Atomic nucleus,
Fundamental Symmetries,
and
Quantum Chaos
Vladimir Zelevinsky NSCL/ Michigan State University
FUSTIPEN, Caen
June 3, 2014
THANKS• 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 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
Cardioid billiard
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
(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
Fragments of sixdifferent spectra50 levels, rescaled
(a), (b), (c) – exact symmetries
(e), (f) – mixed symmetries
Arrows: s < (1/4) D
SPECTRAL STATISTICS
Nearest level spacing distribution
(simplest signature of chaos)
Regular system Disordered spectrum P(s) = exp(-s) = Poisson distributionChaotic system “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
Many other ensembles: GSE, BRM, TBRM, …
Extreme mathematical limit of quantum chaos!
From turbulent to laminar level dynamics
(shell model of 24Mgas a typical example)
Fraction (%) of realistic strength
LEVEL DYNAMICS
Chaos due to particle interactions at high level density
(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
Fragments of sixdifferent spectra50 levels, rescaled
(a), (b), (c) – exact symmetries
(e), (f) – mixed symmetries
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
Nuclear Data Ensemble
1407 resonance energies
30 sequences
For 27 nuclei
Neutron resonancesProton resonances(n,gamma) reactions
SPECTRAL RIGIDITY
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 levelsf=0.44
Data agree with
f=(7/16)=0.44
and
4% missing levels
0, 4% and 10% missing 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 ofgiant quadrupoleresonance
Resolution:(p,p’) 40 keV(e,e’) 50 keV
Unresolved fine structure
D = (2-3) keV
INVISIBLE FINE STRUCTURE, orcatching 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?
GROUND STATE ENERGY OF RANDOM MATRICES
EXPONENTIAL CONVERGENCE
SPECIFIC PROPERTY of RANDOM MATRICES ?
Banded GOE Full GOE
ENERGY CONVERGENCE in SIMPLE MODELS
Tight binding model Shifted harmonic oscillator
REALISTIC SHELL 48 CrMODEL
Excited stateJ=2, T=0
EXPONENTIALCONVERGENCE !
E(n) = E + exp(-an) n ~ 4/N
Local density of statesin condensed matter physics
AVERAGE STRENGTH FUNCTIONBreit-Wigner fit (dashed)Gaussian fit (solid) Exponential tails
REALISTICSHELLMODEL
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
56
EXPONENTIALCONVERGENCEOF SINGLE-PARTICLEOCCUPANCIES
(first excited state J=0)
52 Cr
Orbitals f5/2 and f7/2
CONVERGENCE REGIMES
Fastconvergence
Exponentialconvergence
Power law
Divergence
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).
Shell Model and Nuclear Level Density
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).
Partition structure in the shell model
(a) All 3276 states ; (b) energy centroids
28 Si
Diagonalmatrix elementsof the Hamiltonianin the mean-field representation
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 J=2 J=9
Single – particle occupation numbersThermodynamic behavior identical
in all symmetry classes FERMI-LIQUID PICTURE
28 Si
J=0
Artificially strong interaction (factor of 10) Single-particle thermometer cannot resolve spectral evolution
EFFECTIVE TEMPERATURE of INDIVIDUAL STATES
From occupation numbers in the shell model solution (dots)From thermodynamic entropy defined by level density (lines)
Gaussian level density
839 states (28 Si)
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=0 pairing
stren
gth o
fT=
1 pair
ing
Normal
T=1 pairing
T=0 pairing
realistic nucleus
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 - scalingN – 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
up to
10%
Neutron resonances in heavy nuclei
Kinematic enhancement
235 ULos Alamos dataE=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 fragmentsTransfer 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 of233 Uby coldpolarized neutrons,(Grenoble)
A. Koetzle et al. 2000
Asymmetry determined at the “hot”chaotic stage
CREATIVE CHAOS• 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-symmetriesspin
spin
d dT-reversal
spin
spin
d dP-reversal
Observation of the dipole moment is an indication of parity and time-reversal violation
Limits on EDM for the electron Experiment: < 8.7 x 10-29 e.cmStandard model ~ 10-38 e.cmPhysics beyond SM ~ 10-28 e.cm
Neutron EDM < 2.9 x 10
Observation of the dipole moment is an indication of parity and time-reversal violation
d(199Hg)<3.1x10-29 e.cm
-26
J.F.C. Cocks et al. PRL 78 (1997) 2920.
Half-live
219 Rn 4 s221 Rn 25 m
Half-live223 Rn 24 m223 Ra 11 d
Half-live225 Ra 15 d227 Ra 42 m
Parity-doublet
|+ |- Parity conservation:
Small parity violating interaction W
Perturbed ground state
Non-zero Schiff moment
Mixture by weak interaction W
C O N C L U S I O N
Nuclear ENHANCEMENTS
* Chaotic (statistical) * Kinematic * Structural *accidental
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