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CHAOTIC DYNAMICS AND QUANTUM STATE PATTERNS IN COLLECTIVE MODELS OF NUCLEI
Pavel Stránský
Collaborators:
Michal Macek, Pavel Cejnar
Institute of Particle and Nuclear Phycics,
Faculty of Mathematics and Physics,
Charles University in Prague, Czech Republic
Jan Dobeš
Nuclear Research Institute,
Řež, Czech Republic
Alejandro Frank, Emmanuel Landa,
Irving Morales
Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México
ECT* Seminar*
13 January 2012

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CHAOTIC DYNAMICS AND QUANTUM STATE PATTERNS IN COLLECTIVE MODELS OF NUCLEI
1. Classical chaos
- Stable x unstable trajectories
- Poincaré sections: a manner of visualization
- Fraction of regularity: a measure of chaos
2. Quantum chaos
- Statistics of the quantum spectra, spectral correlations
- 1/f noise: long-range correlations
- Peres lattices: ordering of quantum states
3. Applications in the nuclear physics
- Geometric collective model and Interacting boson model
- Quantum – classical correspondence
- Adiabatic separation of the collective and intrinsic motion

3. (analysis of trajectories)
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Classical Chaos
(analysis of trajectories)

4. Quasiperiodic motion on a toroid
1. Classical chaos
Hamiltonian systems
State of a system: a point in the 4D phase space
Conservative system:
Trajectory restricted to 3D hypersurface
Integrals of motion:
Connected with additional symetries
Integrable system:
Number of independent integrals of motion =
number of degrees of freedom
Canonical transformation to action-angle variables J2
Quasiperiodic motion on a toroid J1

5. property of nonintegrable systems
1. Classical chaos
Hamiltonian systems
State of a system: a point in the 4D phase space
Conservative system:
Trajectory restricted to 3D hypersurface
Integrals of motion:
Connected with additional symetries
Chaotic behaviour:
property of nonintegrable systems
Integrable system:
Number of independent integrals of motion =
number of degrees of freedom
Canonical transformation to action-angle variables J2
Quasiperiodic motion on a toroid J1

6. Poincaré sections px y x chaotic case – “fog” px y = 0 x
1. Classical chaos
Poincaré sections
Generic conservative system of 2 degrees of freedom
We plot a point every time when the trajectory crosses the plane y = 0 px y x
ordered case – “circles”
chaotic case – “fog” x px
2D classical system
Section at
y = 0
Different initial conditions at the same energy

7. Fraction of regularity
1. Classical chaos
Fraction of regularity
Measure of classical chaos
Surface of the section covered with regular trajectories
SALI method x Lyapunov exponent
Other method – random generating of trajectories, we must, however, ensure that the distribution is uniform
Total kinematically accessible surface of the section x px
REGULAR area
CHAOTIC area
freg=0.611

8. Classical chaos –Hypersensitivity to the initial conditions
Quasiperiodic X unstable trajectories
Classical chaos –Hypersensitivity to the initial conditions
1. Lyapunov exponent
Divergence of two neighboring trajectories
Regular: at most polynomial divergence
Chaotic: exponential divergence
2. SALI (Smaller Alignment Index)
two divergencies
fast convergence towards zero for chaotic trajectories
Ch. Skokos, J. Phys. A: Math. Gen 34, (2001); 37 (2004), 6269

9. (analysis of energy spectra)
Quantum Chaos
(analysis of energy spectra)

10. Semiclassical theory of chaos
2. Quantum chaos
Semiclassical theory of chaos
Spectral density:
smooth part
oscillating part
given by the volume of the classical phase space
Gutzwiller formula
(given by the sum of all classical periodic orbits and their repetitions)
The oscillating part of the spectral density can give relevant information about quantum chaos (related to the classical trajectories)
Unfolding:
A transformation of the spectrum
that removes the smooth part of the level density
Note: Improved unfolding procedure using the Empirical Mode Decomposition method in: I. Morales et al., Phys. Rev. E 84, (2011)

