Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF Statistical Nuclear Physics SNP2008 Athens, Ohio July 8, 2008
THANKS B. Alex Brown (NSCL, MSU) Mihai Horoi (Central Michigan University) Declan Mulhall (Scranton University) Alexander Volya (Florida State University) Njema Frazier (NNSA)
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 Observables: energy levels (spectral statistics) wave functions (complexity) transitions (correlations) destruction of symmetries cross sections (correlations) Heavy nuclei dramatic growth of dimensions
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 our of chaos - chaos and thermalization - development of computational tools - new approximations in many-body problem
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? Do we need the exact energy values? Mass predictions Rotational and vibrational spectra Drip line position Level density Astrophysical applications
Banded GOE Full GOE GROUND STATE ENERGY OF RANDOM MATRICES EXPONENTIAL CONVERGENCE SPECIFIC PROPERTY of RANDOM MATRICES?
ENERGY CONVERGENCE in SIMPLE MODELS Tight binding model Shifted harmonic oscillator
REALISTIC SHELL MODEL EXCITED STATES 51Sc 1/2-, 3/2- Faster convergence: E(n) = E + exp(-an) a ~ 6/N
REALISTIC SHELL 48 Cr MODEL Excited state J=2, T=0 EXPONENTIAL CONVERGENCE! E(n) = E + exp(-an) n ~ 4/N
28 Si Diagonal matrix elements of the Hamiltonian in the mean-field representation J=2+, T=0 Partition structure in the shell model (a) All 3276 states ; (b) energy centroids
28Si Energy dispersion for individual states is nearly constant (result of geometric chaoticity!)
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) Many quasi-random paths = J Statistical theory of parentage coefficients? Effective Hamiltonian of classes Interacting boson models, quantum dots,
From turbulent to laminar level dynamics
NEAREST LEVEL SPACING DISTRIBUTION at interaction strength 0.2 of the realistic value WIGNER-DYSON distribution (the weakest signature of quantum chaos)
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) R. Haq et al. 1982 SPECTRAL RIGIDITY Chaotic spectra = a log L +b for L>>1
Spectral rigidity (calculations for 40Ca in the region of ISGQR) [Aiba et al. 2003] Critical dependence on interaction between 2p-2h states
Purity? Mixing levels? 235U, J=3 or 4, 960 lowest levels f=0.44 Data agree with f=(7/16)=0.44 0, 4% and 10% missing D and 4% missing levels D. Mulhall et al.2007
Shell Model 28Si Level curvature distribution for different interaction strengths
EXPONENTIAL DISTRIBUTION : Nuclei (various shell model versions), atoms, IBM
Information entropy is basis-dependent - special role of mean field
INFORMATION ENTROPY AT WEAK INTERACTION
INFORMATION ENTROPY of EIGENSTATES (a) function of energy; (b) function of ordinal number ORDERING of EIGENSTATES of GIVEN SYMMETRY SHANNON ENTROPY AS THERMODYNAMIC VARIABLE
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
1.44 NUMBER of PRINCIPAL COMPONENTS
l=k l=k+1 3 1 l=k+10 l=k+100 l=k+400 1 Correlation functions of the weights W(k)W(l) in comparison with GOE
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
STATISTICAL ENHANCEMENT Parity nonconservation in scattering of slow polarized neutrons up to 10% Coherent part of weak interaction Single-particle mixing Chaotic mixing Neutron resonances in heavy nuclei Kinematic enhancement
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)
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)
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 of 233 U by cold polarized neutrons, (Grenoble) A. Koetzle et al. 2000 Asymmetry determined at the hot chaotic stage
AVERAGE STRENGTH FUNCTION Breit-Wigner fit (solid) Gaussian fit (dashed) Exponential tails
52 Cr Ground and excited states 56 Ni Superdeformed headband
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:
EXPONENTIAL CONVERGENCE OF SINGLE-PARTICLE OCCUPANCIES (first excited state J=0) 52 Cr Orbitals f5/2 and f7/2
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
CONVERGENCE REGIMES Fast convergence Exponential convergence Power law Divergence
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
CLOSED MESOSCOPIC SYSTEM at high level density Two languages: individual wave functions thermal excitation * Mutually exclusive? * Complementary? * Equivalent? Answer depends on thermometer
J=0 J=2 J=9 Single particle occupation numbers Thermodynamic behavior identical in all symmetry classes FERMI-LIQUID PICTURE
J=0 Artificially strong interaction (factor of 10) Single-particle thermometer cannot resolve spectral evolution
Off-diagonal matrix elements of the operator n between the ground state and all excited states J=0, s=0 in the exact solution of the pairing problem for 114Sn
Temperature T(E) T(s.p.) and T(inf) = for 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)
Exp (S) Various measures Level density Information Entropy in units of S(GOE) Single-particle entropy of Fermi-gas Interaction: 0.1 1 10
STATISTICAL MECHANICS of CLOSED MESOSCOPIC SYSTEMS * SPECIAL ROLE OF MEAN FIELD BASIS (separation of regular and chaotic motion; mean field out of chaos) * CHAOTIC INTERACTION as HEAT BATH * SELF CONSISTENCY OF mean field, interaction and thermometer * SIMILARITY OF CHAOTIC WAVE FUNCTIONS * SMEARED PHASE TRANSITIONS * CONTINUUM EFFECTS (IRREVERSIBLE DECAY) new effects when widths are of the order of spacings restoration of symmetries super-radiant and trapped states conductance fluctuations
GLOBAL PROBLEMS 1. Do we understand the role of incoherent interactions in many-body physics? 2. Correlations between classes of states with different symmetry governed by the same Hamiltonian 3. New approach to many-body theory for mesoscopic systems instead of blunt diagonalization - mean field out of chaos, coherent modes plus thermalized chaotic background 4. Internal and external chaos 5. Chaos-free scalable quantum computing
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.