1. Pavel Stránský 1,2 3 rd Chaotic Modeling and Simulation International Conference, Chania, Greece, 2010 3 rd January 2010 1 Institute of Particle and Nuclear Phycics Faculty of Mathematics and Physics Charles University in Prague, Czech Republic M ANIFESTATION OF CHAOS IN COLLECTIVE MODELS OF NUCLEI 2 Instituto de Ciencias Nucleares Universidad Nacional Autonoma de México Collaborators: Michal Macek 1, Pavel Cejnar 1 Alejandro Frank 2, Ruben Fossion 2, Emmanuel Landa 2

2. 1.Model - Geometric Collective Model of nuclei (GCM) (restricted to pure vibrations) 2.Classical chaos in GCM - Measures of regularity - Geometrical method 3.Quantum chaos in GCM - Short-range correlations and Brody parameter - Peres lattices - Long-range correlations and 1/f noise - Comparison of classical and quantum dynamics M ANIFESTATION OF CHAOS IN COLLECTIVE MODELS OF NUCLEI

3. 1.Geometrical Collective Model of nuclei (restricted to pure vibrations)

4. T…Kinetic term V…Potential Hamiltonian Neglect higher order terms neglect Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta 1. Geometric Collective Model of nuclei Surface of homogeneous nuclear matter: Quadrupole deformations = 2 G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969) 4 external parameters

5. T…Kinetic term V…Potential Hamiltonian Neglect higher order terms neglect Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta 1. Geometric Collective Model of nuclei Surface of homogeneous nuclear matter: Quadrupole deformation = 2 Scaling properties 4 external parameters Adjusting 3 independent scales energy (Hamiltonian) 1 “shape” parameter size (deformation) time 1 “classicality” parameter sets absolute density of quantum spectrum (irrelevant in classical case) G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)

6. Principal Axes System (PAS) Shape variables: 1. Geometric Collective Model of nuclei Shape-phase structure Deformed shape Spherical shape V V   B A C=1

7. Nonrotating case J = 0 ! (a) 5D system restricted to 2D (true geometric model of nuclei) (b) 2D system 2 physically important quantization options (with the same classical limit): Classical dynamics – Hamilton equations of motion oportunity to test Bohigas conjecture for different quantization schemes Quantization – Diagonalization in oscillator basis Principal Axes System 1. Geometric Collective Model of nuclei

8. 2. Classical chaos in GCM

9. Fraction of regularity REGULAR area CHAOTIC area f reg =0.611 vxvx x 2. Classical chaos in GCM A = -1, C = K = 1 B = 0.445 Measure of classical chaos Poincaré section

10. Different definitons & comparison Surface of the chosen Poincaré section regular total number of trajectories (with random initial conditions) control parameter E = 0 Statistical measure 2. Classical chaos in GCM

11. Complete map of classical chaos in GCM Integrability Veins of regularity chaotic regular control parameter “Arc of regularity” Global minimum and saddle point Region of phase transition Shape-phase transition 2. Classical chaos in GCM

12. Geometric al method L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007) Hamiltonian in flat Eucleidian space with potential: Hamiltonian of free particle in curved space: Conformal metric Application of methods of Riemannian geometry inside kinematically accesible area induce nonstability. Negative eigenvalues of the matrix 2. Classical chaos in GCM

13. Geometrical criterion = Convex-Concave transition Global minimum and saddle point Region of phase transition Geometric al method - gives good estimation of regularity-chaos transition 2. Classical chaos in GCM

14. y x (d) (c) (b) (a) (b) (c) (d) (a) 1. Classical chaos in GCM Geometric al method Geometrical criterion = Convex-Concave transition Global minimum and saddle point Region of phase transition - gives good estimation of regularity-chaos transition

15. 3. Quantum chaos in GCM

16. Spectral statistics GOE P(s)P(s) s Poisson CHAOTIC system REGULAR system Nearest- neighbor spacing distribution Bohigas conjecture (O. Bohigas, M. J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984), 1) Brody distribution parameter  - Tool to test classical-quantum correspondence - Measure of chaoticity of quantum systems - Artificial interpolation between Poisson and GOE distribution 3. Quantum chaos in GCM

17. Peres lattices Quantum system: A. Peres, Phys. Rev. Lett. 53 (1984), 1711 Infinite number of of integrals of motion can be constructed (time-averaged operators P ): nonintegrable E regular E Integrable chaotic regular B = 0 B = 0.445 Lattice: energy E i versus value of lattice always ordered for any operator P partly ordered, partly disordered 3. Quantum chaos in GCM

