Download Classical Chaos in Geometric Collective Model

Peres Lattices in Nuclear
Structure and Beyond
Pavel Stránský1, Michal Macek1, Pavel Cejnar1, Jan Dobeš2
1Institute
of Particle and Nuclear Phycics
Faculty of Mathematics and Physics
Charles University in Prague, Czech Republic
2Nuclear
Physics Institute Řež
Academy of Sciences of the Czech Republic
CGS-13, Cologne, Germany
26.8.2008
Peres Lattices in Nuclear
Structure and Beyond
1. Visualising and measuring chaos
- Classical and Quantum chaos
- Peres lattices
2. Examples
- Geometric Collective Model (GCM)
- Interacting Boson Model (IBM)
Visualising and Measuring Chaos
Classical chaos
•Trajectories
•Poincaré sections
x
y
Section at
y=0
x
x
Classical chaos
•Fraction of regularity
x
REGULAR area
CHAOTIC area
freg=0.611
x
Quantum chaos
•Spectral statistics
E
Nearest
Neighbour
Spacing
P(s)
distribution
Poisson
GUE
s
REGULAR system
Brody
GOE
distribution
parameter w
CHAOTIC system
GSE
Peres lattices
2D quantum system:
nonintegrable
integrable
P
<P>
regular
E
Fully regular lattice
E
regular
chaotic
A. Peres, Phys. Rev. Lett. 53 (1984), 1711
Examples
1. Geometric Collective Model
GCM Hamiltonian
T…Kinetic term
V…Potential
Principal axes system (PAS)
Nonrotating case J = 0!
Special choice (scaling): A = -1, C = 1
Peres operator
2 physically important
quantization options:
(a) 5D system restricted to 2D
(true geometric model of nuclei)
O(5) invariant (seniority) restricted to J = 0
(b) 2D system
O(2) invariant
Levels and wave functions
<P>
E
Peres lattice
E
Probability
density of wave
function
x
Integrability, Onset of chaos
A=-1, K=C=1
<P>
<P>
E
B=0.005 – small perturbation
E
B = 0 – integrable case
B=0.05 – greater perturbation
Dominion of chaos
<P>
Remnants of
regularity
E
B = 0.24 – the most chaotic case
Island of regularity
• Connection with the arc of
regularity (IBM)
• b – g vibrations resonance
<P>
5D
2D
E
Different quantizations
Peres invariant
classically
Dependence on the classicality parameter
Zoom into sea of levels
PT
E
freg
Classical
1-w
Quantum
E
Classical x quantum view (more examples)
(b)
(a)
(c)
(a) B=0.24
(b) B=0.445
<P>
E
freg
E
(c) B=1.09
Examples
2. Interacting Boson Model
IBM Hamiltonian
a – scaling parameter
3 different dynamical
symmetries
O(6)
0
0
Invariant of O(5)
(seniority)
1
Casten triangle
SU(3)
U(5)
IBM Hamiltonian
a – scaling parameter
3 different dynamical
symmetries
3 different
Peres operators
O(6)
0
0
Invariant of O(5)
(seniority)
1
Casten triangle
SU(3)
U(5)
Different invariants
Arc of regularity
h = 0.5
N = 40
U(5)
SU(3)
O(5)
Variance lattices
• SU(3) invariant
c = -1.0
h = 0.5
N = 30
b - g degeneracies
Variance lattices
• U(5) invariant
c = -1.32
• Phonon calculation
(mean-field approximation)
nb
basis:
nexc
Wave functions components in SU(3) basis
• Phonon calculation
(mean-field approximation)
basis:
L = 0,2,4,6,8
Quasidynamical symmetry
(same amplitude for all low-L states)
Summary – Peres lattices
1. Vivid tool for visualising quantum chaos, especially in 2D systems
2. Capability of distinguishing between „chaotic“ and „regular“ levels
3. Enormous freedom in choosing Peres invariant
4. Peres lattices can be constructed both for mean value and variance
of the chosen operator. Variance lattices can show more subtle
features of the systém.
More results in friendly interactive form on
http://www-ucjf.troja.mff.cuni.cz/~geometric
~stransky
Thank you for your
attention
Peres lattices and invariant
J2
EBK Quantization
quantum
numbers
J1
constant
of motion
Arbitrary 2D system
Difference between eigenvalues of A
(valid for any constant of motion)
constant for each trajectory and
more generally for each torus
A. Peres, Phys. Rev. Lett. 53 (1984), 1711