Symmetry vs. Chaos * in nuclear collective dynamics signatures & consequences

1. 1/15 Symmetry vs. Chaos * in nuclear collective dynamics signatures & consequences Pavel Cejnar Institute of Particle and Nuclear Physics Faculty of Mathematics and Physics Charles University, Prague, Czech Republic cejnar@ipnp.troja.mff.cuni.cz * According to Empedocles (cca.490-430 BC), the real world, Cosmos, is an interference of Sphairos, an exquisite world of perfect order originating in symmetry, and Chaos, a world of complete disorder which results from a lack of symmetry. Kazimierz 2010

2. 2/15 Symmetry • Dynamical symmetry • symmetry of a particular system with respect to a dynamical group – a higher group than the one following from the invariance requirements • invariant symmetry with respect to the dynamical group can be broken, but a number of motion integrals remains preserved • dynamical symmetry integrability • perfect order • Invariant symmetry • commutation of Hamiltonian with generators of a certain group (group of invariant symmetry) • conservation laws Breakdown of dynamical symmetry Signatures of symmetry do not vanish instantly: Quasi dynamical symmetry (Rowe… 1988…) The actual dynamical symmetry of the system is broken but the system partly behaves as if the symmetry were effectively preserved. Partial dynamical symmetry (Leviatan… 1992…) All conserved quantum numbers preserved for a part of states or a part of conserved quantum numbers preserved for all states. • Breakdown of integrability • Regular character of classical motions is preserved for some orbits: KAM theorem (Kolmogorov, Arnold, Moser 1954-63) • “quasi integrability” • “partial regularity” All kinds of dynamical symmetry relevant in physics of nuclear collective motions. These motions are therefore mostly thought to be regular...

3. 3/15 Chaos • Quantum chaos • no genuinely quantum definition of chaos (linearity & quasi periodicity of quantum mechanics) • chaos studied in connection with classical limit • Bohigas conjecture (1984): Chaos on quantum level affects statistical properties of discrete energy spectra. Chaotic systems yield spectral correlations consistent with Gaussian random matrix model. • Classical chaos • exponential sensitivity to initial conditions (“butterfly wing effect”) • practical loss of predictability • quasi ergodic trajectories in the phase space • Nuclei show neat signatures of quantum chaos! • data from neutron and proton resonances: Bohigas, Haq, Pandey (1983) • ensemble of low-energy levels: Von Egidyet al. (1987) (Wigner) Nearest-neighbor spacing Brody distribution interpolates between Poisson (ω=0) … order Wigner(ω=1) … chaos ω=0.62

4. 4/15 Origin of chaos in atomic nuclei Many-body dynamics Complex interactions of all particles in the nucleus too difficult => two complementary simplifications: ? 1) Single-particle dynamics Nucleonic motions in deformed nuclear potentials • Arvieu, Brut, Carbonell, Touchard: PRA 35, 2389 (1987) • Rozmej, Arvieu: NPA 545, C497 (1992) • Heiss, Nazmitdinov, Radu: PRL 72, 2351 (1994); PRL 73, 1235 (1994); PRC 52, 3032 (1995) 2) Collective dynamics Nuclear vibrations and rotations • a) Interacting Boson Model (IBM)Iachello, Arima1975 • Alhassid, Whelan, Novoselsky:PRL 65, 2971 (1990); PRC 43, 2637 (1991); PRC 45, 1677 (1992); PRL 67, 816 (1991); NPA 556, 42 (1993) • Paar, Vorkapic, Dieperink:PLB 205, 7 (1988); PRC 41, 2397 (1990), PRL 69, 2184 (1992) • Mizusaki et al.:PLB 269, 6 (1991) • Canetta, Maino:PLB 483, 55 (2000) • Cejnar, Jolie, Macek, Casten, Dobeš, Stránský:PLB 420, 241 (1998); PRE 58, 387 (1998); PRL 93, 132501 (2004); PRC 75, 064318 (2007), PRC 80, 014319(2009),PRC 82, 014308 (2010), PRL 105, 072503(2010) • b) Geometric Collective Model (GCM) Bohr1952 • Cejnar, Stránský, Kurian, Hruška:PRL 93, 102502 (2004); PRC 74, 014306 (2006); PRE 79, 046202 (2009), PRE 79, 066201(2009), JP Conf.Ser. 239, 012002(2010)

5. 5/15 A. Bohr 1952 Gneusset al. 1969 Geometric Collective Model quadrupole tensor of collective coordinates ( 2 shape + 3 Euler angles = 5D ) …corresponding tensor of momenta Hamiltonian neglect … Angular momentum → 0 • effectively 2D system y x Shape variables Principal Axes System

6. 5/15 A. Bohr 1952 Gneuss et al. 1969 Geometric Collective Model quadrupole tensor of collective coordinates ( 2 shape + 3 Euler angles = 5D ) …corresponding tensor of momenta Hamiltonian neglect higher-order terms neglect … Shape-phase diagram prolate spherical B C >0 y oblate A x Shape variables Principal Axes System

