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Many-Body Quantum Chaos: What did we learn and How can we use it?
Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF INT Workshop Seattle August 14, 2009
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OUTLINE Introduction: Many-body quantum chaos Nuclear case
Thermodynamics and chaos Computational tool (convergence) Experimental tool (invisible structure) Theoretical tool (enhancement of PNC) Mesoscopic phase transitions Chaos and continuum effects Summary
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THANKS B. Alex Brown (NSCL, MSU) Luca Celardo (Tulane University)
Mihai Horoi (Central Michigan University) Felix Izrailev (Puebla, Mexico) Declan Mulhall (Scranton University) Valentin Sokolov (Budker Institute) Alexander Volya (Florida State University)
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Chaotic motion in mesoscopic systems
* Mean field (one-body chaos) * Strong interaction (mainly two-body) * High level density * Mixing of simple configurations * Destruction of quantum numbers, (still conserved energy, J,M;T,T3;parity) * Spectral statistics – Gaussian Orthogonal Ensemble * Correlations between classes of states * Coexistence with (damped) collective motion * Analogy to thermal equilibrium * Continuum effects
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MANY-BODY QUANTUM CHAOS AS AN INSTRUMENT
SPECTRAL STATISTICS – signature of chaos - missing levels - purity of quantum numbers - statistical weight of subsequences - 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 - new approximations in many-body problem - development of computational tools
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Nuclear Shell Model – realistic testing ground
ONE-BODY CHAOS – SHAPE (BOUNDARY CONDITIONS) MANY-BODY CHAOS – INTERACTION BETWEEN PARTICLES Nuclear Shell Model – realistic testing ground Fermi – system with mean field and strong interaction Exact solution in finite space Good agreement with experiment Conservation laws and symmetry classes Variable parameters Sufficiently large dimensions (statistics) Sufficiently low dimensions (easy to analyze) (light nuclei) Observables: energy levels (spectral statistics) wave functions (complexity) transitions (correlations) destruction of symmetries cross sections (correlations and fluctuations) Heavy nuclei – dramatic growth of dimensions
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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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PROPERTY of RANDOM MATRICES ?
GROUND STATE ENERGY OF RANDOM MATRICES EXPONENTIAL CONVERGENCE PROPERTY of RANDOM MATRICES ?
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ENERGY CONVERGENCE in SIMPLE MODELS
Tight binding model Shifted harmonic oscillator
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REALISTIC SHELL Cr MODEL Excited state J=2, T=0 EXPONENTIAL CONVERGENCE ! E(n) = E + exp(-an) n ~ 4/N
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REALISTIC SHELL MODEL EXCITED STATES 51Sc 1/2-, /2- Faster convergence: E(n) = E + exp(-an) a ~ 6/N
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Partition structure in the shell model
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
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Energy dispersion for individual states is nearly constant
(result of geometric chaoticity!)
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EXPONENTIAL DISTRIBUTION :
Nuclei (various shell model versions), atoms, IBM
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From turbulent to laminar level dynamics
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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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Level curvature distribution
for different interaction strengths
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SPECTRAL RIGIDITY Regular spectra = L/15 (universal for small L)
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
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Spectral rigidity (calculations for 40Ca in the region of ISGQR)
[Aiba et al. 2003] Critical dependence on interaction between 2p-2h states
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Purity ? Missing levels ? Data agree with f=(7/16)=0.44 and
235U, I=3 or 4, 960 lowest levels f=0.44 0, 4% and 10% missing D. Mulhall, Z. Huard, V.Z., PRC 76, (2007).
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COMPLEXITY of QUANTUM STATES RELATIVE!
