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Phase Diagram of the Proton-Neutron Interacting Boson Model

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Phase Diagram of the Proton-Neutron Interacting Boson Model PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 21 19 NOVEMBER 2004 Phase Diagram of the Proton-Neutron Interacting Boson Model J. M. Arias,1 J. E. Garcı´a-Ramos,2 and J. Dukelsky3 1Departamento de Fı´sica Ato´mica, Molecular y Nuclear, Facultad de Fı´sica, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain 2Departamento de Fı´sica Aplicada, Universidad de Huelva, 21071 Huelva, Spain 3Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain (Received 2 June 2004; revised manuscript received 3 September 2004; published 15 November 2004) We study the phase diagram of the proton-neutron interacting boson model 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. DOI: 10.1103/PhysRevLett.93.212501 PACS numbers: 21.60.Fw, 05.70.Fh, 21.10.Re, 64.60.Fr Quantum phase transitions (QPT) have become a sub- IBM-1 there are three dynamical symmetries: SU(5), ject of great interest in the study of several quantum O(6), and SU(3). These correspond to well-defined nu- many-body systems in condensed matter, quantum optics, clear shapes: spherical, deformed -unstable, and prolate ultracold quantum gases, and nuclear physics. QPT are axial deformed, respectively. The structure of the IBM-1 structural changes taking place at zero temperature as a Hamiltonian allows to study systematically the transition function of a control parameter (for a recent review, see from one shape to another. There were some pioneering [1]). Examples of control parameters are the magnetic works along these lines in the 1980s [10–12], but it has field in spin systems, quantum Hall systems, and ultra- been the recent introduction of the concept of critical cold gases close to a Feshbach resonance, or the hole- point symmetry that has recalled the attention of the doping in cuprate superconductors. community to the topic of quantum phase transitions in The atomic nucleus is a finite system composed of N nuclei. The phase diagram of the IBM-1 has been studied neutrons and Z protons (Z N 100). Though strictly from several points of view [10–15]. The three different speaking QPT take place for large systems in the thermo- phases are separated by lines of first order phase transi- dynamic limit, finite nuclei can show the precursors of a tion, with a singular point in the transition from spherical phase transition for some particular values of N and Z.In to deformed -unstable shape that is second order. In the these cases, one finds specific patterns in the low energy usual IBM-1, no triaxial shapes appear.These can only be spectrum revealing the strong quantum fluctuations re- stabilized with the inclusion of specific three-body forces. sponsible for the phase transition [2]. Recently the con- A more natural way to generate triaxial deformations is cept of critical point symmetry has been proposed by by explicitly taking into account the proton-neutron de- Iachello and applied to atomic nuclei. First, the transition gree of freedom with the more realistic IBM-2 [16]. from spherical to deformed -unstable shapes was studied In this Letter we will study the phase diagram of the and the corresponding critical point called E 5 [3]. Since IBM-2 using a simplified Hamiltonian that keeps all the then, the interest in nuclear shape-phase transitions has main ingredients of the most general one. This is the been constantly growing. The characteristics of the criti- consistent-Q IBM-2 Hamiltonian [17] cal point in the phase transition from spherical to axially deformed nuclei, called X 5, were presented in Ref. [4]. 1 ÿ x ; ; More recently, the critical point in the phase transition H x nd nd ÿ Q Q ; (1) N from axially deformed to triaxial nuclei, called Y 5,has been analyzed [5]. In all these cases, critical points are P y ; defined in the context of the collective Bohr Hamiltonian where nd dd, Q Q Q with y y ~ 2 y ~ 2 [6]. Using some simplifying approximations, precise Q ds~ sd dd ,andN is the total parameter-free predictions for several observables are number of bosons, which is equal to the number of obtained. This allows to identify nuclei at the critical valence proton plus neutron pairs. The IBM phase dia- points looking at spectroscopic properties. Indeed, some gram studied up to now corresponds to the selection experimental candidates to critical nuclei have already , which produces either spherical, axial, or indepen- been proposed [7,8]. dent shapes. We will extend the previous works on IBM The