Regular and chaotic nuclear vibrations ( m onodromy, bifurcations, regular islands…)

1. Regular andchaotic nuclearvibrations(monodromy, bifurcations, regular islands…) Pavel Cejnar, Michal Macek, Pavel Stránský, Matúš Kurian Institute of Particle & Nuclear Physics, Charles University, Prague, Czech Rep. Thanks to: J. Jolie, S. Heinze(Köln),R. Casten(Yale), J. Dobeš, Z. Pluhař(Prague). CGS12, Notre Dame, 2005 A.D.

2. Geometric Collective Model (GCM) Interacting Boson Model (IBM) description of nuclear collective degrees of freedom (vibrations, rotations) connected with quadrupole deformations Classical limit ! Why classical ? ► trajectories in the phase space of quadrupole deformation parameters visual insight into essential dynamical features ► • classical ↔ quantum correspondence level density spectral correlations • bunching/antibunching of levels • (Gutzwiller, Berry-Tabor formulas) • long-range correlations…

3. Geometric Collective Model (GCM) Interacting Boson Model (IBM) description of nuclear collective degrees of freedom (vibrations, rotations) connected with quadrupole deformations Classical limit ! Why classical ? y ► trajectories in the phase space of quadrupole deformation parameters visual insight into essential dynamical features x ► • classical ↔ quantum correspondence level density spectral correlations • bunching/antibunching of levels • (Gutzwiller, Berry-Tabor formulas) • long-range correlations…

4. Geometric Collective Model (GCM) Interacting Boson Model (IBM) description of nuclear collective degrees of freedom (vibrations, rotations) connected with quadrupole deformations Classical limit ! Why classical ? y ► trajectories in the phase space of quadrupole deformation parameters visual insight into essential dynamical features x ► • classical ↔ quantum correspondence level density spectral correlations • bunching/antibunching of levels • (Gutzwiller, Berry-Tabor formulas) • long-range correlations… E η

5. Order / chaos defined most transparently on the classical level • Important issue in nuclear physics – nuclear motions exhibit an interplay of regular • and chaotic components even at low energies. What is the principal source of chaos? • Lyapunov exponets (sensitivity of motions to initial conditions) • Poincaré sections (organization of trajectories in the phase space) regular chaotic IBM:η=0.4, χ=-0.99 (“arc of regularity”) η=0.4, χ=-0.77 (plane y=0 in the phase space: 30 000 passages of 120 trajectories)

6. GCM classical Hamiltonian quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles ) …corresponding tensor of momenta neglect higher-order terms neglect … B oblate spherical prolate A

7. GCM classical Hamiltonian quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles ) …corresponding tensor of momenta neglect higher-order terms neglect … B With angular momentum =0 oblate spherical prolate motion in principal coordinate frame A 2D system _________________________________________________ For comparison: Hénon-Heiles Hamiltonian ... an archetypal system with competing regular and chaotic features

8. Hénon-Heiles system exhibits rather smooth energy dependence of chaotic measures. Not so the GCM... A=1, B=1.09, C=1 E completely regular completely chaotic transitional

9. Motions near the potential minimum are always regular (oscillator approximation). At some “critical” energy chaos sets in. This happens approx. when the boundary of the accessible area in the x × y plane becomes partlyconcave: concave convex Low energy

10. Regular fraction of the Poincaré section (similar to reg. fraction of entire phase space) E B=C=1 A variable

11. IBM classical limit Method by Hatch, Levit [PRC 25, 614 (1982)] Alhassid, Whelan [PRC 43, 2637 (1991)] ___________________________________________________________ ● use of Glaubercoherent states ● classical Hamiltonian complex variables contain coordinates & momenta (12 real variables) ● boson number conservation (only in average) 10 real variables: (2 quadrupole deformation parameters, 3 Euler angles, 5 associated momenta) fixed ●classical limit: restricted phase-space domain ● angular momentum J=0Euler angles irrelevant only 4D phase space 2 coordinates(x,y) or (β,γ)

12. Consistent-Q Hamiltonian mean field interactions d-boson number operator quadrupole operator scaling constant ħω=1 MeV symmetry triangle control parameters η, χ η 0 1 O(6) U(5) 0 spherical χ Measures of chaos (Lyapunov exponents) inside the triangle: Alhassid et al. [e.g. NP A556, 42 (1993)] deformed -√7 ⁄ 2 SU(3)

13. η=½, χ=- 0.46 η=½, χ=- 0.23 η=½, χ=0 pβ β η=½, χ=- 0.68 IBM Poincaré sections across the triangle (J=0, E=0) η=½, χ=- 0.91 O(6) U(5) “arc of regularity” η=½, χ=-1.16 SU(3)

