Lecture - FUSTIPEN

Report
Atomic nucleus,
Fundamental Symmetries,
and
Quantum Chaos
Vladimir Zelevinsky
NSCL/ Michigan State University
FUSTIPEN, Caen
June 3, 2014
THANKS
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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
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MESO
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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
Cardioid billiard
Angular momentum conserved
No symmetries
CLASSICAL CHAOS
CLASSICAL DETERMINISTIC
CHAOS
• Constants of motion destroyed
• Trajectories labeled by
initial conditions
• Close trajectories exponentially
diverge
• Round-off errors amplified
• Unpredictability = chaos
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)
(b)
(c)
(d)
(e)
(f)
Neutron resonances in 167Er, I=1/2
Proton resonances in 49V, I=1/2
I=2,T=0 shell model states in 24Mg
Poisson spectrum P(s)=exp(-s)
Neutron resonances in 182Ta, I=3 or 4
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
Wigner distribution
Disordered spectrum P(s) = exp(-s)
= Poisson distribution
“Aperiodic crystal” = Wigner P(s)
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)
(b)
(c)
(d)
(e)
(f)
Neutron resonances in 167Er, I=1/2
Proton resonances in 49V, I=1/2
I=2,T=0 shell model states in 24Mg
Poisson spectrum P(s)=exp(-s)
Neutron resonances in 182Ta, I=3 or 4
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)
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
R. Haq et al. 1982
SPECTRAL RIGIDITY
= a log L +b
for L>>1
Purity ?
Missing levels ?
235U, I=3 or 4,
960 lowest levels
f=0.44
Data agree with
f=(7/16)=0.44
and
0, 4% and 10% missing
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:
(1 channel)
Neutron width of neutron resonances as an analyzer
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
MODEL
48 Cr
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)
Exponential tails
Gaussian fit (solid)
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
56
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 !
J=0
28 Si
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)
Is there a pairing phase transition in
mesoscopic system?
Invariant entropy
•Invariant entropy is basis independent
•Indicates the sensitivity of
eigenstate  to parameter G
in interval [G,G+ G]
24Mg
phase diagram
strength of
T= 1 p airing
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 (λT=1) and T=0 pairing (λT=0); three plots indicate phase diagram as
a function of non-pairing matrix elements (λnp) . Realistic case is λT=1=λT=0 =λnp=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
up to
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
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 2nd 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
CREATIVE CHAOS
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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.
This is the creative side of chaos.
Dipole moment and violation of
P- and T-symmetries
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
spin
d
T-reversal
d
spin
-29 e.cm
d(199Hg)<3.1x10
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
d
P-reversal
d
spin
Half-live
219 Rn 4 s
221 Rn 25 m
J.F.C. Cocks et al. PRL 78 (1997) 2920.
Half-live
223 Rn 24 m
223 Ra 11 d
Half-live
225 Ra 15 d
227 Ra 42 m
Parity-doublet
Parity conservation:
|+
|-
Mixture by weak interaction W
Small parity violating interaction W
Perturbed ground state
Non-zero Schiff
moment
CONCLUSION
Nuclear ENHANCEMENTS
* Chaotic (statistical)
* Kinematic
* Structural
*accidental
VERY HARD TIME-CONSUMING
EXPERIMENTS…
SUMMARY
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

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