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Atomic nucleus, Fundamental Symmetries, and Quantum Chaos Vladimir Zelevinsky

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)

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Atomic nucleus, Fundamental Symmetries, and Quantum Chaos Vladimir Zelevinsky

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

  2. 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)

  3. 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

  4. 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

  5. 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, …)

  6. 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

  7. 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

  8. 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

  9. Classical regular billiard Symmetry preserves unfolded momentum

  10. Regular circular billiard

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

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

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

  14. 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

  15. 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

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

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

  18. 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, …

  19. 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

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

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

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

  23. 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 R. Haq et al. 1982 SPECTRAL RIGIDITY

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

  25. Structure of eigenstates Whispering Gallery Bouncing Ball Ergodic behavior With fluctuations

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

  27. 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

  28. 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”

  29. 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?

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

  31. ENERGY CONVERGENCE in SIMPLE MODELS Tight binding model Shifted harmonic oscillator

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

  33. Local density of states in condensed matter physics

  34. AVERAGE STRENGTH FUNCTION Breit-Wignerfit (dashed)‏ Gaussian fit (solid) Exponential tails

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

  36. 52 Cr Ground and excited states 56 56 Ni Superdeformed headband

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

  38. CONVERGENCE REGIMES Fast convergence Exponential convergence Power law Divergence

  39. 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).

  40. 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

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

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

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

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