Symmetry triangle

O(6)SU(3) SU(3)*

U(5) vibrator

rotorγ softrotor

χ

η

0

1

-√7 ⁄ 2 √7 ⁄ 2

Spherical shape

Oblate shape

Prolate shape

Interacting Boson Model 1 (IBM1)

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 arc2

7

2

17

integrable(regular dynamics)

Poincaré sections: integrable cases

SU(3) limit

• 2 independent integrals of motion Ii restrict the motion to surfaces of topological tori

• points lie on “circles” - sections of the tori

• torus characterised by two winding frequencies ωi2

70

x

px y

xE = Emin /2

Poincaré sections: integrable cases

• 2 independent integrals of motion Ii restrict the motion to surfaces of topological tori

• points lie on “circles” - sections of the tori

• torus characterised by two winding frequencies ωi03.0

O(6)-U(5) transition

xE = 0

px

x

y

• no integral of motion besides energy E

• points ergodically fill the accessible phase space

• tori completely destroyed

Poincaré sections: chaotic cases

36.07.0

triangle interior

y

E = Emin /2x

px

x

Poincaré sections: semiregular arc

36.07.0

• 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: 27

217

reg

distinct changes of dynamics in this region of the triangle

px

x

y

x

91.05.0

semiregular arc

Poincaré 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: 27

217

reg

E=0

Fractions of regular area Sreg inPoincare 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)

semiregular arc

91.05.0

• 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).

Digression: mixed dynamics

E5

E4

E3

E2

E1

|chi|>|chireg| chi=chireg |chi|<|chireg|

Energy dependence of regularity at both sides of the semiregular Arc (eta = 0.5)

|chi|>|chireg| chi=chireg |chi|<|chireg|

E10

E9

E8

E7

E6

10 equidistant energy values Ei between Emin and Elim

Crossover of two types of regular trajectories (2a and 2b)

Seen for in the regular arc...65.0,35.0

Coexistence of two species of regular trajectories (“knees and spectacles”) sligthly above E = 0

Increasing the energy, one of them prevails..

E13 E14

0+ states of 40 bosons along the Arcs with k=1..5 by Stefan Heinze

Quantum features: Level Bunching in the semiregular Arc

η = 0.35

η = 0.5

η = 0.65

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.

reg3

k 27

617

k