Phase Diagram of the Proton-Neutron Interacting Boson Model

PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 21 19 NOVEMBER 2004

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 E5 [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 X5, were presented in Ref. [4]. 1 x ; ; More recently, the critical point in the phase transition H xnd nd Q Q ; (1) N from axially deformed to triaxial nuclei, called Y5,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 ! 1 x; p ; 3 7 with (6) 1 1 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 d2; (3) Hamiltonian (1) and present here a selected set of calculations 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 SU3 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 SU3!SU3 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 7=2 are fixed. The relevant control pa- p the perpendicular =2) require a repulsive hexade- rameter is varying from 7=2 (equal and aligned 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 p7=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 1x Ref. [21] are obtained: 2 and 60 . Around E ; ; ; ; ; ;xx 2 ;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)* 2QQ ; (4) where we have used the notation Q0Q0 ; ; and Q2Q2Q2 ; ; Q2 ; ; with s ρ 1 2 x=0 2 θ Q0 2 2 cos cos2 ; U(5) O(6) 1 s7 φ χ = χ =0 (5) x=1 π ν p 1 1 2 Q2 2 2 sin sin2 : 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 SU3 [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. 3. Same as Fig. 2p but for the transition from U(5) to FIG. 2. Transition from SU(3) to SU 3: x 0, p p p SU 3: 7=2 and x varies from 0 (triaxial) to 7=2,and varies from 7=2 to 7=2. In the panels 1 (spherical). are plotted the energy of the ground state in arbitrary units, and the variation of the shape parameters (dimensionless) and (degrees). values for the ’s a nd t he ’s are plotted. Two phase transitions are observed at different values of x. ( > 0:4035). We will call this point ‘‘y.’’ Note that in Starting from x 1 (spherical system), a first transition to axial deformed shape is observed at x 0:8.Atthis this phase transition the order parameter is that changes from 0 in the symmetric phase to a finite value point, the values of and depart from zero but in the nonsymmetric phase [23]. We have minimized the and are zero, indicating a deformed axial symmetry. energy following two inverse paths looking for possible and play the role of order parameters in this phase x : coexistence of minima and the corresponding spinodal transition. For a value of 0 48, a second phase transition is observed. The values of and are different and antispinodal points. Both calculations give exactly from zero in both sides changing smoothly along the the same results. This means that spinodal, critical, and transition. The angular parameters jump from zero to antispinodal points all converge to a single point [2]. finite values, indicating a transition from an axial shape Therefore, the transition from SU(3) to SU3 is second to a triaxial shape. Therefore, and are the order order. Figure 3 shows the transition from U5!SU3 parameters. The different values for the shape parameters through the correspondingp edge in Fig. 1. Along this edge 7=2 are fixed and the relevant control parameter is x changing from one (spherical) to 0 (triax- 0 ial). The values of and are always equal at the 0.0001 -0.5 0 energy minimum, while and are symmetric with -0.0001 respect to 30 axis. In the different panels, the -1 -0.0002 E(a.u.) energy, and the values of and for proton and neutron -1.5 0.8 0.805 shapes are presented. For x 1, 0, implying π a spherical shape.p For x 0, we recover the SU 3 case 1 with 2,and 0 and 60 . A phase β ν transition at x 0:8 is observed. We will call this point 0.5 ‘‘x.’’ As in the preceding case, we have performed two 0 sets of calculations following inverse paths to determine the order of the transition and again we have found no 30 ν region of coexistence, converging at the same place, 20 spinodal, critical, and antispinodal points. Note that in 10 this case, the order parameter is , as well as (degrees) π

γ 0 60 . In Fig. 4, we present the study of a generic 0 0.2 0.4 0.6 0.8 1 transition from U(5) (spherical) to a triaxial shape x through a trajectory within the IBM-2 pyramid. In particular, we have selected the trajectory deﬁned by FIG. 4. Same as Fig. 2 but for a generic transition from U(5) 1:2 and 0:5, using x as the control parameter to a triaxial shape. The structure parameters are 1:2, varying from 1 to 0. The ground state energy, and the 0:5, and the control parameter x varies from 0 to 1. 212501-3 212501-3 PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 21 19 NOVEMBER 2004

