Vladimir Zelevinsky
NSCL/ Michigan State University
FUSTIPEN, Caen
June 3, 2014

• Naftali Auerbach (Tel Aviv)
• B. Alex Brown (NSCL, MSU)
• Mihai Horoi (Central Michigan University)
• Victor Flambaum (Sydney)
• Declan Mulhall (Scranton University)
• Roman Sen’kov (CMU)
• Alexander Volya (Florida State University)

OUTLINE
* Symmetries
* Mesoscopic physics
* From classical to quantum chaos
* Chaos as useful practical tool
* Nuclear level density
* Chaotic enhancement
* Parity violation
* Nuclear structure and EDM

PHYSICS of ATOMIC NUCLEI in XXI CENTURY
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Limits of stability - drip lines, superheavy…
Nucleosynthesis in the Universe; charge asymmetry; dark matter…
Structure of exotic nuclei
Magic numbers
Collective effects – superfluidity, shape transformations, …
Mesoscopic physics – chaos, thermalization, level and width statistics, …
^ random matrix ensembles
^ physics of open and marginally stable systems
^ enhancement of weak perturbations
^ quantum signal transmission
Neutron matter
Applied physics – isotopes, isomers, reactor technology, …
Fundamental physics and violation of symmetries:
^ parity
^ electric dipole moment (parity and time reversal)
^ anapole moment (parity)
^ temporal and spatial variation of fundamental constants

FUNDAMENTAL SYMMETRIES
Uniformity of space = momentum conservation P
Uniformity of time = energy conservation E
Isotropy of space = angular momentum conservation L
Relativistic invariance
Indistinguishability of identical particles
Relation between spin and statistics
Bose – Einstein (integer spin 0,1, …)
Fermi – Dirac (half-integer spin 1/2, 3/2, …)

DISCRETE SYMMETRIES
Coordinate inversion P vectors and pseudovectors, scalars and pseudoscalars
Time reversal T microscopic reversibility, macroscopic irreversibility
Charge conjugation C excess of matter in our Universe
Conserved in strong and electromagnetic interactions
C and P violated in weak interactions
T violated in some special meson decays ( Universe?
)
C P T strictly valid

POSSIBLE NUCLEAR ENHANCEMENT of weak interactions
* Close levels of opposite parity
= near the ground state (accidentally)
= at high level density – very weak mixing?
(statistical = chaotic) enhancement
* Kinematic enhancement
* Coherent mechanisms
= deformation
= parity doublets
= collective modes
* Atomic effects
* Condensed matter effects

MESOSCOPIC SYSTEMS :
MICRO ----- MESO ----- MACRO
• Complex nuclei
• Complex atoms
• Complex molecules (including biological)
• Cold atoms in traps
• Micro- and nano- devices of condensed matter
• --------
• Future quantum computers
Common features : quantum bricks, interaction, complexity; quantum chaos, statistical regularities; at the same time – individual quantum states

Classical regular billiard
Symmetry preserves unfolded momentum

Regular circular billiard

Stadium billiard – no symmetries
A single trajectory fills in phase space

Regular circular billiard
Angular momentum conserved
CLASSICAL CHAOS
Cardioid billiard
No symmetries

MANY-BODY QUANTUM CHAOS AS AN INSTRUMENT
SPECTRAL STATISTICS – signature of chaos
missing levels
purity of quantum numbers *
calculation of level density (given spin-parity) *
presence of time-reversal invariance
EXPERIMENTAL TOOL – unresolved fine structure
width distribution
damping of collective modes
NEW PHYSICS statistical enhancement of weak perturbations
(parity violation in neutron scattering and fission ) *
mass fluctuations
chaos on the border with continuum
THEORETICAL CHALLENGES
order out of chaos
- chaos and thermalization *
- development of computational tools *
new approximations in many-body problem

