Order-to-Chaos transition within the Interacting Boson Model 1

1
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
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 ?i

SU(3) limit
px
y
x
x
E Emin /2
4
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 ?i

O(6)-U(5) transition
px
y
x
x
E 0
5
Poincaré sections 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
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

semiregular arc
px
y
x
x
distinct changes of dynamics in this region of
the triangle
7
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

semiregular arc
E0
Fractions of regular area Sreg in 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)
chigtchireg chichireg
chiltchireg
chigtchireg chichireg
chiltchireg
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 k3.
? 0.65
? 0.5
0 states of 40 bosons along the Arcs with
k1..5 by Stefan Heinze
? 0.35