Symmetry triangle

1. 1 η Symmetry triangle 0 χ vibrator -√7 ⁄ 2 U(5) √7 ⁄ 2 Spherical shape Prolate shape Oblate shape SU(3) O(6) SU(3)* rotor γ soft rotor Interacting Boson Model 1 (IBM1)

2. Regularity / Chaos in IBM1 • Complete integrability at dynamical symmetries due to Cassimir invariants • Also at O(6)-U(5) transition due to underlying O(5) symmetry • What about the triangle interior ? varying degree of chaos initially studied by Alhassid and Whelan integrable (regular dynamics) quasiregular arc

3. Poincarésections: integrable cases • 2 independent integrals of motion Iirestrict the motion to surfaces of topological tori • points lie on “circles” - sections of the tori • torus characterised by two winding frequencies ωi SU(3) limit px y x x E = Emin /2

4. Poincarésections: integrable cases • 2 independent integrals of motion Iirestrict the motion to surfaces of topological tori • points lie on “circles” - sections of the tori • torus characterised by two winding frequencies ωi O(6)-U(5) transition px y x x E = 0

5. Poincarésections: chaotic cases • no integral of motion besides energy E • points ergodically fill the accessible phase space • tori completely destroyed triangle interior px y x x E = Emin /2

6. Poincarésections: semiregular arc • semiregular Arc found by Alhassid and Whelan [Y.Alhassid,N.Whelan, PRL 67 (1991) 816 ] • not connected to any known dynamical symmetry – partial dynamical symmetries possible • linear fit: semiregular arc px y x x distinct changes of dynamics in this region of the triangle

7. Poincarésections: semiregular arc • semiregular Arc found by Alhassid and Whelan [Y.Alhassid,N.Whelan, PRL 67 (1991) 816 ] • not connected to any known dynamical symmetry – partial dynamical symmetries possible • linear fit: semiregular arc E=0 Fractions of regular area Sregin Poincare sections and of regular trajectories Nreg in a random sample (dashed: Nreg/Ntot, full: Sreg/Stot) Method: Ch. Skokos, JPA: Math. Gen. 34, 10029 (2001), P. Stránský, M. Kurian, P. Cejnar, PRC 74, 014306 (2006)

8. Digression: mixed dynamics • Phase space structure of mixed regular-chaotic systems is rather complicated – periodic trajectories crucial As the strength of perturbation to an integrable system increases, the tori start to desintegrate but nevertheless, some survive (KAM – Kolmogorov-Arnold-Moser theorem). Rational tori (i.e. those with periodic trajectories) are the most prone to decay, leaving behind alternating chains of stable and unstable fixed points in Poincaré section (Poincaré-Birkhoff theorem).

9. Energy dependence of regularity at both sides of the semiregular Arc (eta = 0.5) |chi|>|chireg| chi=chireg |chi|<|chireg| |chi|>|chireg| chi=chireg |chi|<|chireg| E10 E5 E9 E4 E8 E3 E7 E2 E6 E1 10 equidistant energy values Ei between Emin and Elim

10. Crossover of two types of regular trajectories(2a and 2b) Seen for in the regular arc... Coexistence of two species of regular trajectories (“knees and spectacles”) sligthly above E = 0 Increasing the energy, one of them prevails.. E13 E14

11. Quantum features: Level Bunching in the semiregular Arc Cosine of action S along the primitive orbits of types 1, 2a, 2b. The shaded region corresponds to the “gap” in the spectrum at k=3. η = 0.65 η = 0.5 0+ states of 40 bosons along the Arcs with k=1..5 by Stefan Heinze η = 0.35