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Atomic nucleus, Fundamental Symmetries, and Quantum Chaos Vladimir Zelevinsky NSCL/ Michigan State University FUSTIPEN, Caen June 3, 2014

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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)

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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

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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

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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, …)

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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

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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

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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

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Classical regular billiard Symmetry preserves unfolded momentum

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Regular circular billiard

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

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

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CLASSICAL DETERMINISTIC CHAOS Constants of motion destroyed Trajectories labeled by initial conditions Close trajectories exponentially diverge Round-off errors amplified Unpredictability = chaos

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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

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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

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(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 six different spectra 50 levels, rescaled (a), (b), (c) – exact symmetries (e), (f) – mixed symmetries Arrows: s < (1/4) D SPECTRAL STATISTICS

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

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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!

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

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(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 six different spectra 50 levels, rescaled (a), (b), (c) – exact symmetries (e), (f) – mixed symmetries Arrows: s < (1/4) D

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

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

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

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Purity ?Missing levels ? 235U, I=3 or 4, 960 lowest levels f=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).

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Structure of eigenstates Whispering Gallery Bouncing Ball Ergodic behavior With fluctuations

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

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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

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

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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?

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

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ENERGY CONVERGENCE in SIMPLE MODELS Tight binding model Shifted harmonic oscillator

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

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Local density of states in condensed matter physics

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

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

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52 Cr Ground and excited states 56 Ni Superdeformed headband 56

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

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CONVERGENCE REGIMES Fast convergence Exponential convergence Power law Divergence

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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).

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

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

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

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

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

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

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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)

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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]

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24 Mg phase diagram 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

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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

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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

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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)

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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)

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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

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

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

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Boris V. CHIRIKOV (1928 – 2008)

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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.

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Dipole moment and violation of P- and T-symmetries spin d d T-reversal spin d d P-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.cm Standard model ~ 10 -38 e.cm Physics 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)<3.1 x 10 -29 e.cm -26

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

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Half-live 223 Rn 24 m 223 Ra 11 d

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Half-live 225 Ra 15 d 227 Ra 42 m

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

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C O N C L U S I O N Nuclear ENHANCEMENTS * Chaotic (statistical) * Kinematic * Structural *accidental VERY HARD TIME-CONSUMING EXPERIMENTS…

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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

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