11. Quantum chaos: Spectral statistics
no level interaction
level repulsion E
Nearest-neighbor spacing distribution
Gaussian OrthogonalEnsemble
Gaussian Unitary
Ensemble
Gaussian Symplectic
Ensemble
Remember: Black – regular, Red – chaotic
Brody parameter – artificially interpolates between Poisson and GOE distribution
1-omega – adjunct of the Brody parameter
Properties of ensembles, not of individual levels
Connection with eigenvalues of random matrix ensembles
Poisson ensembles – uncorrelated levels, level clustering
GOE – level repulsion
GOE – T-invariance, R-invariance (symmetric matrix)
GSE – T-invariance, half-spin size
Symplectic matrices - O=M^T O M, where M = (0 1//-1 0)
Wigner surmise (1950)
P(s)
REGULAR system
CHAOTIC systems
Ensembles of random matrices
Transformation H
T invariance
Angular momentum
R invariance
GOE
Orthogonal
Symmetric
YES n
n/2
GUE
Unitary
Hermitian NO
GSE
Symplectic
O. Bohigas, M. J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984), 1
M.V. Berry, M.Tabor, Proc. Roy. Soc. A 356, 375 (1977)

12. Spectral statistics Poisson Wigner REGULAR system CHAOTIC system
2. Quantum chaos
Spectral statistics
Nearest-neighbor spacing distribution
Poisson
Wigner
P(s) s
REGULAR system
CHAOTIC system
Properties of ensembles, not of individual levels
distribution
parameter w
Brody
- Artificial interpolation between Poisson and GOE distribution
- Measure of chaoticity of quantum systems
- Tool to test classical-quantum correspondence

13. They are also extensively studied experimentally
2. Quantum chaos
Quantum chaos - examples
Billiards
Dirichlet boundary condition – E
Neumann boundary condition - B
They are also extensively studied experimentally
Schrödinger equation:
(for wave function)
Helmholtz equation:
(for intensity of el. field)

14. Quantum chaos - applications
Riemann z function:
Prime numbers
Riemann hypothesis:
All points z(s)=0 in the complex plane lie on the line s=½+iy (except trivial zeros on the real exis s=–2,–4,–6,…)
GUE
Zeros
of z function

15. Quantum chaos - applications
GOE
Correlation matrix
of the human EEG signal
P. Šeba, Phys. Rev. Lett. 91 (2003),

16. 1/f noise Power spectrum a = 2 a = 1 a = 2 CHAOTIC system
2. Quantum chaos
Ubiquitous in the nature (many time signals or space characteristics of complex systems have 1/f power spectrum)
1/f noise
- Fourier transformation of the time series constructed from energy levels fluctuations
dn = 0 dk d4
Power spectrum d3 k d2
d1 = 0
a = 2
a = 1
a = 2
CHAOTIC system
REGULAR system
a = 1
Direct comparison of
3 measures of chaos
A. Relaño et al., Phys. Rev. Lett. 89, (2002)
E. Faleiro et al., Phys. Rev. Lett. 93, (2004)
J. M. G. Gómez et al., Phys. Rev. Lett. 94, (2005)

17. Peres lattices Integrable nonintegrable Quantum system:
2. Quantum chaos
Peres lattices
Quantum system:
Infinite number of of integrals of motion can be constructed
(time-averaged operators P):
Lattice: energy Ei versus value of
lattice always ordered for any operator P
partly ordered,
partly disordered
Integrable
nonintegrable
Tori remnants in the classical chaotic system
Einstein-Brillouin-Keller quantization
3000 dots in each figure, no degeneration in energies
B = 0
B = 0.445
<P>
regular
<P>
chaotic
regular E E
A. Peres, Phys. Rev. Lett. 53, 1711 (1984)

18. 3. Application to the collective models of nuclei
K – meaning od mass
Slightly more general Hamiltonian than that was presented by Dimitris.
Spherical tensor of rank 2
This Hamiltonian may be regarded as …
Scalar.