Typical eigenstate: GOE: Porter-Thomas distribution for weights: (1 channel) Neutron width of neutron resonances as an analyzer
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l=k l=k+1 1 3 l=k+10 l=k+100 l=k+400 1 Correlation functions of the weights W(k)W(l) in comparison with GOE
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Reduced neutron widths
(energy dependence eliminated) Distribution for strengths of E2 transitions in (sd)-shell model for 6 particles (using local average) Porter – Thomas distribution for single channels
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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/<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”
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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
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N - scaling Single – particle matrix element
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
up to 10% 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
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Transmission coefficients for two helicity states
235 U Los Alamos data E=63.5 eV 10.2 eV (0.08)% (0.37) (0.40) * (0.86) (0.11) (1.30) (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 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)
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Parity non-conservation in fission by polarized neutrons –
Parity violating asymmetry Parity preserving asymmetry [Grenoble] A. Alexandrovich et al Parity non-conservation in fission by polarized neutrons – on the level up to 0.001
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ILL experiment Koetzle et al. (2000) PNC asymmetry in fission of 235 U by cold polarized neutrons is insensitive to distribution of total kinetic energy or to the mass distribution
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AVERAGE STRENGTH FUNCTION
Breit-Wigner fit (dashed) Gaussian fit (solid) Exponential tails
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52 Cr Ground and excited states 56 Ni 56 Superdeformed headband
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OTHER OBSERVABLES ? Occupation numbers Add a new partition of dimension d , Corrections to wave functions where Occupation numbers are diagonal in a new partition The same exponential convergence:
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Off-diagonal matrix elements of the operator n (j=5/2) between
the ground state and all excited states J=0, s=0 in the exact solution of the pairing problem for 114Sn
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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 exponents
10 particles on 10 doubly-degenerate orbitals 252 s=0 states Fast convergence at weak interaction G Pairing phase transition at G=0.25
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CONVERGENCE REGIMES Fast convergence Exponential convergence Power law Divergence
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CHAOS versus THERMALIZATION
L. BOLTZMANN – Stosszahlansatz = MOLECULAR CHAOS N. BOHR - Compound nucleus = MANY-BODY CHAOS N. S. KRYLOV - Foundations of statistical mechanics L. Van HOVE – Quantum ergodicity L. D. LANDAU and E. M. LIFSHITZ – “Statistical Physics” Average over the equilibrium ensemble should coincide with the expectation value in a generic individual eigenstate of the same energy – the results of measurements in a closed system do not depend on exact microscopic conditions or phase relationships if the eigenstates at the same energy have similar macroscopic properties TOOL: MANY-BODY QUANTUM CHAOS
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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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Single – particle occupation numbers
J= J= J=9 Single – particle occupation numbers Thermodynamic behavior identical in all symmetry classes FERMI-LIQUID PICTURE
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J=0 Artificially strong interaction (factor of 10) Single-particle thermometer cannot resolve spectral evolution
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Shell model level density (28Si, J=0, T=0)
Averaging over 10 levels 40 levels (distorted edges) Shell model versus Fermi-gas a = 1.4/MeV a (F-G) = 2/MeV (two parities?) J = 2, T = 0
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EFFECTIVE TEMPERATURE of INDIVIDUAL STATES
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)
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Information entropy is basis-dependent
- special role of mean field
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INFORMATION ENTROPY AT WEAK INTERACTION
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INFORMATION ENTROPY of EIGENSTATES
(a) function of energy; (b) function of ordinal number ORDERING of EIGENSTATES of GIVEN SYMMETRY SHANNON ENTROPY AS THERMODYNAMIC VARIABLE
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12 C 1183 states Smart information entropy (separation of center-of-mass excitations of lower complexity shifted up in energy) CROSS-SHELL MIXING WITH SPURIOUS STATES
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NUMBER of PRINCIPAL COMPONENTS
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Exp (S) Various measures Level density Information Entropy in units of S(GOE) Single-particle entropy of Fermi-gas Interaction:
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Invariant correlational entropy as signature of phase transitions
Eigenstates in an arbitrary basis (Hamiltonian with random parameters) Density matrix of a given state (averaged over the ensemble) Correlational entropy has clear maximum at phase transition (extreme sensitivity) Pure state: eigenvalues of the density matrix are 1 (one) and 0 (N-1), S=0 Mixed state: between 0 and 1, S up to ln N For two discrete points