χ = −χ =− x=0 π ν 7/2 discussed in which the transition is second order, the SU(3)* behavior of the corresponding order parameter near the critical point is consistent with a critical exponent 1=2 as given by the Landau theory [23]. We would like to stress T y that the critical surface separating spherical and axially deformed nuclei is almost a sphere with a radius equal to x* 0:2 and centered in U5. The straight line plotted x=0 inside the figure gives an idea of the trajectory followed U(5) O(6) S χ = χ =0 by the transition discussed in Fig. 4. We would like to x=1 e π ν A emphasize that we have found a new critical point (x)at x the phase transition changing directly from spherical to triaxial shapes. We are currently studying the spectroscopic properties of this critical point. The results will be SU(3) χ = χ =− presented elsewhere. Finally, this scheme of analysis can x=0 π ν 7/2 be easily extended to positive values of to obtain the dynamical symmetry limits SU3 and SU3. FIG. 5. Schematic phase diagram for IBM-2. S stands for We acknowledge discussions with F. Iachello and M. spherical, A for axial, and T for triaxial phases. The critical points ‘‘x’’ and ‘‘y’’ studied here and those already known for Caprio. This work was supported in part by the Spanish the IBM-1 phase diagram, ‘‘x,’’ ‘‘e,’’ and ‘‘O(6)’’ are marked DGI under Projects No. BFM2002-03315, with dots. No. BFM2000-1320-C02-02, No. BFM2003-05316- C02-02, and No. FPA2003-05958. for protons and neutrons are due to the selection of the structure parameters and for this trajectory. As in preceding cases, we have performed two sets of calcula- [1] M. Vojta, Rep. Prog. Phys. 66, 2069 (2003). tions following inverse paths to determine the order of the [2] F. Iachello and N.V. Zamfir, Phys. Rev. Lett. 92, 212501 phase transition. The transition at x 0:8 can be ana- (2004). lyzed looking at the behavior of the ground state energy [3] F. Iachello, Phys. Rev. Lett. 85, 3580 (2000). (inset). Where the full line corresponds to a forward [4] F. Iachello, Phys. Rev. Lett. 87, 052502 (2001). [5] F. Iachello, Phys. Rev. Lett. 91, 132502 (2003). x x calculation, starting at 0, increasing , and the [6] A. Bohr and B. Mottelson, Nuclear Structure (Benjamin, dashed line to a backward calculation, starting at x 1, New York, 1975), Vol. II. decreasing x. The inset shows that there are two minima [7] R. F. Casten and N.V. Zamfir, Phys. Rev. Lett. 85, 3584 competing, one spherical and one deformed. If the system (2000). comes from the spherical region, it keeps spherical for a [8] R. F. Casten and N.V. Zamfir, Phys. Rev. Lett. 87, 052503 while even if there is another deformed minimum with (2001). slightly lower energy. On the other side, if the system [9] F. Iachello and A. Arima, The Interacting Boson Model comes from the deformed region, it keeps deformed (look (Cambridge University Press, Cambridge, England, at the small peak in the full line in the inset at x 0:802) 1987). although another spherical minimum has slightly lower [10] A. E. L. Dieperink et al., Phys. Rev. Lett. 44, 1747 (1980). energy. This coexistence of deformed and spherical min- [11] D. H. Feng et al., Phys. Rev. C 23, 1254 (1981). [12] A. Frank, Phys. Rev. C 39, 652 (1989). ima in a small region around x 0:8 is the signature for a [13] E. Lo´pez-Moreno and O. Castanõs, Phys. Rev. C 54, 2374 first order phase transition. The phase transition at x (1996). 0:48 has been studied with forward and backward calcu- [14] J. Jolie et al., Phys. Rev. Lett. 89, 182502 (2002). lations. We have not found any coexistence region. The [15] J. M. Arias et al., Phys. Rev. Lett. 91, 162502 (2003). antispinodal, critical, and spinodal points come together [16] A. Arima et al., Phys. Lett. B 66, 205 (1977). to a single point as corresponds to a second order phase [17] R. F. Casten and D. D. Warner, Rev. Mod. Phys. 60, 389 transition. We have explored the parameter space of the (1988). IBM-2 Hamiltonian (1) in Fig. 1. The resulting phase [18] J. N. Ginocchio and M.W. Kirson, Nucl. Phys. A350,31 diagram of the proton-neutron IBM as described by the (1980). Hamiltonian (1) is depicted in Fig. 5. There are three [19] A. Bohr and B. Mottelson, Phys. Scr. 22, 468 (1980). well-defined phases: spherical, axially deformed (prolate [20] J. N. Ginocchio and A. Leviatan, Ann. Phys. (N.Y.) 216, 152 (1992). in the schematic presentation of Fig. 5), and triaxial. The [21] A. E. L. Dieperink and R. Bijker Phys. Lett. 116B,77 critical surface separating spherical and axially deformed (1982). shapes (e-x -x-e)isfirst order, while the surface separat- [22] J. Jolie et al., Phys. Rev. Lett. 87, 162501 (2001). ing axially deformed and triaxial shapes (e-O6-y-x -e) [23] J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. is second order, including the common line between both Newman, The Theory of Critical Phenomena surfaces (e-x). We have checked that in all the cases (Clarendon Press, Oxford, 1992). 212501-4 212501-4