1. Q UANTUM CHAOS AND P ERES LATTICES FOR 0+ STATES IN THE G EOMETRIC COLLECTIVE MODEL Pavel Stránský, Pavel Cejnar, Michal Macek 5th Workshop on Shape-Phase Transitions and Critical Point Phenomena in Nuclei Istanbul, Turkey (2009) 16 th September 2009 Institute of Particle and Nuclear Phycics Faculty of Mathematics and Physics Charles University in Prague, Czech Republic P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202 and 066201 P. Stránský, M. Kurian, P. Cejnar, Phys. Rev. C 74 (2006), 014306 P. Cejnar, P. Stránský, Phys. Rev. Lett. 93 (2004), 102502

2. T…Kinetic term V…Potential GCM Hamiltonian neglect higher order terms neglect Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta Principal axes system (PAS) Scaling properties Deformed shape Spherical shape V V   B A C=1

3. Mapping classical chaos “Arc of regularity” B = 0.62 Integrability Regular fraction of the phase space Region of harmonicity

4. Brody distribution Classical Quantum 1-  f reg B = 0.62 B = 1.09 measures of chaos Classical f reg Brody good qualitative agreement A = -1, K = C = 1

5. Peres lattices Quantum system: A. Peres, Phys. Rev. Lett. 53 (1984), 1711 E Integrable lattice always ordered for any operator P Infinite number of of integrals of motion can be constructed: Lattice: energy E i versus value of nonintegrable E partly ordered, partly disordered chaotic regular

6. B = 0.62 B = 1.09 <L2><L2> 1-  f reg Peres operators: More regular structures in the lattices Increase in measures of regularity

7. Increasing perturbation E A=-1, K=C=1 Integrability x Onset of chaos <L2><L2> B = 0 B = 0.001 B = 0.05 B = 0.24 Integrable Empire of chaos

8. http://www-ucjf.troja.mff.cuni.cz/~geometric Summary The geometric collective model of nuclei Exhibits complex behaviour encoded in simple dynamical equation Provides excellent tools for studying various phenomena (manifestations of chaos in classical and quantum physics…) More results in clickable form on ~stransky P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202 and 066201 P. Stránský, M. Kurian, P. Cejnar, Phys. Rev. C 74 (2006), 014306 P. Cejnar, P. Stránský, Phys. Rev. Lett. 93 (2004), 102502 Related papers