Symmetry vs. Chaos *

in nuclear collective dynamics

signatures & consequences

Pavel CejnarInstitute of Particle and Nuclear PhysicsFaculty of Mathematics and Physics Charles University, Prague, Czech Republic

cejnar @ ipnp.troja.mff.cuni.cz

Kazimierz 2010

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* According to Empedocles (cca.490-430 BC), the real world, Cosmos, is an interference of Sphairos, an exquisite world of perfect order originating in symmetry, and Chaos, a world of complete disorder which results from a lack of symmetry.

Invariant symmetryÞ commutation of Hamiltonian

with generators of a certain group (group of invariant symmetry)

Þ conservation laws

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Breakdown of dynamical symmetry

Signatures of symmetry do not vanish instantly:

Quasi dynamical symmetry (Rowe… 1988…)The actual dynamical symmetry of the system is broken but the system partly behaves as if the symmetry were effectively preserved.

Partial dynamical symmetry (Leviatan… 1992…)All conserved quantum numbers preserved for a part of states or a part of conserved quantum numbers preserved for all states. All kinds of dynamical symmetry

relevant in physics of nuclear collective motions. These motions are therefore mostly thought to be regular...

Symmetry

Breakdown of integrability

Regular character of classical motions is preserved for some orbits: KAM theorem (Kolmogorov, Arnold, Moser 1954-63)Þ “quasi integrability”Þ “partial regularity”

Dynamical symmetryÞ symmetry of a particular system with respect to

a dynamical group – a higher group than the one following from the invariance requirements

Þ invariant symmetry with respect to the dynamical group can be broken, but a number of motion integrals remains preserved

Þ dynamical symmetry integrability

perfect order

Quantum chaosÞ no genuinely quantum definition of chaos

(linearity & quasi periodicity of quantum mechanics)

Þ chaos studied in connection with classical limitÞ Bohigas conjecture (1984): Chaos on

quantum level affects statistical properties of discrete energy spectra. Chaotic systems yield spectral correlations consistent with Gaussian random matrix model.

Classical chaosÞ exponential sensitivity to initial

conditions (“butterfly wing effect”)Þ practical loss of predictabilityÞ quasi ergodic trajectories in the

phase space

Chaos

ω=0.62

Nuclei show neat signatures of quantum chaos!• data from neutron and

proton resonances: Bohigas, Haq, Pandey (1983)

• ensemble of low-energy levels: Von Egidy et al. (1987)

(Wigner)

112 )(

Nearest-neighbor spacing

1

1

ii

iii EE

EEs

Brody distribution interpolates between Poisson (ω=0) … orderWigner (ω=1) … chaos

1

e)(

sssP

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1) Single-particle dynamics Nucleonic motions in deformed nuclear potentials

?

Origin of chaos in atomic nuclei

2) Collective dynamics Nuclear vibrations and rotations

a) Interacting Boson Model (IBM) Iachello, Arima 1975

• Alhassid, Whelan, Novoselsky: PRL 65, 2971 (1990); PRC 43, 2637 (1991); PRC 45, 1677 (1992); PRL 67, 816 (1991); NPA 556, 42 (1993)

• Paar, Vorkapic, Dieperink: PLB 205, 7 (1988); PRC 41, 2397 (1990), PRL 69, 2184 (1992)• Mizusaki et al.: PLB 269, 6 (1991) • Canetta, Maino: PLB 483, 55 (2000)• Cejnar, Jolie, Macek, Casten, Dobeš, Stránský: PLB 420, 241 (1998); PRE 58, 387 (1998); PRL 93,

132501 (2004); PRC 75, 064318 (2007), PRC 80, 014319 (2009), PRC 82, 014308 (2010), PRL 105, 072503 (2010)

b) Geometric Collective Model (GCM) Bohr 1952

• Cejnar, Stránský, Kurian, Hruška: PRL 93, 102502 (2004); PRC 74, 014306 (2006); PRE 79, 046202 (2009), PRE 79, 066201 (2009), JP Conf.Ser. 239, 012002 (2010)

• Arvieu, Brut, Carbonell, Touchard: PRA 35, 2389 (1987)• Rozmej, Arvieu: NPA 545, C497 (1992)• Heiss, Nazmitdinov, Radu: PRL 72, 2351 (1994); PRL 73, 1235 (1994); PRC 52, 3032

(1995)

Many-body dynamics Complex interactions of all particles in the nucleustoo difficult => two complementary simplifications:

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Angular momentum → 0

.....][5][2

35][5.....][

2

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

H

)1(* ][10 mm iJ

Geometric Collective Model

V

T

zyx

CBAK

JH ...3cos...

2

1

),(2432

2

22

,,

2'

rot

22vib 2

1yxK

T

Þ effectively 2D system

PAS2 |Re2sin yPrincipal Axes System

PAS0 |Recos x

Shape variables

…corresponding tensor of momenta

quadrupole tensor of collective coordinates ( 2 shape + 3 Euler angles = 5D )

Hamiltonian

A. Bohr 1952Gneuss et al. 1969

neglect …

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x

y

Geometric Collective Model

neglect higher-order terms

…corresponding tensor of momenta

quadrupole tensor of collective coordinates ( 2 shape + 3 Euler angles = 5D )

.....][5][2

35][5.....][

2

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

H

PAS2 |Re2sin yPrincipal Axes System

PAS0 |Recos x

A. Bohr 1952Gneuss et al. 1969

x

y

Hamiltonian

V

T

zyx

CBAK

JH ...3cos...

