitation energy gained by the deformed nucleus at large separation distances is crucial in determining the conditions at the barrier. Study of ¹⁵⁴Sm + ¹⁶O and ²³⁸U + ¹⁶O systems involving heavier deformed nuclei shows that the extent of reorientation also depends on the moment of inertia of the deformed nucleus.     

Description of Rotation Bands in 157,158Er at Ultrahigh Spin

Properties of the four rotation bands, ¹⁵⁷Er(1,2) and ¹⁵⁸Er(1,2), at ultrahigh spin are investigated within the supersymmetry scheme including many-body interactions and possessing the SO(5) (or SU(5)) symmetry on the rotational symmetry. Quantitatively good results of the  γ-ray energies and the dynamical moments of inertia in the rotation bands in ¹⁵⁷Er and ¹⁵⁸Er at ultrahigh spin are obtained. We theoretically predict that the competition between the anti-pairing and pairing effects may exist in ¹⁵⁷Er(1,2) and ¹⁵⁸Er(2) bands states. In ¹⁵⁸Er(1) band state, the favoure-pairing effects may exist and the SO(5) (or SU(5)) symmetry play a dominant role. There may be sphere coexisting with headecupole deformed in ¹⁵⁸Er(1) rotation band state.     

Validity of an algebraic-variational approach to the theory of collective motion for an exactly soluble shell model with R(5) symmetry

An algebraic-variational approach to the theory of collective motion previously applied in variant forms to pairing and monopole interaction models is here developed for an exactly soluble shell model Hamiltonian with R(5) symmetry. The spectrum of this class of Hamiltonian operators has previously been shown to represent a two-dimensional vibrator-rotator. The approximation scheme developed yields almost exact results up to the two-phonon level in the spherical region and goes over smoothly into a theory of the lowest states of the ground state rotational band in the deformed regime.     

Hydrogen atom on null-plane and Melosh transformation

The null-plane dynamics of hydrogen-like atoms is studied in approximations depending on c, the velocity of light, being large. Neglecting terms in the Hamiltonian of order c^?3 (relative to electron rest energy) a symmetry SU (2)_W appears which is analogous to the SU (6)_W of hadron classification. This symmetry, if accurate, would dictate zero ground state magnetic moment. The symmetry is broken by terms of third order, which can, however, be transformed a way by the appropriate approximation to the Melosh transformation. There then emerges a better symmetry, SU (2)_M, broken only at fourth order. The ground state magnetic moment acquires its usual non-relativistic value. The symmetry SU (2)_M corresponds to a subgroup of a symmetry [U (2) × U (2)]_FW which appears in the old Foldy-Wouthuysen approach when spin-orbit coupling is neglected. As well as “current” and “constituent” pictures, “classification” pictures are distinguished; it is to one of the latter that the Melosh transformation transforms.     

Nonabelian electric and magnetic flux in unified SU(5) gauge theory

We study the long-distance behaviour of pure unified SU(5) gauge theory in the limit when the electroweak subgroup is unbroken. We show that the symmetry breaking pattern SU(5)→SU(3)_c×SU(2)×U(1)_Y, with SU(3)_c and SU(2) ×U(1)_Y realized, respectively, in confining and coulombic phases, is a possible dynamical phase of the SU(5) theory. The proof relies on showing that the duality equation of 't Hooft, relating the electric and magnetic flux, is exactly satisfied for the above symmetry breaking pattern. The infrared structure of SU(5), broken down to SU(3)_c×SU(2)×U(1)_Y, is not self-dual.     

