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
………
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
REALISTICSHELLMODEL
EXCITED STATES 51Sc
1/2-, 3/2-
Faster convergence:E(n) = E + exp(-an) a ~ 6/N
REALISTIC SHELL 48 CrMODEL
Excited stateJ=2, T=0
EXPONENTIALCONVERGENCE !
E(n) = E + exp(-an) n ~ 4/N
Partition structure in the shell model
(a) All 3276 states ; (b) energy centroids
28 Si
Diagonalmatrix elementsof the Hamiltonianin the mean-field representation
J=2+, T=0
Energy dispersion for individual states is nearly constant (result of geometric chaoticity!)
28Si
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)
… = JMany quasi-random paths
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)
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
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 levelsf=0.44
Data agree with
f=(7/16)=0.44
and
4% missing levels
0, 4% and 10% missing
D D. Mulhall et al.2007
Level curvature distribution for different interaction strengths
Shell Model 28Si
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
Smart information entropy (separation of center-of-mass excitationsof lower complexity shifted up in energy)
12C
CROSS-SHELL MIXING WITH SPURIOUS STATES
1183 states
NUMBER of PRINCIPAL COMPONENTS
1.44
l=k l=k+1
l=k+10 l=k+100 l=k+400
31
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
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
AVERAGE STRENGTH FUNCTIONBreit-Wigner fit (solid)Gaussian fit (dashed) Exponential tails
52 Cr
Ground and excited states
56 Ni
Superdeformed headband
56
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:
EXPONENTIALCONVERGENCEOF SINGLE-PARTICLEOCCUPANCIES
(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
Fastconvergence
Exponentialconvergence
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 !
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)
Interaction: 0.1 1 10
Exp (S)Various measures
Level density
Information Entropy inunits of S(GOE)
Single-particle entropyof Fermi-gas
* 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 …
STATISTICAL MECHANICS of CLOSED MESOSCOPIC SYSTEMS
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
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.
B. V. CHIRIKOV :
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