MANY-BODY QUANTUM CHAOS AS AN INSTRUMENT
SPECTRAL STATISTICS – signature of chaos
missing levels
purity of quantum numbers *
calculation of level density (given spin-parity) *
presence of time-reversal invariance
EXPERIMENTAL TOOL – unresolved fine structure
width distribution
damping of collective modes
NEW PHYSICS statistical enhancement of weak perturbations
(parity violation in neutron scattering and fission ) *
mass fluctuations
chaos on the border with continuum
THEORETICAL CHALLENGES
order out of chaos
- chaos and thermalization *
- development of computational tools *
new approximations in many-body problem

Fragments of six different spectra
50 levels, rescaled
(a), (b), (c) – exact symmetries
(e), (f) – mixed symmetries
(a) Neutron resonances in 167Er, I=1/2
(b) Proton resonances in 49V, I=1/2
(c) I=2,T=0 shell model states in 24Mg
(d) Poisson spectrum P(s)=exp(-s)
(e) Neutron resonances in 182Ta, I=3 or 4
(f) Shell model states in 63Cu, I=1/2,…,19/2
SPECTRAL STATISTICS
Arrows: s < (1/4) D

Nearest level spacing distribution
( simplest signature of chaos )
Regular system
Chaotic system
Disordered spectrum P(s) = exp(-s)
= Poisson distribution
“Aperiodic crystal” = Wigner P(s)
Wigner distribution

RANDOM MATRIX ENSEMBLES
• universality classes
• all states of similar complexity
• local spectral properties
• uncorrelated independent matrix elements
Gaussian Orthogonal Ensemble (GOE) – real symmetric
Gaussian Unitary Ensemble (GUE) – Hermitian complex
Extreme mathematical limit of quantum chaos!
Many other ensembles: GSE, BRM, TBRM, …

LEVEL DYNAMICS
(shell model of 24Mg as a typical example)
Fraction (%) of realistic strength
From turbulent to laminar level dynamics
Chaos due to particle interactions at high level density

Fragments of six different spectra
50 levels, rescaled
(a), (b), (c) – exact symmetries
(e), (f) – mixed symmetries
(a) Neutron resonances in 167Er, I=1/2
(b) Proton resonances in 49V, I=1/2
(c) I=2,T=0 shell model states in 24Mg
(d) Poisson spectrum P(s)=exp(-s)
(e) Neutron resonances in 182Ta, I=3 or 4
(f) Shell model states in 63Cu, I=1/2,…,19/2
Arrows: s < (1/4) D

Nearest level spacing distributions for the same cases
(all available levels)

NEAREST LEVEL SPACING DISTRIBUTION at interaction strength 0.2 of the realistic value
WIGNER-DYSON distribution
( the weakest signature of quantum chaos )

R. Haq et al. 1982
SPECTRAL RIGIDITY
Nuclear Data Ensemble
1407 resonance energies
30 sequences
For 27 nuclei
Neutron resonances
Proton resonances
(n,gamma) reactions
Regular spectra = L/15
(universal for small L)
Chaotic spectra
= a log L +b for L>>1

Purity ?
Missing levels ?
235U, I=3 or 4,
960 lowest levels f=0.44
0, 4% and 10% missing
Data agree with f=(7/16)=0.44
and
4% missing levels
D. Mulhall, Z. Huard, V.Z.,
PRC 76, 064611 (2007).

Structure of eigenstates
Whispering Gallery
Bouncing Ball
Ergodic behavior
With fluctuations

COMPLEXITY of QUANTUM STATES
RELATIVE!
Typical eigenstate:
GOE:
Porter-Thomas distribution for weights:
Neutron width of neutron resonances as an analyzer
(1 channel)

Cross sections in the region of giant quadrupole resonance
Resolution:
(p,p’) 40 keV
(e,e’) 50 keV
Unresolved fine structure
D = (2-3) keV

INVISIBLE FINE STRUCTURE , or catching the missing strength with poor resolution
Assumptions : Level spacing distribution ( Wigner )
Transition strength distribution ( Porter-Thomas )
Parameters : s=D/<D>, I=(strength)/<strength>
Two ways of statistical analysis: <D(2+)>= 2.7 (0.9) keV and
3.1 (1.1) keV.
“Fairly sofisticated, time consuming and finally successful analysis”

TYPICAL COMPUTATIONAL PROBLEM
DIAGONALIZATION OF HUGE MATRICES
( dimensions dramatically grow with the particle number )
Practically we need not more than few dozens – is the rest just useless garbage ?
Process of progressive truncation –
* how to order ?
* is it convergent ?
* how rapidly ?
* in what basis ?
* which observables ?