collective Bohr Hamiltonian, underlying this ap- phase transitions by exploring the transitions from axial proach to critical point symmetries, is closely related to to triaxial shapes within the mean field or intrinsic state the interacting boson model (IBM) [9]. The simplest formalism. The trial wave function is the most general version of the IBM is called IBM-1 since in it no explicit proton-neutron boson condensate [18–20], jgi distinction is made between protons and neutrons. In jN;N; ; ; ; ; i 212501-1 0031-9007=04=93(21)=212501(4)$22.50 2004 The American Physical Society 212501-1 PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 21 19 NOVEMBER 2004 y y ÿ N R^ ÿ N tion to polar coordinates (see Fig. 1) jgi p3 j0i; (2) N !N ! ÿ 1 ÿ x; ÿ p ; 3 7 with (6) 1 1 ÿ p : y p y y 3 7 ÿ s cos d0 p sin 1 2 2 We have explored the IBM-2 parameter space of y y d2 dÿ2; (3) Hamiltonian (1) and present here a selected set of calcu- lations in order to establish the IBM-2 phase diagram (a ^ where ; and R3 is the three-dimensional ro- more detailed presentation will be given in a forthcoming tation operator with fixing the relative orientation publication).We have not found traces of phase transitions (Euler angles) between the proton and neutron conden- in the transition from O(6) to SU 3 in a parallel way as sates. N and N are the numbers of valence proton and the already known transition from O(6) to SU(3) in IBM- neutron pairs, respectively. The equilibrium values of the 1. The O(6) symmetry is in fact very unstable against structure parameters ( ; ; ; ; ) and the energy small perturbations driving the system out of the dy- of the system for given values of the control parameters in namical symmetry either to axial deformed or to triaxial the Hamiltonian (x; ;) can be obtained by minimiz- shapes depending on the interaction. The O(6) symmetry ing the expectation value of the Hamiltonian (1) in the itself has been proposed as a critical dynamical symme- intrinsic state (2): hgjHjgi0. Although there is an try [22]. explicit dependence of the energy on the Euler angles, it In Fig. 2 we show the transition SU 3!SU 3 has been shown [20] that oblique configurations (relative through the edge plotted in Fig. 1. Along this line x orientation angles different from the aligned 0 or p 0 and ÿ 7=2 are fixed. The relevant control pa- p the perpendicular =2) require a repulsive hexade- rameter is varying from ÿ 7=2 (equal and aligned ÿ capole interaction. Therefore, since our Hamil- quadrupole prolate shapes for protons and neutrons) to tonian (1) has no hexadecapole terms, we do not expect p 7=2 (quadrupole prolate shape for protons and quadru- oblique configurations. We can then safely assume that pole oblate shape for neutrons with perpendicular axis of any arbitrary local minimum will have 0; symmetry [21]). In Fig. 2 we present the results for the 0 (or equal to 60 ) for the aligned configurations ground state energy (in arbitrary units) and the shape or 0; 0 ; 60 (or 60 ; 0 ) parameters ( ; ). The resulting proton parameters for the perpendicular configurations. In both cases, p 0, and the rotation operator disappears from the intrinsic are 2 and 0 for allp values of the control state (2). In that situation, the energy per boson in the parameter . In the limit ÿp7=2, we recover the results known from IBM-1: 2 and 0.Inthe limit N;N !1reduces to p opposite limit 7=2 the results known from X 2 p 1ÿx Ref. [21] are obtained: 2 and 60 . Around E ; ; ; ;;;xx 2 ÿ ;1 4 0:4035, a clear shape-phase transition is observed, X X changing the system from axial ( < 0:4035) to triaxial 2 Q 0;2 ; χ = −χ =− x=0 π ν 7/2 SU(3)* 2Q Qÿ ; (4) where we have used the notation Q0 Q0 ; ; and Q2 Qÿ2 Q2 ; ; Qÿ2 ; ; with s ρ 1 2 x=0 2 θ Q0 2 2 cos ÿ cos 2 ; U(5) O(6) 1 s7 φ χ = χ =0 (5) x=1 π ν p 1 1 2 Q2 2 2 sin sin 2 : 1 7 As a natural extension of the Casten triangle for IBM-1 [17], the geometrical representation of the IBM-2 is a SU(3) χ = χ =− pyramid with the new triaxial dynamical symmetry x=0 π ν 7/2 SU 3 [21] in the upper vertex. Figure 1 shows a pictorial representation of the IBM-2 parameter space. Any point FIG. 1. Pictorial representation of the IBM-2 parameter in this space is obtained with the following transforma- space with a dynamical symmetry in each of the four vertices. 212501-2 212501-2 PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 21 19 NOVEMBER 2004 0 -1.4 -0.5 -1.6 -1 -1.8 E(a.u.) E (a.u.) -1.5 -2 1.5 1 ν 1 π,ν β 0.5 β 0.5 60 60 ν 40 40 20 (degrees) 20 π (degrees) ν π,ν 0 γ 0 γ 0 0.2 0.4 0.6 0.8 1 -1.2 -0.8 -0.4 χ 0 0.4 0.8 1.2 x ν FIG.
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