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Model of two levels with pair transfer Capacity 16 + 16, N=16
Critical value 0.3 (in BCS ¼) Averaging interval 0.01 (seniority 0) First excited state “pair vibration” No instability in the exact solution Softening at the same point 0.3
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Shell model 48Ca Ground state invariant entropy; phase transition depends on non-pairing interactions Occupancy of f7/2 shell Correlation energy ~ 2 MeV
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48 Ca Invariant entropy and the line of phase transitions Occupancy of the f7/2 orbital Effective number of T=1 pairs
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24 Mg Isovector against isoscalar pairing Dependence on non-pairing interactions (phase transitions smeared, absolute values of entropy suppressed) Critical value for T=0 phase transition: ~ 3 /Bertsch, 2009/
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PAIR CORRELATOR (b) Only pairing (d) Non-pairing interactions (f) All interactions
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PAIRING PHASE TRANSITION PAIR CORRELATOR as a THERMODYNAMIC FUNCTION
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Pair correlator as a function of J Yrast states
Average over all states Old semiclassical theory (Grin’& Larkin, 1965) (too small) Geometry of orbital space rather than Coriolis force
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GLOBAL BEHAVIOR Pair correlator in 24 Mg for all states of various spins Central part of the spectrum is well described by statistical model with mean occupation numbers
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J=0, T=0 states in 24 Mg Degenerate s.-p. energies + realistic interactions Growing level density quickly leads to chaos In the absence of the mean-field skeleton, pairing works for lowest states only Realistic single-particle energies + random interactions (Gaussian matrix elements with zero mean and the same variance as in realistic interaction) Enhancement – for the states of lowest complexity
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RESULTS Regular behavior of pair correlator in a mesoscopic system Long tail beyond “phase transition” Similar picture for all spin and isospin classes In the middle – semiclassical picture with average occupation numbers of single-particle orbitals Pairing is considerably influenced by non-pairing interaction Are the shell model results generic? - exact solution - rotational invariance - isospin invariance - well tested at low energy - with growing level density leads to many-body quantum chaos in agreement with random matrix theory - loosely bound systems and effects of continuum
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Ingredients Intrinsic states: P-space Continuum states: Q-space
States of fixed symmetry Unperturbed energies e1; some e1>0 Hermitian interaction V Continuum states: Q-space Channels and their thresholds Ecth Scattering matrix Sab(E) Coupling with continuum Decay amplitudes Ac1(E) Typical partial width =|A|2 Resonance overlaps: level spacing vs. width
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EFFECTIVE HAMILTONIAN
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Specific features of the continuum shell model
Remnants of traditional shell model Non-Hermitian Hamiltonian Energy-dependent Hamiltonian Decay chains New effective interaction – unknown…
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Interplay of collectivities
Definitions n - labels particle-hole state n – excitation energy of state n dn - dipole operator An – decay amplitude of n Model Hamiltonian Driving GDR externally (doing scattering) Everything depends on angle between multi dimensional vectors A and d
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Most effective excitation
Pygmy resonance Orthogonal: GDR not seen Parallel: Most effective excitation of GDR from continuum At angle: excitation of GDR and pigmy A model of 20 equally distant levels is used
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Particle in Many-Well Potential
Hamiltonian Matrix: Solutions: No continuum coupling - analytic solution Weak decay - perturbative treatment of decay Strong decay – localization of decaying states at the edges
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Typical Example N=1000 e=0 and v=1 Critical decay strength g about 2
Decay width as a function of energy Location of particle
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Disordered problem
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Example: disorder + localization
e=random number and v=1 Critical decay strength g about 2 Location of a particle
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Distribution of widths as a function of decay strength
Weak decay: Random Distribution Transitional region: Formation of superradiant states Strong decay: Superradiance
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DO WE UNDERSTAND ROLE of INCOHERENT INTERACTIONS ? Ground state predominantly J=0 (even A) Ordered structure of wave functions ? New aspects of quantum chaos: - correlations between different symmetry sectors governed by the same Hamiltonian geometry of a mesoscopic system “random” mean field effects of time-reversal invariance exploration of interaction space manifestations of collective phenomena
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QUESTIONS and PROBLEMS
Geometric chaoticity Extension to continuum: - level densities - correlations and fluctuations of cross sections - mesoscopic universal conductance fluctuations - dependence on intrinsic chaos - loosely bound nuclei Microscopic picture of shape phase transitions New approximations for large systems: pairing + collective motion + incoherent chaos
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STATISTICS of GROUND STATE SPINS ?