Banded GOE Full GOE
GROUND STATE ENERGY OF RANDOM MATRICES
EXPONENTIAL CONVERGENCE
SPECIFIC PROPERTY of RANDOM
MATRICES ?

ENERGY CONVERGENCE in SIMPLE MODELS
Tight binding model Shifted harmonic oscillator

REALISTIC
SHELL 48 Cr
MODEL
Excited state
J=2, T=0
EXPONENTIAL
CONVERGENCE !
E(n) = E + exp(-an) n ~ 4/N

Local density of states in condensed matter physics

AVERAGE STRENGTH FUNCTION
Breit-Wigner fit (dashed)
Gaussian fit (solid)
Exponential tails

REALISTIC
SHELL
MODEL
EXCITED STATES
51Sc
1/2-, 3/2-
Faster convergence:
E(n) = E + exp(-an) a ~ 6/N

52
Cr
Ground and excited states
Ni
Superdeformed headband

EXPONENTIAL
CONVERGENCE
OF SINGLE-PARTICLE
OCCUPANCIES
(first excited state J=0)
52
Cr
Orbitals f5/2 and f7/2

CONVERGENCE REGIMES
Fast convergence
Exponential convergence
Power law
Divergence

Shell Model and Nuclear Level Density
M. Horoi, J. Kaiser, and V. Zelevinsky, Phys. Rev. C 67 , 054309 (2003).
M. Horoi, M. Ghita, and V. Zelevinsky, Phys. Rev. C 69 , 041307(R) (2004).
M. Horoi, M. Ghita, and V. Zelevinsky, Nucl. Phys. A785 , 142c (2005).
M. Scott and M. Horoi, EPL 91 , 52001 (2010).
R.A. Sen’kov and M. Horoi, Phys. Rev. C 82, 024304 (2010).
R.A. Sen’kov, M. Horoi, and V. Zelevinsky, Phys. Lett. B702 , 413 (2011).
R. Sen’kov , M. Horoi, and V. Zelevinsky, Computer Physics Communications
184 , 215 (2013).
Statistical Spectroscopy
:
S. S. M. Wong , Nuclear Statistical Spectroscopy (Oxford, University Press, 1986).
V.K.B. Kota and R.U. Haq , eds., Spectral Distributions in Nuclei and
Statistical Spectroscopy (World Scientific,
Singapore, 2010).

28
Si
Diagonal matrix elements of the Hamiltonian in the mean-field representation
Partition structure in the shell model
(a) All 3276 states ; (b) energy centroids

Energy dispersion for individual states is nearly constant
(result of geometric chaoticity !)
Also in multiconfigurational method (hybrid of shell model and density functional)

CLOSED MESOSCOPIC SYSTEM at high level density
Two languages : individual wave functions thermal excitation
* Mutually exclusive ?
* Complementary ?
* Equivalent ?
Answer depends on thermometer

Temperature T(E)
T(s.p.) and T(inf) = for individual states !

28 Si
J=0 J=2 J=9
Single – particle occupation numbers
Thermodynamic behavior identical in all symmetry classes
FERMI-LIQUID PICTURE

J=0
Artificially strong interaction (factor of 10)
Single-particle thermometer cannot resolve spectral evolution

Gaussian level density
839 states (28 Si )
EFFECTIVE TEMPERATURE of INDIVIDUAL STATES
From occupation numbers in the shell model solution ( dots )
From thermodynamic entropy defined by level density ( lines )

Invariant entropy
• Invariant entropy is basis independent
• Indicates the sensitivity of eigenstate
 to parameter G in interval [G,G+