19. Geometric collective model
3a. Geometric collective model
Geometric collective model
Surface of homogeneous nuclear matter:
(even-even nuclei – collective character of the lowest excitations)
Monopole deformations l = 0
- “breathing” mode
- Does not contribute due to the incompressibility of the nuclear matter
Dipole deformations l = 1
Related to the motion of the center of mass
- Zero due to momentum conservation

20. Geometric collective model
3a. Geometric collective model
Geometric collective model
Surface of homogeneous nuclear matter:
Quadrupole deformations l = 2
Quadrupole tensor of collective coordinates
(2 shape parameters, 3 Euler angles)
Corresponding tensor of momenta
T…Kinetic term
V…Potential
Neglect higher order terms
neglect
4 external parameters
G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)

21. Geometric collective model
3a. Geometric collective model
Geometric collective model
Surface of homogeneous nuclear matter:
Quadrupole deformations l = 2
Quadrupole tensor of collective coordinates
(2 shape parameters, 3 Euler angles)
Corresponding tensor of momenta
T…Kinetic term
V…Potential
Neglect higher order terms
neglect
4 external parameters
Adjusting 3 independent scales
Scaling properties
energy (Hamiltonian)
size (deformation)
time
1 “shape” parameter
1 “classicality” parameter
(order parameter)
sets absolute density of quantum spectrum (irrelevant in classical case)
P. Stránský, M. Kurian, P. Cejnar, Phys. Rev. C 74, (2006)

22. Principal Axes System (PAS)
3a. Geometric collective model
Principal Axes System (PAS) g
Shape variables: b
Beta, gamma – polar coordinates in an abstract plane
Shape-phase structure
Phase separatrix B V V A b
C=1
Deformed shape b
Spherical shape

23. (a) 5D system restricted to 2D (true geometric model of nuclei)
3a. Geometric collective model
Dynamics of the GCM
Nonrotating case J = 0!
Classical dynamics
– Hamilton equations of motion
Quantization
– Diagonalization in the oscillator basis
2 physically important quantization options
(with the same classical limit):
An opportunity to test the Bohigas conjecture in different quantization schemes
(a) 5D system restricted to 2D (true geometric model of nuclei)
(b) 2D system

24. (a) 5D system restricted to 2D (true geometric model of nuclei)
3a. Geometric collective model
Peres operators
Independent
Peres operators in GCM H’ L2 2D 5D
Nonrotating case J = 0!
(a) 5D system restricted to 2D (true geometric model of nuclei)
(b) 2D system
P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79, (2009); (2009)

25. Complete map of classical chaos in GCM
3a. Geometric collective model
Complete map of classical chaos in GCM
Integrability
Veins of regularity
chaotic
Shape-phase transition
regular
“Arc of regularity”
control parameter
Global minimum and saddle point
HO approximation
Region of phase transition

26. Peres lattices in GCM Integrable Increasing perturbation
3a. Geometric collective model
Peres lattices in GCM
Small perturbation affects only a localized part of the lattice
(The place of strong level interaction)
B = 0
B = 0.005
B = 0.05
B = 0.24
<L2>
Peres lattices for two different operators
Remnants of
regularity
<H’> E
Integrable
Increasing perturbation
Empire of chaos

27. “Arc of regularity” B = 0.62 b – g vibrations resonance 2D
3a. Geometric collective model
“Arc of regularity” B = 0.62
b – g vibrations resonance
<L2>
<VB> 2D
(different quantizations) 5D E
Connection with IBM: M. Macek et al., Phys. Rev. C 75, (2007)

28. Zoom into the sea of levels
3a. Geometric collective model
Dependence on the classicality parameter E
<L2>
Zoom into the sea of levels
Dependence of the Brody parameter on energy

29. Peres invariant classically
3a. Geometric collective model
Peres operators & Wavefunctions 2D
Peres invariant classically
Poincaré section
E = 0.2
Selected squared wave functions:
<L2>
<VB> E

30. Quantum measure (Brody)
3a. Geometric collective model
Classical and quantum
measures - comparison
B = 0.24
Classical measure
B = 1.09
Quantum measure (Brody)