2

1

),(2432

2

22

,,

2'

rot

22vib 2

1yxK

T

sphericalprolate

oblate

B

AC >0

Shape-phase diagram

Shape variables

neglect …

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.....][5][2

35][5.....][

2

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

H

independent scales

energy time coordinates

External parameters

ACB2

… but a fixed value of Planck constant

K

2

1) “shape” parameter

2) “classicality” parameter

path crossing all parabolas Þ all equivalent classes of Hamiltonians

integrability

Geometric Collective Model

Two essential parameters

B

A

prolate

oblate

spherical

C >0

Hamiltonian

Stránský, Cejnar… 2004 ……. 2010

neglect …

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Geometric Collective Model

totregreg /f

chaos order

Stránský, Cejnar… 2004 ……. 2010

Classical chaos

Regular phase space fraction

map of the degree of chaos

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Geometric Collective Model

totregreg /f

chaos order

Stránský, Cejnar… 2004 ……. 2010

Classical chaos

Regular phase space fraction

map of the degree of chaos

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x

y

x

y

x

y

x

y

convex

concave

convex

concave

change of the shape of the border of the accessible domain in the xy plane

Classical chaos

J = 0, E = 0, A = −5.05, B = C = K = 1

50,000 passages of 52 randomly chosen trajectories throughthe section y=0

Geometric Collective Model

x

yx

Stránský, Cejnar… 2004 ……. 2010

Poincaré section

examples of trajectories and a Poincaré section

x

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J = 0

2

22

22vib

21

21

K

KT yx

Two quantization options

432

2222322

3cos

)()3()(

CBA

yxCxyxByxAV

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

H

2

2

2

2

2

2

2

22

vib

112

2

K

yxKT

3sin

3sin11

2 24

4

2

vib KT

The 2 options differ also in the metric (measure) for calculating matrix elements.

► Possibility to test Bohigas conjecture in different quantization schemes.

with additional constraints

(to avoid quasi-degeneracies due to the symmetry of V)

),(),(

),(),( 32

k

even / odd

K

2 classicality parameter

Geometric Collective ModelStránský, Cejnar… 2004 ……. 2010

Quantum chaos

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Geometric Collective ModelStránský, Cejnar… 2004 ……. 2010

Quantum chaoscomparison of classical and quantal measures

freg … classical regular fraction 1−ω … adjunct of Brody parameter

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1) Quasi-2D angular momentum

2) Hamiltonian perturbation

Choice of P

Geometric Collective ModelStránský, Cejnar… 2004 ……. 2010

Quantum chaosdeparture from integrable regime

2) <H’>

1) <L2>

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iii PP

iE

Visual method by A. Peres (1984)

B=0.62

Geometric Collective ModelStránský, Cejnar… 2004 ……. 2010

Quantum chaos

1) <L2>

complex mixture of regular and chaotic patterns: ordered vs. chaotic states

2) <H’>

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Visual method by A. Peres (1984)

■ ● ♦

B=0.62

Geometric Collective ModelStránský, Cejnar… 2004 ……. 2010

Quantum chaoscomplex mixture of regular and chaotic patterns: ordered vs. chaotic states

1) <L2>

2) <H’>

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Visual method by A. Peres (1984)

Macek, Cejnar, Jolie… 200713/15

B

ddn~

d

)()(1

),(2d QQ

Nn

NH

)2(]~

[~

)( dddssdQ

Interacting Boson ModelMore than 1 classical control parameter

But there exist regions of almost full compatibility

Þ multi-dimensional chaotic map.

J = 0, E = 0

integrableregime

with the GCM.

Interacting Boson ModelMacek, Dobeš, Cejnar… 201014/15

Consequences of regularity for the adiabatic separation of intrinsic and collective motions: rotational bands exist even at very high excitation energies if the corresponding region of the J =0 spectrum is regular.

• Product of 0+-2+ and 0+-4+

correlation coefficients for intrinsic wave

functions in the given “band”

• Energy ratio for 4+ and 2+ states in

a given “band”

• Selection of hypothetical bands of rotational states based on the maximal correlation of the intrinsic SU(3) structures.

• Classical regular fraction in the

respective energy region

0 15 30 J

N = 30

0

2

intrinsic wave functions for various band members in the SU(3)

basis

),(

integrable regime

G C M J = 0, E = 0

Conclusions• Nuclear collective motions exhibit an intricate interplay or regular and

chaotic features (despite the presumption that collective = regular).• Models of collective motions may serve as a laboratory for general

studies of chaos (profit from the coexistence of simplicity and complexity).

• Order/Chaos have relevant nuclear-structure consequences (e.g. for the adiabatic separation of collective and intrinsic motions etc.).

Thanks to Pavel Stránský (now UNAM Mexico) Michal Macek (soon Uni Jerusalem)

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Appendices

J=0, A=−0.84, B=C=K=1

Dependenceon energy

32 105.2/

1

09.1

1

K

C

B

A

GCM

GCM

32 105.2/

1

09.1

1

K

C

B

A

Peres latticesA visual method to study quantum chaos in 2D systems due to Asher Peres, PRL 53, 1711 (1984)

► In any system there exists infinite number of integrals of motions: e.g. time averages of an arbitrary quantity along individual orbits (note: in fully or partly chaotic systems, these integrals do not allow one to build action-angle variables since they are strongly nonanalytic)

0,)(lim0

H1

HPdttPPT

TT

(1934-2005)

► One can construct a lattice: energy versus value of

► Quantum counterparts of such observables can be found:

► In a fully regular (integrable) system, the lattice is always ordered (the new integral of motion is constant on tori => it is a function of actions)

► In a chaotic system, the lattice is disordered► In a mixed regular & chaotic system, the lattice is partly ordered & disordered

P

E

regular mixed chaotic

iiii PPP iE

Relation to the regular fraction GCM

4104

41025

410100

K/2classicality parameter

B=1.09 2D even