Peres Lattices in Nuclear Structure and Beyond

Pavel Stránský1, Michal Macek1, Pavel Cejnar1, Jan Dobeš2

CGS-13, Cologne, Germany 26.8.2008

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

2. Examples

- Geometric Collective Model (GCM)

- Interacting Boson Model (IBM)

1. Visualising and measuring chaos

- Classical and Quantum chaos

- Peres lattices

Peres Lattices in Nuclear Structure and Beyond

Visualising and Measuring Chaos

Classical chaos

Section aty = 0

•Trajectories•Poincaré sections

x

y

x

x

Classical chaos

REGULAR area

CHAOTIC area

freg=0.611 x

x

•Fraction of regularity

E

GOE

GUE

GSE

P(s)

s

Poisson

CHAOTIC systemREGULAR system

Quantum chaos •Spectral statistics

Nearest Neighbour Spacing

distribution

Brodydistributionparameter

nonintegrable

<P>

E

Peres lattices2D quantum system:

A. Peres, Phys. Rev. Lett. 53 (1984), 1711

E

integrable

P

Fully regular lattice regular chaotic

regular

Examples 1. Geometric Collective

Model

GCM HamiltonianT…Kinetic term

V…Potential

Nonrotating case J = 0!

Principal axes system (PAS)

Special choice (scaling): A = -1, C = 1

(a) 5D system restricted to 2D (true geometric model

of nuclei)

(b) 2D system

2 physically important quantization options:

O(5) invariant (seniority) restricted to J = 0 O(2) invariant

Peres operator

Levels and wave functions

Probability density of

wave function

Peres lattice

E

x

E

<P>

B=0.005 – small perturbation

B=0.05 – greater perturbation

<P>

E

B = 0 – integrable case

<P>

E

A=-1, K=C=1Integrability, Onset of chaos

B = 0.24 – the most chaotic case

<P>

E

Dominion of chaos

Remnants of regularity

E

Island of regularity

<P>

Different quantizations

2D

5D

Peres invariant classically

• Connection with the arc of regularity (IBM)

• – vibrations resonance

E

PT

Zoom into sea of levels

Dependence on the classicality parameter

E

1- Quantum

Classical

freg

Classical x quantum view (more examples)

(a)

(b)

(c)

(b) B=0.445 (c) B=1.09(a) B=0.24

<P>

freg

E

E

Examples2. Interacting Boson Model

IBM Hamiltonian

3 different dynamical symmetries

U(5)SU(3)

O(6)

0 0

1

Casten triangle

a – scaling parameter

Invariant of O(5) (seniority)

3 different dynamical symmetries

U(5)SU(3)

O(6)

IBM Hamiltonian

0 0

1

Casten triangle

Invariant of O(5) (seniority)

a – scaling parameter

3 different Peres

operators

Different invariants

= 0.5

N = 40

U(5)

SU(3)

O(5)

Arc of regularity

Variance lattices

- degeneracies

• SU(3) invariant

= -1.0

N = 30

= 0.5

Variance lattices • U(5) invariant

• Phonon calculationn

nexc

(mean-field approximation)

basis:

= -1.32

Wave functions components in SU(3) basis

• Phonon calculation(mean-field

approximation)basis:

Quasidynamical symmetry(same amplitude for all low-L states)

L = 0,2,4,6,8

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

A. Peres, Phys. Rev. Lett. 53 (1984), 1711

constant of motion

J1

J2

Arbitrary 2D system

constant for each trajectory and more generally for each torus

EBK Quantization quantu

m numbers

Difference between eigenvalues of A

(valid for any constant of motion)