31. Shortest periodic classical orbit
3a. Geometric collective model
1/f noise
Integrable case: a = 2 expected
(averaged over 4 successive sets of 8192 levels, starting from level 8000)
(512 successive sets of 64 levels)
log<S> x
Correlations we are interested in x x
Averaging of smaller intervals
Universal region
log f
Shortest periodic classical orbit

32. 1/f noise Calculation of a: a - 1 1 - w freg Mixed dynamics A = 0.25
3a. Geometric collective model
1/f noise
Mixed dynamics A = 0.25
Calculation of a:
a - 1
Each point –averaging over 32 successive sets of 64 levels in an energy window
1 - w
regularity
freg E

33. Interacting Boson Model
3b. Interacting boson model
Interacting Boson Model
K – meaning od mass
Slightly more general Hamiltonian than that was presented by Dimitris.
Spherical tensor of rank 2
This Hamiltonian may be regarded as …
Scalar.

34. IBM Hamiltonian Symmetry Dynamical symmetries (group chains)
3b. Interacting boson model
IBM Hamiltonian
- Valence nucleon pairs with l = 0, 2
s-bosons (l=0)
d-bosons (l=2)
- quanta of quadrupole collective excitations
Symmetry
U(6) with 36 generators
total number of bosons is conserved
SO(3) – total angular momentum L is conserved
Dynamical symmetries (group chains)
vibrational
g-unstable
nuclei
rotational
The most general Hamiltonian (constructed from Casimir invariants of the subgoups)

35. Classical limit via coherent states
3b. Interacting boson model
Consistent-Q Hamiltonian
d-boson number operator
quadrupole operator
U(5)
SU(3)
SO(6) 1
Arc of regularity
a – scaling parameter
Classical limit via coherent states
integrable cases
Invariant of SO(5) (seniority)
Shape phase transition
F. Iachello, A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987)

36. 3 independent Peres operators
3b. Interacting boson model
Consistent-Q Hamiltonian
d-boson number operator
quadrupole operator
U(5)
SU(3)
SO(6) 1
Casten triangle
a – scaling parameter
3 independent Peres operators
integrable cases
Invariant of SO(5) (seniority)

37. Regular lattices in integrable case
3b. Interacting boson model
Regular lattices in integrable case
- even the operators non-commuting with Casimirs of U(5) create regular lattices !
commuting
non-commuting 40 30
-10
SU(3) quasidynamical symmetry
Classsical f_reg line doesn’t reach the highest ‘quantum energies’ because of the peculiarity of the classical system – it becomes unbound near E_max – see the classical potential. This becomes more significant as chi goes away from zero.
Classical and quantum energy scales are otherwise directly comparable in the fig. 20
U(5)
limit
-20 10
-30
-40
-10
N = 40
-20
L = 0
-30
-40

38. Different invariants h = 0.5 N = 40 Arc of regularity U(5) SU(3) O(5)
3b. Interacting boson model
Arc of regularity
Different invariants
classical regularity
h = 0.5
N = 40
U(5)
SU(3) quasidynamical symmetry
Classsical f_reg line doesn’t reach the highest ‘quantum energies’ because of the peculiarity of the classical system – it becomes unbound near E_max – see the classical potential. This becomes more significant as chi goes away from zero.
Classical and quantum energy scales are otherwise directly comparable in the fig.
SU(3)
O(5)
M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 80, (2009)

39. Correspondence with GCM
3b. Interacting boson model
Arc of regularity
Different invariants
Correspondence with GCM
<L2>
classical regularity
h = 0.5
N = 40
U(5)
SU(3) quasidynamical symmetry
Classsical f_reg line doesn’t reach the highest ‘quantum energies’ because of the peculiarity of the classical system – it becomes unbound near E_max – see the classical potential. This becomes more significant as chi goes away from zero.
Classical and quantum energy scales are otherwise directly comparable in the fig.
SU(3)
O(5)
M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 80, (2009)

