Pavel Stránský1,2

XXXIII Symposium on Nuclear Physics, Cocoyoc, Mexico, 2010 5th January 2010

1Institute of Particle and Nuclear PhycicsFaculty of Mathematics and PhysicsCharles University in Prague, Czech Republic

CHAOTIC DYNAMICS IN COLLECTIVE MODELS OF

NUCLEI

2Instituto de Ciencias NuclearesUniversidad Nacional Autonoma de México

Collaborators:

Michal Macek1, Pavel Cejnar1

Alejandro Frank2, Ruben Fossion2, Emmanuel Landa2

Introduction- Basics of Geometric Collective Model (GCM) (restricted to nonrotating motions)

1. Classical chaos in GCM- Measures of regularity- Geometrical method

2. Quantum chaos in GCM- Short-range correlations and Brody parameter- Peres lattices- Long-range correlations and 1/f noise- Comparison of classical and quantum dynamics

CHAOTIC DYNAMICS IN COLLECTIVE MODELS OF

NUCLEI

3. Interacting Boson Model (IBM)- Application of the above mentioned methods

Geometrical Collective Model(restricted to nonrotating motions)

T…Kinetic term V…Potential

Hamiltonian of GCM

Neglect higher order terms neglect

Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles)

Corresponding tensor of momenta

Principal Axes System

Shape variables:

Introduction: Geometric Collective Model

B … strength of nonintegrability(B = 0 – integrable quartic oscillator)

T…Kinetic term V…Potential

Neglect higher order terms neglect

Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles)

Corresponding tensor of momenta

Principal Axes System

Introduction: Geometric Collective Model

Phase structure

Deformed shape Spherical shape

VV

B

A

C=1

Hamiltonian of GCM

Principal Axes System

Introduction: Geometric Collective Model

Nonrotating case J = 0!

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

of nuclei)

(b) 2D system

2 physically important quantization options(with the same classical limit):

Classical dynamics– Hamilton equations of motion

• oportunity to test Bohigas conjecture for different quantization schemes

Quantization– Diagonalization in oscillator basis

Hamiltonian of GCM

1. Classical chaos in GCM

Fraction of regularity

REGULAR area

CHAOTIC area

freg=0.611

vx

x

1. Classical chaos in GCM

A = -1, C = K = 1B = 0.445

Measure of classical chaos

Poincaré section

Different definitons & comparison

1. Classical chaos in GCM

Surface of chosen Poincaré section

regular

totalnumber of

trajectories (with random initial conditions)

control parameter

E = 0

Statistical measure

Complete map of classical chaos in GCM IntegrabilityIntegrability

Veins ofVeins of regularityregularity

chaotichaoticc

regularegularr

control parameter

““ Arc

of

Arc

of

regula

rity

”re

gula

rity

”

1. Classical chaos in GCM

Global minimum and saddle pointRegion of phase transition

Sh

ap

e-p

hase

Sh

ap

e-p

hase

tr

ansi

tion

transi

tion

Geometrical method

L. Horwitz et al., Phys. Rev. Lett. 98 (2007), 234301

Hamiltonian in flat Eucleidian space with potential:

Hamiltonian of free particle in curved space:

Conformal metric

Application of methods of Riemannian geometry

inside kinematically accesible area induce nonstability.

Negative eigenvalues of the matrix

1. Classical chaos in GCM

Convex-Concave transition

y

x

(d)

(c)

(b)

(a)

(b)

(c)

(d)

(a)

Global minimum and saddle pointRegion of phase transition

1. Classical chaos in GCM

Geometrical method

2. Quantum chaos in GCM

Spectral statistics

GOE

P(s)

s

Poisson

CHAOTIC systemREGULAR system

Nearest-neighbor spacing distribution

Bohigas conjecture (O. Bohigas, M. J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984), 1)

Brodydistributionparameter

- Tool to test classical-quantum correspondence

- Measure of chaoticity of quantum systems- Artificial interpolation between Poisson and GOE distribution

2. Quantum chaos in GCM

Peres lattices Quantum system:

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

Infinite number of of integrals of motion can be constructed (time-averaged operators P):

nonintegrable

E

<P>

regular

E

Integrable

<P>

chaoticregular

B = 0 B = 0.445

Lattice: energy Ei versus value of

lattice always ordered for any operator P

partly ordered, partly disordered

2. Quantum chaos in GCM

Principal Axes System

Nonrotating case J = 0!