18. Principal Axes System Nonrotating case J = 0 ! (a) 5D system restricted to 2D (true geometric model of nuclei) (b) 2D system Independent Peres operators in GCM H’ L2L2 2D L2L2 5D Hamiltonian of GCM 3. Quantum chaos in GCM

19. Increasing perturbation E Nonintegrable perturbation <L2><L2> B = 0 B = 0.005 Integrable Empire of chaos Small perturbation affects only localized part of the lattice B = 0.05 B = 0.24 Remnants of regularity 3. Quantum chaos in GCM

20. Island of high regularity B = 0.62 2D 5D (different quantizations) E  –  vibrations resonance 3. Quantum chaos in GCM

21. Zoom into sea of levels Dependence on the classicality parameter E 3. Quantum chaos in GCM

22. Selected squared wave functions: E Peres operators & Wavefunctions 2D Peres invariant classically Poincaré section E = 0.2 3. Quantum chaos in GCM

23. Classical and quantum measure - comparison Classical measure Quantum measure (Brody) B = 0.24 B = 1.09 3. Quantum chaos in GCM

24. 1/f noise Power spectrum 2. Quantum chaos in GCM A. Relaño et al., Phys. Rev. Lett. 89, 244102 (2002) CHAOTIC system  = 1  = 2 - Direct comparison of REGULAR system  = 2  = 1  1 = 0 22 33 44  n = 0 kk k - Fourier transformation of the time series

25. Integrable case:  = 2 expected 3.0 - 1.92x 6.0 - 1.93x Shortest periodic classical orbit Universal region (averaged over 4 successive sets of 8192 levels, starting from level 8000) (512 successive sets of 64 levels) 2.0 - 1.94x log log f 1/f noise 3. Quantum chaos in GCM

26. Mixed dynamics A = 0.25 regularity f reg  - 1 1 -  E Calculation of  : Each point – averaging over 32 successive sets of 64 levels in an energy window 1/f noise 3. Quantum chaos in GCM

27. Summary 1.Geometrical Collective Model of nuclei Complex behavior encoded in simple equations Possibility of studying manifestations of both classical and quantum chaos and their relation 2.Peres lattices Allow visualising quantum chaos Capable of distinguishing between chaotic and regular parts of the spectra Freedom in choosing Peres operator 3.Methods of Riemannian geometry Approximate location of the onset of chaoticity in classical systems 4.1/f Noise Effective method to introduce measure of chaos using long-range correlations in quantum spectra 5.Other models studied Interacting boson model, Double pendulum Thank you for your attention http://www-ucjf.troja.mff.cuni.cz/~geometric ~stransky This is the last slide

28. Appendix. Double pendulum 3. Chaos in IBM

29. Angular momenta Quantization: Peres operators: Ambiguous procedure (noncommuting in the kintetic term) Hamiltonian Double pendulum

30. f reg - Double pendulum (a) E = 1 (b) E = 5 (c) E = 14 (c) (a) (b) Double pendulum - results  = l =  = 1

31. Double pendulum in ISS No gravity Integrable case m = l = 1 Libration Rotation in distinguishing different classes of motion Peres lattices Double pendulum - results

32. Introducing gravity  = 0  = 1 Chaotic band Double pendulum - results

33. Classical-Quantum Correspondence (a) E = 1 (b) E = 5 (c) E = 14 (c) (a) (b) Harmonic approximation Empire of chaos Prevalence of rotations regularity f reg  - 1 1 - 

34. IBM Hamiltonian 3 different dynamical symmetries U(5) SU(3) O(6) 0 0 1 Casten triangle a – scaling parameter Invariant of O(5) (seniority) 3. Chaos in IBM

35. 3 different dynamical symmetries U(5) SU(3) O(6) 0 0 1 Casten triangle Invariant of O(5) (seniority) a – scaling parameter 3 different Peres operators 3. Chaos in IBM IBM Hamiltonian

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

37. Different invariants  = 0.5 N = 40 U(5) SU(3) O(5) Arc of regularity classical regularity 3. Chaos in IBM

38. Different invariants  = 0.5 N = 40 U(5) SU(3) O(5) Arc of regularity classical regularity 3. Chaos in IBM GOE

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

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

41. N = 30 L = 0,2,4 η = 0.5, χ= -1.04 (arc of regularity) 3. Chaos in IBM

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

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

45. Wave functions E Probability densities regular chaotic 2. Quantum chaos in GCM

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

47. Long-range correlations number variace  3 („spectral rigidity“) Short-range correlations – nearest neighbor spacing distribution Only 1 realization of the ensemble in GCM – averaging impossible Chaoticity of the system changes with energy – nontrivial dependence on both L and E 2. Quantum chaos in GCM