40. High-lying rotational bands
3b. Interacting boson model
High-lying rotational bands
η = 0.5, χ= (arc of regularity)
N = 30
L = 0
SU(3) quasidynamical symmetry
Classsical f_reg line doesn’t reach the highest ‘quantum energies’ because of the peculiarity of the classical system – it becomes unbound near E_max – see the classical potential. This becomes more significant as chi goes away from zero.
Classical and quantum energy scales are otherwise directly comparable in the fig. E

41. High-lying rotational bands
3b. Interacting boson model
High-lying rotational bands
η = 0.5, χ= (arc of regularity)
N = 30
L = 0,2
SU(3) quasidynamical symmetry
Classsical f_reg line doesn’t reach the highest ‘quantum energies’ because of the peculiarity of the classical system – it becomes unbound near E_max – see the classical potential. This becomes more significant as chi goes away from zero.
Classical and quantum energy scales are otherwise directly comparable in the fig. E

42. High-lying rotational bands
3b. Interacting boson model
High-lying rotational bands
η = 0.5, χ= (arc of regularity)
N = 30
L = 0,2,4
SU(3) quasidynamical symmetry
Classsical f_reg line doesn’t reach the highest ‘quantum energies’ because of the peculiarity of the classical system – it becomes unbound near E_max – see the classical potential. This becomes more significant as chi goes away from zero.
Classical and quantum energy scales are otherwise directly comparable in the fig. E

43. High-lying rotational bands
3b. Interacting boson model
High-lying rotational bands
η = 0.5, χ= (arc of regularity)
N = 30
L = 0,2,4,6
SU(3) quasidynamical symmetry
Classsical f_reg line doesn’t reach the highest ‘quantum energies’ because of the peculiarity of the classical system – it becomes unbound near E_max – see the classical potential. This becomes more significant as chi goes away from zero.
Classical and quantum energy scales are otherwise directly comparable in the fig.
Regular areas:
Adiabatic separation of the intrinsic and collective motion E

44. Numerical evidence of the rotational bands
3b. Interacting boson model
Numerical evidence of the rotational bands
Pearson correlation coefficient
SU(3) quasidynamical symmetry
Classsical f_reg line doesn’t reach the highest ‘quantum energies’ because of the peculiarity of the classical system – it becomes unbound near E_max – see the classical potential. This becomes more significant as chi goes away from zero.
Classical and quantum energy scales are otherwise directly comparable in the fig.
=10/3 for rotational band
Classical fraction of regularity
M. Macek, J. Dobeš, P. Stránský, P. Cejnar, Phys. Rev. Lett. 105, (2010)
M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 81, (2010)

45. Components of eigenvectors in SU(3) basis
3b. Interacting boson model
Components of eigenvectors in SU(3) basis
li – i-th eigenstate with angular momentum l
RB Appears naturally in the SU(3) basis
low-lying band
highly excited band
Quasidynamical symmetry
The characteristic features of a dynamical symmetry (the existence of the rotational bands here) survive despite the dynamical symmetry is broken
SU(3) quasidynamical symmetry
Classsical f_reg line doesn’t reach the highest ‘quantum energies’ because of the peculiarity of the classical system – it becomes unbound near E_max – see the classical potential. This becomes more significant as chi goes away from zero.
Classical and quantum energy scales are otherwise directly comparable in the fig.
Non-rotational sequence of states
indices labeling the intrinsic b, g excitations (SU(3) basis states)

46. Thank you for your attention
Enjoy the last slide!
Summary
Thank you for your attention
Peres lattices
Allow visualising quantum chaos
Capable of distinguishing between chaotic and regular parts of the spectra
Freedom in choosing Peres operator
1/f Noise
Effective method to introduce a measure of chaos using long-range correlations in quantum spectra
Geometrical Collective Model
Complex behavior encoded in simple equations
(order-chaos-order transition)
Possibility of studying manifestations of both classical and quantum chaos and their relation
Good classical-quantum correspondence found even in the mixed dynamics regime
Interacting boson model
Peres operators come naturally from the Casimirs of the dynamical symmetries groups
Evidence of connection between chaoticity and separation of collective and intrinsic motions
~stransky