Symmetry triangle O(6)SU(3)SU(3)* U(5) vibrator rotorγ softrotor χ η 0 1 -√7 ⁄ 2√7 ⁄ 2...

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Transcript of Symmetry triangle O(6)SU(3)SU(3)* U(5) vibrator rotorγ softrotor χ η 0 1 -√7 ⁄ 2√7 ⁄ 2...

  • Slide 1
  • Symmetry triangle O(6)SU(3)SU(3)* U(5) vibrator rotor softrotor 0 1 -7 27 2 Spherical shape Oblate shape Prolate shape Interacting Boson Model 1 (IBM1)
  • Slide 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 quasiregular arc integrable (regular dynamics)
  • Slide 3
  • Poincar sections: integrable cases SU(3) limit 2 independent integrals of motion I i restrict the motion to surfaces of topological tori points lie on circles - sections of the tori torus characterised by two winding frequencies i x pxpx y x E = E min /2
  • Slide 4
  • Poincar sections: integrable cases 2 independent integrals of motion I i restrict the motion to surfaces of topological tori points lie on circles - sections of the tori torus characterised by two winding frequencies i O(6)-U(5) transition x E = 0 pxpx x y
  • Slide 5
  • no integral of motion besides energy E points ergodically fill the accessible phase space tori completely destroyed Poincar sections: chaotic cases triangle interior y E = E min /2 x pxpx x
  • Slide 6
  • Poincar sections: 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: distinct changes of dynamics in this region of the triangle pxpx x y x semiregular arc
  • Slide 7
  • Poincar sections: 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: E=0 Fractions of regular area S reg in Poincare sections and of regular trajectories N reg in a random sample (dashed: N reg /N tot, full: S reg /S tot ) Method: Ch. Skokos, JPA: Math. Gen. 34, 10029 (2001), P. Strnsk, M. Kurian, P. Cejnar, PRC 74, 014306 (2006) semiregular arc
  • Slide 8
  • 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 Poincar section (Poincar-Birkhoff theorem). Digression: mixed dynamics
  • Slide 9
  • E5 E4 E3 E2 E1 |chi|>|chi reg | chi=chi reg |chi||chi reg | chi=chi reg |chi|