7. 6/15 Stránský, Cejnar… 2004 ……. 2010 Geometric Collective Model independent scales energy time coordinates … but a fixed value of Planck constant Hamiltonian neglect … External parameters Two essential parameters 1) “shape” parameter 2) “classicality” parameter prolate spherical B C >0 oblate A integrability • path crossing all parabolas • all equivalent classes of Hamiltonians

8. 7/15 Stránský, Cejnar… 2004 ……. 2010 Geometric Collective Model Classical chaos map ofthedegreeof chaos Regular phase space fraction chaos order

9. 7/15 Stránský, Cejnar… 2004 ……. 2010 Geometric Collective Model Classical chaos map ofthedegreeof chaos change of the shape of the border of the accessible domain in the xy plane convex concave y y x x Regular phase space fraction concave convex y y chaos order x x

10. 8/15 Stránský, Cejnar… 2004 ……. 2010 Geometric Collective Model Classical chaos J = 0, E = 0, A = −5.05, B = C = K = 1 examples of trajectories and a Poincaré section Poincaré section y x 50,000 passages of 52 randomly chosen trajectories through the section y=0 x

11. 9/15 Stránský, Cejnar… 2004 ……. 2010 Geometric Collective Model J=0 Quantum chaos classicality parameter Two quantization options (b) 5D system restricted to 2D (true geometric model of nuclei) (a) 2D system The 2 options differ also in the metric (measure) for calculating matrix elements. ► Possibility to test Bohigas conjecture in different quantization schemes. with additional constraints (to avoid quasi-degeneracies due to the symmetry of V) even/odd

12. 10/15 Stránský, Cejnar… 2004 ……. 2010 Geometric Collective Model Quantum chaos comparison of classical and quantal measures freg … classical regular fraction 1−ω … adjunct of Brody parameter

13. 11/15 Stránský, Cejnar… 2004 ……. 2010 Geometric Collective Model Quantum chaos 1) <L2> 2) <H’> departure from integrable regime Visualmethod by A. Peres (1984) ChoiceofP 1) Quasi-2D angularmomentum 2) Hamiltonian perturbation

14. 12/15 Stránský, Cejnar… 2004 ……. 2010 Geometric Collective Model Quantum chaos complex mixture of regular and chaotic patterns: ordered vs. chaotic states B=0.62 1) <L2> 2) <H’> Visualmethod by A. Peres (1984)

15. 12/15 Stránský, Cejnar… 2004 ……. 2010 Geometric Collective Model Quantum chaos complex mixture of regular and chaotic patterns: ordered vs. chaotic states B=0.62 1) <L2> 2) <H’> Visualmethod by A. Peres (1984) ■ ♦ ●

16. 13/15 Macek, Cejnar, Jolie… 2007 Interacting Boson Model More than 1 classical control parameter But there exist regions of almost full compatibility • multi-dimensional chaotic map. with the GCM. integrable regime J = 0, E = 0

17. 14/15 Macek, Dobeš, Cejnar… 2010 Interacting Boson Model Consequences of regularity for the adiabatic separation of intrinsic and collective motions: rotational bandsexist even at very high excitation energies if the corresponding region of the J =0 spectrum is regular. intrinsic wave functions for various band members in the SU(3) basis • Selection of hypothetical bands of rotational states based on the maximal correlation of the intrinsic SU(3) structures. N = 30 0 0 15 30 J • Product of 0+-2+ and 0+-4+ correlation coefficients for intrinsic wave functions in the given “band” • Energy ratio for 4+ and 2+ states in a given “band” • Classical regular fraction in the respective energy region

18. 15/15 • Conclusions • Nuclear collective motions exhibit an intricate interplay or regular and chaotic features (despite the presumption that collective = regular). • Models of collective motions may serve as a laboratory for general studies of chaos (profit from the coexistence of simplicity and complexity). • Order/Chaos have relevant nuclear-structure consequences (e.g. for the adiabatic separation of collective and intrinsic motions etc.). • Thanks to PavelStránský(now UNAM Mexico) • Michal Macek (soonUni Jerusalem) GCMJ = 0, E = 0 integrable regime

19. Appendices

20. Dependence on energy J=0, A=−0.84, B=C=K=1

21. GCM

22. GCM

23. Peres lattices A visual method to study quantum chaos in 2D systems due to Asher Peres, PRL 53, 1711 (1984) ► In any system there exists infinite number of integrals of motions: e.g. time averages of an arbitrary quantity along individual orbits (note: in fully or partly chaotic systems, these integrals do not allow one to build action-angle variables since they are strongly nonanalytic) ► Quantum counterparts of such observables can be found: (1934-2005) ► One can construct a lattice: energy versus value of ► In a fully regular (integrable) system, the lattice is always ordered (the new integral of motion is constant on tori => it is a function of actions) ► In a chaotic system, the lattice is disordered ► In a mixed regular & chaotic system, the lattice is partly ordered & disordered regular mixed chaotic

24. 2D even GCM Relation to the regular fraction B=1.09 classicality parameter