Pavel Stránský1,2

3rd Chaotic Modeling and Simulation International Conference, Chania, Greece, 2010

3rd January 2010

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

MANIFESTATION OF CHAOS IN COLLECTIVE MODELS OF

NUCLEI

2Instituto de Ciencias NuclearesUniversidad Nacional Autonoma de México

Collaborators:

Michal Macek1, Pavel Cejnar1

Alejandro Frank2, Ruben Fossion2, Emmanuel Landa2

1. Model- Geometric Collective Model of nuclei (GCM) (restricted to pure vibrations)

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

3. 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

MANIFESTATION OF CHAOS IN COLLECTIVE MODELS OF

NUCLEI

1. Geometrical Collective Model of nuclei

(restricted to pure vibrations)

T…Kinetic term V…Potential

Hamiltonian

Neglect higher order terms

neglect

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

Corresponding tensor of momenta

1. Geometric Collective Model of nuclei

Surface of homogeneous nuclear matter:

Quadrupole deformations = 2

G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)

4 external parameters

T…Kinetic term V…Potential

Hamiltonian

Neglect higher order terms

neglect

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

Corresponding tensor of momenta

1. Geometric Collective Model of nuclei

Surface of homogeneous nuclear matter:

Quadrupole deformation = 2

Scaling properties

4 external parametersAdjusting 3 independent scalesenergy

(Hamiltonian)

1 “shape” parameter

size (deformation)

time

1 “classicality” parametersets absolute density of quantum spectrum (irrelevant in classical case)

G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)

Principal Axes System (PAS)

Shape variables:

1. Geometric Collective Model of nuclei

Shape-phase structure

Deformed shape Spherical shape

VV

B

A

C=1

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

Principal Axes System

1. Geometric Collective Model of nuclei

2. Classical chaos in GCM

Fraction of regularity

REGULAR area

CHAOTIC area

freg=0.611

vx

x

2. Classical chaos in GCM

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

Measure of classical chaos

Poincaré section

Different definitons & comparison

Surface of the chosen Poincaré section

regular

totalnumber of

trajectories (with random initial conditions)

control parameter

E = 0

Statistical measure

2. Classical chaos in GCM

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

”

Global minimum and saddle pointRegion of phase transition

Sh

ap

e-p

hase

Sh

ap

e-p

hase

tr

ansi

tion

transi

tion

2. Classical chaos in GCM

Geometrical method

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

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

2. Classical chaos in GCM

Geometrical criterion= Convex-Concave transition

Global minimum and saddle point

Region of phase transition

Geometrical method- gives good estimation of regularity-chaos transition

2. Classical chaos in GCM

y

x

(d)

(c)

(b)

(a)

(b)

(c)

(d)

(a)

1. Classical chaos in GCM

Geometrical method

Geometrical criterion= Convex-Concave transition

Global minimum and saddle point

Region of phase transition

- gives good estimation of regularity-chaos transition

3. 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

3. 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

3. 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

Hamiltonian of GCM

3. Quantum chaos in 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

3. Quantum chaos in GCM

Island of high regularity B = 0.62

<L2>

2D

<VB>

5D

(different quantizations)

E

• – vibrations resonance

3. Quantum chaos in GCM

Zoom into sea of levels

Dependence on the classicality parameter

E

<L2>

3. Quantum chaos in GCM

Selected squared wave functions:

E

Peres operators & Wavefunctions

<L2>

<VB>

2D

Peres invariant classically

Poincaré sectionE = 0.2

3. Quantum chaos in GCM

Classical and quantum measure - comparison Classical

measure

Quantum measure (Brody)

B = 0.24 B = 1.09

3. Quantum chaos in GCM

1/f noise

Power spectrum

2. Quantum chaos in GCM

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

CHAOTIC system = 1 = 2

- Direct comparison of

REGULAR system

= 2

= 1

1 = 0

2

3

4

n = 0 k

k

- Fourier transformation of the time series

Integrable case: = 2 expected

3.0 - 1.92x

6.0 - 1.93x

Shortest periodic classical orbit

Universal region

(averaged over 4 successive sets of 8192 levels, starting from level 8000)

(512 successive sets of 64 levels)

2.0 - 1.94x

log<S>

log f

1/f noise

3. Quantum chaos in GCM

Mixed dynamics A = 0.25

reg

ula

rity

freg

- 11 -

E

Calculation of :Each point –

averaging over 32 successive sets of

64 levels in an energy window

1/f noise

3. Quantum chaos in GCM

Summary

1. Geometrical Collective Model of nuclei • Complex behavior encoded in simple equations• 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 regular parts of the spectra• Freedom in choosing Peres operator

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

chaoticity in classical systems

4. 1/f Noise• Effective method to introduce measure of

chaos using long-range correlations in quantum spectra

5. Other models studied• Interacting boson model, Double

pendulum

Thank you for your attention

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

~stransky

This is the last slide

Appendix. Double pendulum

3. Chaos in IBM

Angular momenta

Quantization:

Peres operators:

Ambiguous procedure

(noncommuting in the kintetic term)

Hamiltonian

Double pendulum

freg - Double pendulum

(a) E = 1

(b) E = 5

(c) E = 14

(c)

(a)

(b)

Double pendulum - results

= l = = 1

Double pendulum in ISS • No gravity• Integrable case• m = l = 1

Libration

Rotationin

distinguishing different classes of motion

Peres lattices

Double pendulum - results

Introducing gravity = 0 = 1

Chaotic band

Double pendulum - results

Classical-Quantum Correspondence

(a) E = 1

(b) E = 5

(c) E = 14

(c)

(a)(b)

Harm

onic

ap

pro

xim

ati

on

Em

pir

e o

f ch

aos

Pre

vale

nce

of

rota

tions

reg

ula

rity

freg

- 1

1 -

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

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