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