G]

24
rea listic nucleus
T= 1
pa iring
Normal
T= 0
pa iring strength of T= 0 pairing
Contour plot of invariant correlational entropy showing a phase diagram as a function of T=1 pairing ( λ
T=1
) and T=0 pairing ( λ
T=0
); three plots indicate phase diagram as a function of non-pairing matrix elements ( λ np
) . Realistic case is λ
T=1
= λ
T=0
= λ np
=1

N - scaling
N – large number of “simple” components in a typical wave function
Q – “simple” operator
Single – particle matrix element
Between a simple and a chaotic state
Between two fully chaotic states

STATISTICAL ENHANCEMENT
Parity nonconservation in scattering of slow polarized neutrons
Coherent part of weak interaction
Single-particle mixing
Chaotic mixing
Neutron resonances in heavy nuclei
Kinematic enhancement up to
10%

235 U
Los Alamos data
E=63.5 eV
10.2 eV -0.16(0.08)%
11.3 0.67(0.37)
63.5 2.63(0.40) *
83.7 1.96(0.86)
89.2 -0.24(0.11)
98.0 -2.8 (1.30)
125.0 1.08(0.86)
Transmission coefficients for two helicity states
(longitudinally polarized neutrons)

Parity nonconservation in fission
Correlation of neutron spin and momentum of fragments
Transfer of elementary asymmetry to ALMOST MACROSCOPIC LEVEL –
What about 2 nd law of thermodynamics?
Statistical enhancement – “hot” stage ~
- mixing of parity doublets
Angular asymmetry – “cold” stage,
fission channels , memory preserved
Complexity refers to the natural basis (mean field)

Parity violating asymmetry
Parity preserving asymmetry
[Grenoble]
A. Alexandrovich et al . 1994
Parity non-conservation in fission by polarized neutrons – on the level up to 0.001

Fission of
233 U by cold polarized neutrons,
(Grenoble)
A. Koetzle et al. 2000
Asymmetry determined at the “hot” chaotic stage

• STATISTICAL MECHANICS
• PHASE TRANSITIONS
• COMPLEXITY
• INFORMATICS
• CRYPTOGRAPHY
• LARGE FACILITIES
• LIVING ORGANISMS
• HUMAN BRAIN
• ECONOPHYSICS
• FUNDAMENTAL SYMMETRIES
• PARTICLE PHYSICS
• COSMOLOGY

Boris V. CHIRIKOV (1928 – 2008)

B. V. CHIRIKOV :
… The source of new information is always chaotic. Assuming farther that any creative activity, science including, is supposed to be such a source, we come to an interesting conclusion that any such activity has to be
(partly!) chaotic.

spin
T-reversal d
Observation of the dipole moment is
Observation of the dipole moment is an indication of parity and timean indication of parity and timereversal violation reversal violation d(199Hg)<3.1
x
10
-29 e.cm
Limits on EDM for the electron
Experiment: < 8.7 x 10 -29 e.cm
Standard model ~ 10 -38 e.cm
Physics beyond SM ~ 10 -28 e.cm
Neutron
EDM < 2.9 x 10
-26 spin
P-reversal d d spin d spin

J.F.C. Cocks et al. PRL 78 (1997) 2920.
Half-live
219 Rn 4 s
221 Rn 25 m

Half-live
223 Rn 24 m
223 Ra 11 d

Half-live
225 Ra 15 d
227 Ra 42 m

|+

|-

Parity conservation :
Mixture by weak interaction W
Small parity violating interaction W
Perturbed ground state
Non-zero Schiff moment

C O N C L U S I O N
Nuclear ENHANCEMENTS
VERY HARD TIME-CONSUMING
EXPERIMENTS…

S U M M A R Y
1. Many-body quantum chaos as universal phenomenon at high level density
2. Experimental, theoretical and computational tool
3. Role of incoherent interactions not fully understood
4. Chaotic paradigm of statistical thermodynamics
5. Nuclear structure mechanisms for enhancement of tiny effects, chaoric and regular