14. η=½, χ=- 0.46 η=½, χ=- 0.23 η=½, χ=0 pβ β η=½, χ=- 0.68 IBM Poincaré sections across the triangle (J=0, E=0) η=½, χ=- 0.91 O(6) U(5) “arc of regularity” • More info: • CGS12 poster: M. Macek, P. Cejnar • http://www-ucjf.troja.mff.cuni.cz/~geometric/ η=½, χ=-1.16 SU(3)

15. O(6)-U(5) transition (χ=0) J=0 η=0.6 kinetic energy Tcl potential energy Vcl

16. … the system isintegrable ! • 2 compatible integrals of motions: • energy • J=0 projected O(5) “angular momentum” 0 Classification of trajectories by the ratio of periods associated with oscillations in β and γ directions. For rational the trajectory is periodic:

17. Spectrum of orbits (obtained in a numerical simulation involving ≈ 50000 randomly selected trajectories) E η=0.6 E=0 R

18. Spectrum of orbits (obtained in a numerical simulation involving ≈ 50000 randomly selected trajectories) E η=0.6 The mechanism responsible for narrowing of the band: inverse bifurcations (2 separate branches of orbit with the same R “annihilate”) E=0 R

19. R≈2 “bouncing-ball orbits” (like in spherical oscillator) Spectrum of orbits (obtained in a numerical simulation involving ≈ 50000 randomly selected trajectories) E η=0.6 At E=0 the motions change their character from O(6)- to U(5)-like type of trajectories E=0 R>3 “flower-like orbits” (Mexican-hat potential) R

20. What about the quantum case ? O(6) transitional U(5) energy Lattice of J=0 states (energy vs. seniority) N=40 →seniority

21. U(5) limit n2=nrad+2v/3 Analogy with standard isotropic 2D harmonic oscillator: _________________________ n1=nrad+v/3 angular-momentum quantum numberm 0 1 2 3 4 4 radial quantum numbernrad 3 2 * 1 0 … yes, but only for nd=3k 4 3 principal quantum number N=2nrad+m 2 1 0 * differences between the O(2)andJ=0 projected O(5) angular momenta

22. O(6) limit O(6) quantum number: σ O(5) quantum number: seniority v

23. Transitional case U(5)-like type of cells E=0 O(6)-like type of cells Redistribution of levels between O(6) and U(5) multiplets

24. Transitional case Μονοδρoμια (monodromy) Singular bundle of E=v=0 orbits connected with the unstable equilibrium at β=0 U(5)-like type of cells E=0 O(6)-like type of cells Redistribution of levels between O(6) and U(5) multiplets

25. J=0 level dynamics across the O(6)-U(5) transition (all v’s) N=40 E=0

26. J=0 level dynamics across the O(6)-U(5) transition U(5)-like N=40 most probably a real phase transition involving excited states (nonzero temperatures) O(6)-like E=0

27. Conclusions: • IBM & GCM hide extremely rich variety • of behaviors. Here we discussed: • nontrivial dependence of chaos on energy & control parameters (unexpected islands of regularity) • emergence / decay of various types of regular orbits (consequences for level bunching patterns) • abrupt changes of dynamics with energy & control parameters (signatures of structural phase transitions) Physicum Magia Maxima * * GCM: A=-1, B=0.62, C=K=1, E=3.6 “…it’s kind of magic!” • More info: • CGS12 poster: M. Macek, P. Cejnar • http://www-ucjf.troja.mff.cuni.cz/~geometric/ • nucl-th/0504016, nucl-th/0504017 (to be published)

28. APPENDICES

29. Scaling properties of the classical GCM 4 parameters – 3 scaling constants = 1 essential parameter A= -1, C=K=1 Relevant combination of parameters E=0 A= -1/B2 B=C=K=1

30. Energy dependence of freg A=-5.05 A=-0.842 A=0 A=0.25 (phase transition)

31. Dependence of Freg on angular momentum Regular fraction of the available phase-space volume

32. Phase-transitional region

33. Berry-Tabor trace formula (an analog of the Gutzwiller formula, but for 2D integrable systems) … fluctuating part of level density …pair of integers characterizing periodic orbit with ratio of frequencies …number of repetitions …period of the primitive orbit …function defined by …action per period …Maslov index of the primitive orbit

34. Bifurcations

35. Monodromy Discovered: Duistermaat (1980) Elaborated: Cushman, Bates: Global Aspects of Classical Integrable Systems (1997) unstable equilibrium The simplest example: spherical pendulum • Other examples of monodromy: • Mexican-hat (Champagne bottle) potentials • two-center potentials • coupled rotators • hydrogen in orthogonal E/M fields • …………… Figures taken from: