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Pavel Str ánský 1,2

CH AOTIC DYNAMICS IN COLLECTIVE MODELS OF NUCLEI. Pavel Str ánský 1,2. 1 Institute of Particle and Nuclear Phycics Faculty of Mathematics and Physics Charles University in Prague, Czech Republic. 2 Institut o de Ciencias Nucleares Universidad Nacional Autonoma de M éxico. Collaborators:.

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Pavel Str ánský 1,2

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  1. CHAOTIC DYNAMICS IN COLLECTIVE MODELS OF NUCLEI Pavel Stránský1,2 1Institute of Particle and Nuclear Phycics Faculty of Mathematics and Physics Charles University in Prague, Czech Republic 2Institutode Ciencias Nucleares Universidad Nacional Autonoma de México Collaborators: Michal Macek1, Pavel Cejnar1 Alejandro Frank2, Ruben Fossion2, Emmanuel Landa2 5th January 2010 XXXIII Symposium on Nuclear Physics, Cocoyoc, Mexico, 2010

  2. CHAOTIC DYNAMICS IN COLLECTIVE MODELS OF NUCLEI Introduction - Basicsof Geometric Collective Model (GCM) (restricted to nonrotating motions) 1. Classical chaosin GCM - Measures of regularity - Geometrical method 2. Quantum chaos in GCM - Short-range correlationsand Brody parameter - Peres lattices - Long-range correlations and 1/f noise - Comparison of classical and quantum dynamics 3. Interacting Boson Model(IBM) - Application of the above mentioned methods

  3. Geometrical Collective Model (restricted to nonrotating motions)

  4. Introduction: Geometric Collective Model Hamiltonianof GCM T…Kinetic term V…Potential Neglect higher order terms neglect Quadrupole tensor ofcollective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta Principal Axes System Shape variables: B … strength of nonintegrability (B = 0 – integrable quartic oscillator)

  5. Introduction: Geometric Collective Model Hamiltonianof GCM T…Kinetic term V…Potential Neglect higher order terms neglect Quadrupole tensor ofcollective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta Principal Axes System Phase structure B V V A b C=1 b Deformedshape Spherical shape

  6. Nonrotating case J = 0! Introduction: Geometric Collective Model Hamiltonianof GCM Principal Axes System Classical dynamics – Hamilton equations of motion Quantization – Diagonalizationinoscillator basis 2 physically importantquantization options (with the same classical limit): • oportunity to test Bohigas conjecture for different quantization schemes (a) 5D system restricted to 2D (true geometric model of nuclei) (b) 2D system

  7. 1. Classical chaos in GCM

  8. 1. Classical chaos in GCM Fraction of regularity Measure of classical chaos Poincaré section vx x REGULARarea CHAOTICarea A = -1, C = K = 1 B = 0.445 freg=0.611

  9. 1. Classical chaos in GCM Different definitons & comparison regular Surface of chosen Poincaré section (with random initial conditions) trajectories number of total Statistical measure E = 0 control parameter

  10. 1. Classical chaos in GCM Complete map of classical chaos in GCM Integrability Veins of regularity chaotic Shape-phase transition regular “Arc of regularity” control parameter Global minimum and saddle point Region of phase transition

  11. 1. Classical chaos in GCM Geometrical method Hamiltonian inflat Eucleidian space with potential: Conformal metric Hamiltonianof free particle in curved space: Application of methods of Riemannian geometry Negative eigenvalues of the matrix inside kinematically accesible area induce nonstability. L. Horwitz et al., Phys. Rev. Lett. 98 (2007), 234301

  12. (a) (b) (d) (c) y (c) (d) x (b) (a) 1. Classical chaos in GCM Geometrical method Global minimum and saddle point Convex-Concave transition Region of phase transition

  13. 2. Quantum chaos in GCM

  14. 2. Quantum chaos in GCM Spectralstatistics Nearest-neighbor spacing distribution Poisson GOE P(s) s REGULAR system CHAOTIC system distribution parameter w Brody - Artificial interpolationbetween Poisson andGOEdistribution - Measure of chaoticity of quantum systems - Tool to test classical-quantum correspondence Bohigas conjecture (O. Bohigas, M. J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984), 1)

  15. regular regular chaotic 2. Quantum chaos in GCM Peres lattices Quantumsystem: Infinite number of of integrals of motion can be constructed (time-averaged operators P): Lattice: energy Ei versus value of partly ordered, partly disordered lattice always ordered for any operator P Integrable nonintegrable B = 0 B = 0.445 <P> <P> E E A. Peres, Phys. Rev. Lett.53 (1984), 1711

  16. H’ Independent Peresoperators in GCM Nonrotating case J = 0! L2 5D L2 2D 2. Quantum chaos in GCM Hamiltonianof GCM Principal Axes System (a) 5D system restricted to 2D (true geometric model of nuclei) (b) 2D system

  17. 2. Quantum chaos in GCM Nonintegrableperturbation Small perturbationaffects only localized part of the lattice B = 0 B = 0.005 B = 0.05 B = 0.24 <L2> Remnants of regularity <H’> E Integrable Increasing perturbation Empire of chaos

  18. 5D 2. Quantum chaos in GCM “Arcof regularity”B= 0.62 <L2> <VB> 2D (different quantizations) E

  19. 5D 2. Quantum chaos in GCM • Connection with the arc of regularity (IBM) • b – g vibrations resonance “Arcof regularity”B= 0.62 <L2> <VB> 2D (different quantizations) E

  20. <L2> Zoom into sea of levels E 2. Quantum chaos in GCM Dependenceon the classicality parameter

  21. 2. Quantum chaos in GCM Classical and quantum measure - comparison B= 1.09 B= 0.24 Classical measure Quantum measure (Brody)

  22. - In GCM we cannot average over ensembles!!! 2. Quantum chaos in GCM 1/f noise - Fourier transformation of the time series Power spectrum a= 2 a= 1 CHAOTIC system REGULAR system -Direct comparison of A. Relañoet al., Phys. Rev. Lett. 89, 244102 (2002)

  23. 2. Quantum chaos in GCM 1/f noise Integrable case: a = 2 expected (4096 levels starting from level 2000) A = +1 A = -1

  24. 2. Quantum chaos in GCM 1/f noise Comparison of measures B= 0.24 B= 0.62

  25. 3. Chaos inIBM 3. Interacting Boson Model

  26. 3 different dynamical symmetries O(6) 0 0 Invariant of O(5) (seniority) 1 Casten triangle U(5) SU(3) 3. Chaos inIBM IBMHamiltonian a – scaling parameter

  27. 3 different Peres operators 3. Chaos inIBM IBMHamiltonian a – scaling parameter 3 different dynamical symmetries O(6) 0 0 Invariant of O(5) (seniority) 1 Casten triangle U(5) SU(3)

  28. 3. Chaos inIBM Regular lattices in integrable case - even the operators non-commuting with Casimirs of U(5) create regular lattices ! commuting non-commuting 0 40 -10 30 U(5) limit 20 -20 10 -30 0 -40 0 -10 N = 40 -20 L = 0 -30 -40

  29. Arc of regularity 3. Chaos inIBM Different invariants classical regularity h = 0.5 N = 40 U(5) SU(3) O(5)

  30. Arc of regularity <L2> GOE 3. Chaos inIBM Different invariants classical regularity h = 0.5 N = 40 U(5) SU(3) O(5)

  31. 3. Chaos inIBM Application: Rotational bands η = 0.5, χ= -1.04 (arc of regularity) N = 30 L = 0

  32. 3. Chaos inIBM Application: Rotational bands η = 0.5, χ= -1.04 (arc of regularity) N = 30 L = 0,2

  33. 3. Chaos inIBM Application: Rotational bands η = 0.5, χ= -1.04 (arc of regularity) N = 30 L = 0,2,4

  34. 3. Chaos inIBM Application: Rotational bands η = 0.5, χ= -1.04 (arc of regularity) N = 30 L = 0,2,4,6

  35. This is the last slide Thank you for your attention Summary • Collective models of nuclei • Complex behavior encoded in simple dynamical equation • Possibility of studying manifestations of both classical and quantum chaos and their relation • Peres lattices • Allow visualising quantum chaos • Capable of distinguishing between chaotic and egular parts of the spectra • Freedom in choosing Peres operator • Methods of Riemannian geometry • Determine location of the onset of chaoticity in classical systems • 1/f Noise • Preliminary results, deeper investigation should be done http://www-ucjf.troja.mff.cuni.cz/~geometric ~stransky

  36. 1. Lyapunov exponent 1. Classical chaos in GCM How to distinguish quasiperiodic and unstable trajectories numerically? Divergence of two neighboring trajectories 2. SALI (Smaller Alignment Index) • twodivergencies • fast convergence towardszero for chaotic trajectories Ch. Skokos, J. Phys. A: Math. Gen 34 (2001), 10029; 37 (2004), 6269

  37. regular chaotic regular 2. Quantum chaos in GCM Wave functions <L2> <VB> E Probability densities

  38. 2. Quantum chaos in GCM Wave functions and Peres lattice B= 1.09 OT E Peres lattice E convex → concave (regular → chaotic) Probability density of wave functions b

  39. 2. Quantum chaos in GCM Long-rangecorrelations • Short-rangecorrelations – nearest neighbor spacing distribution • number variace • D3(„spectral rigidity“) Only 1 realization of the ensemble in GCM – averaging impossible Chaoticityof the system changes with energy – nontrivial dependence on both LandE

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