Regular andRegular and chaotic chaotic nuclearnuclear vibrationsvibrations

((mmonodromy, bifurcations, regular islands…)onodromy, bifurcations, regular islands…)

Pavel CejnarPavel Cejnar, Michal Macek, , Michal Macek, Pavel StrPavel Stránský, Matúš Kurianánský, Matúš Kurian

Institute of Particle & Nuclear Physics, Charles University, Prague, Czech Rep.

CGS12, Notre Dame, 2005 A.D.

Thanks to: J. Jolie, S. HeinzeJ. Jolie, S. Heinze (Köln), R. CastenR. Casten (Yale),

J. DobeJ. Dobešš, Z. Pluha, Z. Pluhařř (Prague).

classical ↔ quantum correspondence level density spectral correlations

• bunching/antibunching of levels (Gutzwiller, Berry-Tabor formulas) • long-range correlations…

trajectories in the phase space of quadrupole deformation parameters visual insight into essential dynamical features

Classical limit ! Why classical ?

description of nuclear collective degrees of freedom (vibrations, rotations)

connected with quadrupole deformations

►

►

Geometric Collective Model (GCM)

Interacting Boson Model (IBM)

classical ↔ quantum correspondence level density spectral correlations

• bunching/antibunching of levels (Gutzwiller, Berry-Tabor formulas) • long-range correlations…

trajectories in the phase space of quadrupole deformation parameters visual insight into essential dynamical features

Classical limit ! Why classical ?

description of nuclear collective degrees of freedom (vibrations, rotations)

connected with quadrupole deformations

►

►

Geometric Collective Model (GCM)

Interacting Boson Model (IBM)

y

x

classical ↔ quantum correspondence level density spectral correlations

• bunching/antibunching of levels (Gutzwiller, Berry-Tabor formulas) • long-range correlations…

trajectories in the phase space of quadrupole deformation parameters visual insight into essential dynamical features

Classical limit ! Why classical ?

description of nuclear collective degrees of freedom (vibrations, rotations)

connected with quadrupole deformations

►

►

Geometric Collective Model (GCM)

Interacting Boson Model (IBM)

E

η

y

x

Order / chaos defined most transparently on the classical level

Important issue in nuclear physics – nuclear motions exhibit an interplay of regular and chaotic components even at low energies. What is the principal source of chaos?

• Lyapunov exponets (sensitivity of motions to initial conditions)• Poincaré sections (organization of trajectories in the phase space)

regular chaotic

IBM: η=0.4, χ=-0.99 (“arc of regularity”) η=0.4, χ=-0.77 min114

min )2( VVVE

(plane y=0 in the phase space: 30 000 passages of 120 trajectories)

GCM classical Hamiltonian

.....][5][2

35][5.....][

2

5 2)2()0()2()0()0( CBAK

H

…corresponding tensor of momenta

neglect higher-order terms neglect …

A

B

spherical

oblate

prolate

quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles )

GCM classical Hamiltonian

.....][5][2

35][5.....][

25 2)2()0()2()0()0( CBAK

H

222232222 )()3()()(2

1yxCxyxByxA

KH yx

432 3cos CBAV

)3(3

1)(

2

1)( 232222 xyxyxH yx

quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles )

…corresponding tensor of momenta

With angular momentum

neglect higher-order terms neglect …

)1(*][10 iJ = 0

For comparison: Hénon-Heiles Hamiltonian

motion in principal coordinate frame

sinRe2

cosRe

2

y

x

A

B

2D system

... an archetypal system with competing regular and chaotic features

_________________________________________________

oblate

prolate

spherical

A=1, B=1.09, C=1

Hénon-Heiles system

exhibits rather smooth

energy dependence

of chaotic measures.

Not so the GCM...

E

completely regular

completely chaotic

transitional

Motions near the potential minimum are always regular (oscillator approximation).

At some “critical” energy chaos sets in. This happens approx. when the boundary

of the accessible area in the x × y plane becomes partly concave:

convex

concave

Low energy

Regular fraction

B=C=1

A variable

of the Poincaré section

tot

regreg S

Sf

(similar to reg. fraction

of entire phase space)

E

IBM classical limit

Method by Hatch, Levit [PRC 25, 614 (1982)] Alhassid, Whelan [PRC 43, 2637 (1991)]___________________________________________________________

● use of Glauber coherent states

HH cl

● classical Hamiltonian

22s ||||NN

● boson number conservation (only in average)

complex variables contain coordinates & momenta

● classical limit:

ipq

N)(

21

2

qp

N

fixed 10 real variables:

(2 quadrupole deformation

parameters, 3 Euler angles,

5 associated momenta)

0)exp( s|| 2

21

dse

● angular momentum J=0 Euler angles irrelevant only 4D phase space

(12 real variables)

2 coordinates (x,y) or (β,γ)

restricted phase-space domain

)()(

1),(

2d QQN

nN

H

ddn~

d

)2(]~

[~

)( dddssdQ

Consistent-Q Hamiltonian

d-boson number operator

quadrupole operator

mean field interactions

scaling constant ħω=1 MeV

control parameters η, χ

SU(3)

O(6) U(5)

χ

η0 10

-√7 ⁄ 2

symmetry triangle

deformed

spherical

Measures of chaos (Lyapunov exponents)

inside the triangle:

Alhassid et al. [e.g. NP A556, 42 (1993)]

SU(3)

O(6) U(5)

η=½, χ=0

η=½, χ=- 0.68

η=½, χ=- 0.46

η=½, χ=- 0.91

η=½, χ=- 0.23

η=½, χ=-1.16

pβ

β

“arc of regularity”

IBM Poincaré sections

across the triangle

(J=0, E=0)

SU(3)

O(6) U(5)

η=½, χ=0

η=½, χ=- 0.68

η=½, χ=- 0.46

η=½, χ=- 0.91

η=½, χ=- 0.23

η=½, χ=-1.16

pβ

β

“arc of regularity”

IBM Poincaré sections

across the triangle

(J=0, E=0)

More info:• CGS12 poster: M. Macek, P. Cejnar

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

O(6)-U(5) transition (χ=0)η=0.6

42222 )1(2

45)1(

2 H

kinetic energy Tcl potential energy Vcl

2

2222

yx

222 yx

J=0

… the system is integrable !

