Quantum%20Chaos

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Quantum Chaos as a Practical
Tool in Many-Body Physics Vladimir
Zelevinsky NSCL/ Michigan State University
Supported by NSF CHIRIKOV Memorial Seminar
Budker Institute, Novosibirsk May 23, 2008
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THANKS

• B. Alex Brown (NSCL, MSU)
• Mihai Horoi (Central Michigan University)
• Declan Mulhall (Scranton University)
• Alexander Volya (Florida State University)
• Njema Frazier (Congress USA!)

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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 d imensions (statistics)
• Sufficiently low diimensions
• Observables
• energy levels (spectral statistics)
• wave functions (complexity)
• transitions (correlations)
• destruction of symmetries
• cross sections (correlations)
• Heavy nuclei dramatic growth of
  dimensions

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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 our of chaos
- chaos and thermalization
- development of computational tools
- new
approximations in many-body problem
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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?

• Do we need the exact energy values?
• Mass predictions
• Rotational and vibrational spectra
• Drip line position
• Level density
• Astrophysical applications

Process of progressive truncation how to
order? is it convergent? how rapidly?
in what basis? which observables?
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Banded GOE
Full GOE
GROUND STATE ENERGY OF RANDOM MATRICES
EXPONENTIAL CONVERGENCE
SPECIFIC 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 MODEL
EXCITED STATES 51Sc
1/2-, 3/2-
Faster convergence E(n) E exp(-an) a
6/N
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REALISTIC SHELL 48 Cr MODEL
Excited state J2, T0
EXPONENTIAL CONVERGENCE !
E(n) E exp(-an) n 4/N
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28
Si
Diagonal matrix elements of the Hamiltonian in
the mean-field representation
J2, T0
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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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,
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Off-diagonal matrix elements of the operator n
between the ground state and all
excited states J0, s0 in the exact solution
of the pairing problem for 114Sn
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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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EXPONENTIAL DISTRIBUTION Nuclei (various shell
model versions), atoms, IBM
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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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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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1.44
NUMBER of PRINCIPAL
COMPONENTS
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lk
lk1
1
3
lk10
lk100
lk400
1
Correlation functions of the weights W(k)W(l) in
comparison with GOE
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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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OUR TRADITION
PRIMKNUVSHIE
L. Barkov M. Zolotorev I. Khriplovich A.
Vainshtein V. Flambaum O. Sushkov D. Budker V.
Novikov V. Dzuba
N. Auerbach V. Spevak G. Gribakin M. Kozlov J.
Ginges A. Lisetskiy A. Volya
V.Dmitriev V. Telitsin M. Pospelov V.
Khatsymovsky A. Yelkhovsky O. Vorov P.
Silvestrov R. Senkov ..
1978 - 2008
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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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235 U Los Alamos data E63.5 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 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 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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AVERAGE STRENGTH FUNCTION Breit-Wigner fit
(solid) Gaussian fit (dashed)
Exponential tails
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52 Cr
Ground and excited states
56 Ni
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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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EXPONENTIAL CONVERGENCE OF SINGLE-PARTICLE OCCUPAN
CIES
(first excited state J0)
52 Cr
Orbitals f5/2 and f7/2
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Convergence exponents 10 particles
on 10 doubly-degenerate orbitals
252 s0 states
Fast convergence at weak interaction G Pairing
phase transition at G0.25
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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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J0 J2 J9

Single particle occupation numbers

• Thermodynamic behavior
• identical in all symmetry classes

FERMI-LIQUID PICTURE
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J0
Artificially strong interaction
(factor of 10)

• Single-particle thermometer cannot resolve
• spectral evolution

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Temperature T(E)
T(s.p.) and T(inf) for individual states !
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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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Exp (S) Various measures
Level density
Information Entropy in units of S(GOE)
Single-particle entropy of Fermi-gas
Interaction 0.1 1
10
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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
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GLOBAL PROBLEMS

• New approach to many-body theory for
• mesoscopic systems
• instead of blunt diagonalization -
• mean field out of chaos,
• coherent modes plus
• thermalized chaotic background
• Chaos-free scalable quantum computing
• (internal and external chaos)

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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.