ORDER FROM RANDOM INTERACTIONS ? FULL ROTATIONAL INVARIANCE FERMI-STATISTICS RANDOM AMPLITUDES V(L) SYMMETRIC ENSEMBLE STATISTICS of GROUND STATE SPINS ? Non-equivalence of particle-particle and particle-hole channels
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Spectra are chaotic: Gaussian level density, Wigner-Dyson level spacing distribution, Exponential distribution of off-diagonal many-body matrix elements (average over many realizations)
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Distribution of ground state spins
6 particles, j=11/2
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Fraction of ground states of spin J=0 and J=J(max)
(single j model)
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GROUND STATE SPIN DISTRIBUTION (6 particles, j=21/2)
Natural multiplicity (b) Boson approximation (c) Uniform V(L) from –1 to (d) Gaussian V(L), dispersion 1 (e) Uniform V(L) scaled 1/(2L+1) (f) [Zhao et al., 2002] (g) Uniform V(L) except V(0)=-1 (h) As (g) but V(0)=+1 (i) As (g) and (h) but V(0)=0
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Degenerate orbitals 63 random m.e. Realistic orbitals 63 random m.e. Realistic orbitals and 6 pairing m.e., 57 random Degenerate orbitals, 6 random pairing m.e.
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Do we understand the role of incoherent interactions
in many-body physics ? -- Random interactions prefer ground state spin 0 -- Probability of maximum spin enhanced -- Ordered wave functions? Collectivity? -- New aspect of quantum chaos: correlations between the symmetry classes -- Geometric chaoticity of angular momentum coupling -- Bosonization of fermion pairs? -- Role of time-reversal invariance
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Widths of level distributions in the J-class for a single-j model
(6 particles)
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Statistical theory of parentage coefficients ?
IDEA of GEOMETRIC CHAOTICITY Angular momentum coupling as a random process Bethe (1936) j(a) + j(b) = J(ab) + j(c) = J(abc) + j(d) = J(abcd) … = J Many quasi-random paths Statistical theory of parentage coefficients ? Effective Hamiltonian of classes Interacting boson models, quantum dots, … Papenbrock, Weidenmueller 2007
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Effective Hamiltonian for
N particles and given M=J explained by geometry: j + j = L Cranking frequency is linear in M Typical predictions for f(0)
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Dotted lines – statistical predictions for the state M=J
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Predictions for energy of individual states with J=0 and J=J(max)
compared to exact diagonalization (6 particles, j = 21/2)
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Collectivity of low-lying states
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Distribution of overlaps
|0> ground state of spin J = 0 in the random ensemble |s = 0> fully paired state of seniority s = 0 4 particles on j = 15/2 – dimension d(0) = 3 (left) 6 particles on j = 15/2 – dimension d(0) = 4 (right) Completely random overlaps:
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Collectivity out of chaos: Johnson, Dean, Bertsch 1998
V.Z., Volya Johson, Nam Horoi, V.Z Predominance of prolate deformations : Teller, Wheeler – alpha-carcass Bohr, Wheeler liquid drop Lemmer extra kinetic energy of large orbital momenta Castel, Goeke the same in terms of collective energy Castel, Rowe, Zamick adding self-consistency Frisk single-particle level density Arita et al periodic orbits and their bifurcations Deleplanque et al – Hamamoto, Mottelson metallic clusters 2009 – surface properties of deformed field “The nature of the parameters responsible for the prolate dominance has not yet been adequately understood”
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ALAGA RATIO (take sequences J=0, J=2)
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N= Distribution of Alaga ratio 4 neutrons + 4 protons 0f7/2 + 1p3/2 4.0 0.5 4.0 Interaction: weak strong Selection by E(4)/E(2) [3.0, 3.6] All cases J=0, J=2 Here A(rot) = 4.10 Selection N(rot): A between 3.90 and 4.30 Selection N(prolate): Q(2)<0
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4 protons + 6 neutrons N(rot) lower, N(prolate) higher 4 neutrons + 4 protons 1p3/2 + 0f7/2 (inverted sequence) 4 neutrons + 4 protons 0f7/2 + 0g9/2 (opposite parity)
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Strong interaction 4.0 Matrix elements 9-12: pf mixing, 16: quadrupole pair transfer, 20-24: quadrupole-quadrupole forces in particle-hole channel = formation of the mean field
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