2 compatible integrals of motions: • energy• J=0 projected O(5) “angular momentum”

xy yx

2clas.limit335

12 2)(2)()5(O TTJJC

0

T

TR

Classification of trajectories

by the ratio

of periods associated with

oscillations in β and γ

directions. For rational

the trajectory is periodic:

R

R

E

E=0

Spectrum of orbits(obtained in a numerical simulation

involving ≈ 50000 randomly selected

trajectories)

η=0.6

R

E

E=0

Spectrum of orbits(obtained in a numerical simulation

involving ≈ 50000 randomly selected

trajectories)

η=0.6

The mechanism responsible

for narrowing of the band:

inverse bifurcations

(2 separate branches of orbit

with the same R “annihilate”)

R

E

R>3

“flower-like orbits”

(Mexican-hat potential)

R≈2

“bouncing-ball orbits” (like in spherical oscillator)

E=0

Spectrum of orbits(obtained in a numerical simulation

involving ≈ 50000 randomly selected

trajectories)

η=0.6

At E=0 the motions change their

character from O(6)- to U(5)-like

type of trajectories

O(6) transitional U(5)

→seniority

ener

gy

Lattice of J=0 states(energy vs. seniority)

N=40

What about the

quantum case ?

U(5) limit

Analogy with standard isotropic

2D harmonic oscillator:

_________________________n1=nrad+v/3

n2=nrad+2v/32212

21 H

radial quantum number nrad

principal quantum number N=2nrad+m

angular-momentum quantum number m0 2 341

0

01

1

2

2

3

3

4

4… yes, but only for nd=3k

differences between the O(2) and J=0 projected O(5) angular momenta

*

*

O(5) quantum number: seniority v

O(6) quantum number: σ

O(6) limit

422422 H

Transitional case

O(6)-like

type of cells

U(5)-like

type of cells

Redistribution of levels

between O(6) and U(5)

multiplets

E=0

Transitional case

O(6)-like

type of cells

U(5)-like

type of cells

Μονοδρoμια

(monodromy)Singular bundle of E=v=0

orbits connected with the

unstable equilibrium at β=0

Redistribution of levels

between O(6) and U(5)

multiplets

E=0

E=0

J=0 level dynamics across the O(6)-U(5) transition

N=40

(all v’s)

E=0

U(5)-like

O(6)-like

N=40

most probably a real

phase transitioninvolving excited states

(nonzero temperatures)

J=0 level dynamics across the O(6)-U(5) transition

PhysicPhysicumum

Magia Magia MaximaMaxima

Conclusions:

IBM & GCM hide extremely rich variety

of behaviors. Here we discussed:

• nontrivial dependence of chaos on energy & control parameters (unexpected islands of regularity)

• emergence / decay of various types of regular orbits (consequences for level bunching patterns)

• abrupt changes of dynamics with energy & control parameters (signatures of structural phase transitions)

More info:• CGS12 poster: M. Macek, P. Cejnar• http://www-ucjf.troja.mff.cuni.cz/~geometric/• nucl-th/0504016, nucl-th/0504017 (to be published)

GC

M: A

=-1

, B=

0.6

2, C

=K

=1

, E=

3.6

*

*

“…it’s kin

d o

f magic!”

APPENDICES

B=C=K=1

A= -1, C=K=1

E=0A= -1/B2

Scaling properties of the classical GCM

ttt

b

hHHH

~

~

~

432222 3cos21 CBAβK

H

4 parameters – 3 scaling constants = 1 essential parameter

2BAC

R

Relevant combination

of parameters

A=-0.842

A=0

A=0.25 (phase transition)

A=-5.05

Energy dependence of freg

Dependence of Freg on angular momentum

Regular fraction of the

available phase-space volume

),(max BEJJ

j

Phase-transitional region

421

1,,3

2

1fluct cos

rESrgr

TE

r E

Berry-Tabor trace formula(an analog of the Gutzwiller formula, but for 2D integrable systems)

Efluct

T

IES 2

21,

2

1

2

1

… fluctuating part of level density

…pair of integers characterizing periodic orbit with

ratio of frequencies

…period of the primitive orbit

r…number of repetitions

…action per period

1IgE 2

1,21,

EE gEgIIH…function defined by

…Maslov index of the primitive orbit

Bifurcations

Monodromy

The simplest example: spherical pendulum

Discovered: Duistermaat (1980)

Elaborated: Cushman, Bates: Global Aspects of Classical Integrable

Systems (1997)

unstable equilibrium

Figures taken from:

Other examples of monodromy:• Mexican-hat (Champagne bottle) potentials• two-center potentials• coupled rotators• hydrogen in orthogonal E/M fields• ……………