原子核过渡区的量子相变

在相互作用玻色子模型(IBM) 基础上用O(6) 高阶项代替传统的SU(3) 四极-四极相互作用来研究原子核从振动到转动过渡区的量子相变行为。利用U(5)-SU(3) 和UQ 两种方案,在玻色子数分别为N = 8 和N = 20 不同情况下,研究了原子核的一些低激发态的能级比和电四极跃迁比。结果表明:随着玻色子数N 的增大,系统的量子相变行为得到加强;两种方案都可以用来描述从振动到转动过渡区的原子核的量子相变特征且O(6) 高阶项方案下量子相变行为更为明显。在此基础上,进一步应用这两种方案具体讨论了152Sm 核的低激发态能级和电四极跃迁性质并与实验数据进行对比分析。结果表明,用O(6) 高阶项可以更为合理地描述带内跃迁及不同带之间的带间跃迁性质。With the framework of Interacting Boson Model(IBM), transitional patterns from the spherical to the axially deformed limit of the IBM with a schematic Hamiltonian are studied by replacing the SU(3) quadrupole-quadrupole term with O(6) cubic interaction. But, we use the two schemes to investigate some energy ratios and B(E2) ratios for different bosons N = 8 and N = 20. The results show that with the increasing of the numbers of bosons, the transitional behaviors can be enhanced; the transitional behaviors are very similar in the two schemes. However, there are some distinctive differences for some quantities across the entire transitional region, such as energy levels and ratios, B(E2) values and ratios, and expectation values of the shape variables. Generally speaking, the transition is smoother and the nuclear shape is less well defined in the new scheme. Then we apply the two schemes to the critical point symmetry candidate, such as 152Sm, and find the overall fitting quality of the UQ scheme is better than that of the U(5)-SU(3) scheme, especially for the inter-band E2 transitions in 152Sm.     

The SU(8) GUT for chiral families

We argue that there is a natural inclusion of the standard model with the family-unified chiral local symmetry SU(3)_H in a new SU(8) GUT inspired by Georgi's SU(11) model. The SU(8) symmetry breaking along the chain with the intermediate electro-weak-horizontal unification SU(8)→SU(5)_EWH×SU(3)_c×U(1) is proved d to be the distinguished pattern. The model predicts a long-lived proton (10³³–10³⁵yr) and practically the observed value of the Weinberg angle, in sharp contrast to the ordinary GUTs. The observation of the specific flavour-changing baryon decays (p→π,μ,Ke,…) could unambiguously single out the flavour unified GUTs as well as confirm the existence of the elementary horizontal forces at small distances.     

Entropies and IPR as Markers for a Phase Transition in a Two-Level Model for Atom–Diatomic Molecule Coexistence

A quantum phase transition (QPT) in a simple model that describes the coexistence of atoms and diatomic molecules is studied. The model, which is briefly discussed, presents a second-order ground state phase transition in the thermodynamic (or large particle number) limit, changing from a molecular condensate in one phase to an equilibrium of diatomic molecules–atoms in coexistence in the other one. The usual markers for this phase transition are the ground state energy and the expected value of the number of atoms (alternatively, the number of molecules) in the ground state. In this work, other markers for the QPT, such as the inverse participation ratio (IPR), and particularly, the Rényi entropy, are analyzed and proposed as QPT markers. Both magnitudes present abrupt changes at the critical point of the QPT.     

Oblique-basis shell-model calculations

A one-dimensional harmonic oscillator in a box is used to introduce the oblique-basis concept. The method is extended to the nuclear shell model by combining traditional spherical shell model states, which yield a diagonal representation of the usual single-particle interaction, with SU(3) shell model collective configurations that track deformation. An application to ²⁴Mg, using the realistic two-body interaction of Wildenthal, is used to explore the validity of this mixed-mode shell-model scheme. The theory is also applied to lower pf-shell nuclei, ^44–48Ti and ⁴⁸Cr, using the Kuo-Brown-3 interaction. These nuclei show strong SU(3) symmetry breaking due mainly to the single-particle spin-orbit splitting. Nevertheless, the results also show that yrast band B(E2) values are insensitive to fragmentation of SU(3) symmetry. Specifically, the quadrupole collectivity as measured by B(E2) strengths remains high even though the SU(3) symmetry is rather badly broken. The results suggest that an oblique-basis mixed-mode shell-model theory may be useful in situations where competing degrees of freedom dominate the dynamics.     

Description of the Superdeformed Bands of Odd-Odd Nuclei in A ～ 80 Region

Properties of the superdeformed bands of odd-odd nuclei in A～80 mass region are investigated systematically within the supersymmetry scheme including many-body interactions and a perturbation possessing the SO(5) (or SU(5)) symmetry on the rotational symmetry. The obtained γ-ray energies, and the dynamical moments of inertia agree with experimental data. It shows that this approach is quite powerful in describing odd-odd nuclei in A～80 mass region.     

SU(5) gut phase transitions via 't Hooft's duality relation

't Hooft's duality relation is used to investigate the possible dynamical symmetry breaking pattern SU(5)→ SU(4) ? U(1) where the gauge fields of SU(4) may be in one of four possible phases: (i) confinement phase, (ii) Higgs phase, (iii) “self-dual phase”, (iv) Coulomb phase. It is found that the duality relation involving the electric and magnetic free energies is satisfied in all these cases.