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Peres Lattices in Nuclear Structure and Beyond

Peres Lattices in Nuclear Structure and Beyond. Pavel Str ánský 1 , Michal Macek 1 , Pavel Cejnar 1 , Jan Dobe š 2. 1 Institute of Particle and Nuclear Phycics Faculty of Mathematics and Physics Charles University in Prague, Czech Republic. 2 Nuclear Physics Institute Řež

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Peres Lattices in Nuclear Structure and Beyond

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  1. Peres Lattices in Nuclear Structure and Beyond Pavel Stránský1, Michal Macek1, Pavel Cejnar1, Jan Dobeš2 1Institute of Particle and Nuclear Phycics Faculty of Mathematics and Physics Charles University in Prague, Czech Republic 2Nuclear Physics Institute Řež Academy of Sciences of the Czech Republic 26.8.2008 CGS-13, Cologne, Germany

  2. Peres Lattices in Nuclear Structure and Beyond 2. Examples - Geometric Collective Model (GCM) - Interacting Boson Model (IBM) 1. Visualising and measuring chaos - Classical and Quantum chaos - Peres lattices

  3. Visualising and Measuring Chaos

  4. Section at y = 0 x x • Trajectories • Poincaré sections Classical chaos x y

  5. x x Classical chaos • Fraction of regularity REGULARarea CHAOTICarea freg=0.611

  6. E distribution parameter w Brody Quantum chaos • Spectral statistics Nearest Neighbour Spacing distribution Poisson GOE GUE GSE P(s) s REGULAR system CHAOTIC system

  7. nonintegrable integrable P <P> regular E E Fully regular lattice regular chaotic Peres lattices 2D quantum system: A. Peres, Phys. Rev. Lett.53 (1984), 1711

  8. Examples • 1. Geometric Collective Model

  9. Peres operator 2 physically importantquantization options: (a) 5D system restricted to 2D (true geometric model of nuclei) (b) 2D system O(2) invariant O(5) invariant (seniority) restricted toJ = 0 GCMHamiltonian T…Kinetic term V…Potential Principal axes system (PAS) Special choice (scaling):A = -1, C = 1 Nonrotating case J = 0!

  10. Levels and wave functions <P> E Peres lattice E Probability densityof wave function x

  11. Integrability, Onset of chaos A=-1, K=C=1 <P> <P> E B=0.005 – small perturbation E B=0.05 – greater perturbation B = 0 – integrable case

  12. Dominion of chaos <P> Remnants of regularity E B = 0.24 – the most chaotic case

  13. Island of regularity • Connection with the arc of regularity (IBM) • b – g vibrations resonance <P> 5D 2D E Peres invariant classically Different quantizations

  14. Zoom into sea of levels PT freg Classical 1-w Quantum E E Dependence on the classicality parameter

  15. (b) (c) (a) (c) B=1.09 (a) B=0.24 (b) B=0.445 <P> E freg E Classical x quantumview (more examples)

  16. Examples 2. Interacting Boson Model

  17. 3 different dynamical symmetries O(6) 0 0 Invariant of O(5) (seniority) 1 Casten triangle U(5) SU(3) IBMHamiltonian a – scaling parameter

  18. 3 different Peres operators IBMHamiltonian a – scaling parameter 3 different dynamical symmetries O(6) 0 0 Invariant of O(5) (seniority) 1 Casten triangle U(5) SU(3)

  19. Different invariants Arc of regularity h = 0.5 N = 40 U(5) SU(3) O(5)

  20. Variance lattices • SU(3) invariant c = -1.0 h = 0.5 N = 30 b - g degeneracies

  21. Variance lattices • U(5) invariant c = -1.32 • Phonon calculation (mean-field approximation) nb basis: nexc

  22. Wave functions components in SU(3) basis • Phonon calculation (mean-field approximation) basis: L = 0,2,4,6,8 Quasidynamical symmetry (same amplitude for all low-L states)

  23. Summary – Peres lattices • Vivid tool for visualising quantum chaos, especially in 2D systems • Capability of distinguishing between „chaotic“ and „regular“ levels • Enormous freedom in choosing Peres invariant • Peres lattices can be constructed both for mean value and variance of the chosen operator. Variance lattices can show more subtle features of the systém. More results in friendly interactive form on http://www-ucjf.troja.mff.cuni.cz/~geometric ~stransky

  24. Thank you for your attention

  25. Peres lattices and invariant J2 EBK Quantization quantum numbers J1 constant of motion Arbitrary 2D system Difference between eigenvalues of A (valid for any constant of motion) constant for each trajectory and more generally for each torus A. Peres, Phys. Rev. Lett.53 (1984), 1711

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