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

of nuclei)

(b) 2D system

IndependentPeres operators in

GCM

H’

L22DL2

5D

2. Quantum chaos in GCM

Hamiltonian of GCM

Increasing perturbation

E

Nonintegrable perturbation

<L2>

B = 0 B = 0.005

<H’>

Integrable Empire of chaos

Small perturbation affects only localized part of the lattice

B = 0.05 B = 0.24

Remnants ofregularity

2. Quantum chaos in GCM

“Arc of regularity” B = 0.62<L2>

2D

<VB>

5D

2. Quantum chaos in GCM

(different quantizations)

E

“Arc of regularity” B = 0.62<L2>

2D

<VB>

5D

E

(different quantizations)

2. Quantum chaos in GCM

• Connection with the arc of regularity (IBM)

• – vibrations resonance

Zoom into sea of levels

Dependence on the classicality parameter

E

<L2>

2. Quantum chaos in GCM

Classical and quantum measure - comparison Classical

measure

Quantum measure (Brody)

B = 0.24 B = 1.09

2. Quantum chaos in GCM

1/f noise- Fourier transformation of the time series

Power spectrum

CHAOTIC system REGULAR system

= 1 = 2

- Direct comparison of

- In GCM we cannot average over ensembles!!!

2. Quantum chaos in GCM

A. Relaño et al., Phys. Rev. Lett. 89, 244102 (2002)

1/f noiseIntegrable case: = 2 expected

2. Quantum chaos in GCM

A = +1

(4096 levels starting from level 2000)

A = -1

1/f noiseComparison of measures

B = 0.24 B = 0.62

2. Quantum chaos in GCM

3. Interacting Boson Model

3. Chaos in IBM

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. Chaos in IBM

3 different dynamical symmetries

U(5)SU(3)

O(6)

0 0

1

Casten triangle

Invariant of O(5) (seniority)

a – scaling parameter

3 different Peres

operators

3. Chaos in IBM

IBM Hamiltonian

Regular lattices in integrable case

3ˆ.ˆ SUQQ

dn̂v

- even the operators non-commuting with Casimirs of U(5) create regular lattices !

40

-40

-2020

10

30 -10

-30

0

-40

-20

-10

-30

0

0

3ˆ.ˆ SUQQ

6ˆ.ˆ OQQ

dn̂

v

L = 0

commuting non-commuting

U(5)

limit

N = 40

3. Chaos in IBM

Different invariants

= 0.5

N = 40

U(5)

SU(3)

O(5)

Arc of regularityArc of regularity

classical regularity

3. Chaos in IBM

Different invariants

= 0.5

N = 40

U(5)

SU(3)

O(5)

Arc of regularityArc of regularity

classical regularity

3. Chaos in IBM

GOE<L2>

Application: Rotational bands

dn̂

N = 30L = 0

η = 0.5, χ= -1.04 (arc of regularity)

3ˆ.ˆ SUQQdn̂

3. Chaos in IBM

N = 30L = 0,2

η = 0.5, χ= -1.04 (arc of regularity)

3ˆ.ˆ SUQQdn̂

3. Chaos in IBM

Application: Rotational bands

Application: Rotational bands

N = 30L = 0,2,4

η = 0.5, χ= -1.04 (arc of regularity)

3ˆ.ˆ SUQQdn̂

3. Chaos in IBM

3ˆ.ˆ SUQQ

N = 30L = 0,2,4,6

η = 0.5, χ= -1.04 (arc of regularity)

dn̂

3. Chaos in IBM

Application: Rotational bands

Summary

1. Collective models of nuclei • Complex behavior encoded in simple dynamical

equation• Possibility of studying manifestations of both

classical and quantum chaos and their relation

2. Peres lattices• Allow visualising quantum chaos• Capable of distinguishing between chaotic

and egular parts of the spectra• Freedom in choosing Peres operator

3. Methods of Riemannian geometry• Determine location of the onset of

chaoticity in classical systems

4. 1/f Noise• Preliminary results, deeper investigation

should be done

Thank you for your attention

http://www-ucjf.troja.mff.cuni.cz/~geometric

~stransky

This is the last slide

How to distinguish quasiperiodic and unstable trajectories

numerically?1. Lyapunov

exponent

Divergence of two neighboring trajectories

2. SALI (Smaller Alignment Index)

• fast convergence towards zero for chaotic trajectories

Ch. Skokos, J. Phys. A: Math. Gen 34 (2001), 10029; 37 (2004), 6269

• two divergencies

1. Classical chaos in GCM

Wave functions<L2>

E

<VB>

Probability densities

regular regularchaotic

2. Quantum chaos in GCM

Wave functions and Peres lattice

convex → concave (regular → chaotic)

E

E

OT

Probability density of wave

functions

Peres lattice

B = 1.09

2. Quantum chaos in GCM

Long-range correlations

• number variace

• 3 („spectral rigidity“)

• Short-range correlations – nearest neighbor spacing distribution

Only 1 realization of the ensemble in GCM – averaging impossibleChaoticity of the system changes with energy – nontrivial dependence on both L and E

2. Quantum chaos in GCM