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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
THANKS
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B. Alex Brown (NSCL, MSU)
Mihai Horoi (Central Michigan University)
Declan Mulhall (Scranton University)
Alexander Volya (Florida State University)
Njema Frazier (Congress USA!)
ONE-BODY CHAOS – SHAPE (BOUNDARY CONDITIONS)
MANY-BODY CHAOS – INTERACTION BETWEEN PARTICLES
Nuclear Shell Model – realistic testing ground
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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
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 –
Do we need the exact energy values?
* how to order?
• Mass predictions
* is it convergent?
• Rotational and vibrational spectra
• Drip line position
* how rapidly?
• Level density
* in what basis?
* which observables?
• 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
MODEL
48 Cr
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
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)
…=J
Many quasi-random paths
Statistical theory of parentage coefficients ?
Effective Hamiltonian of classes
Interacting boson models, quantum dots, …
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
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)
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
12C
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
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
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
OUR TRADITION
L. Barkov
M. Zolotorev
I. Khriplovich
A. Vainshtein
V. Flambaum
O. Sushkov
D. Budker
V. Novikov
V. Dzuba
PRIMKNUVSHIE
V.Dmitriev
V. Telitsin
M. Pospelov
V. Khatsymovsky
A. Yelkhovsky
O. Vorov
P. Silvestrov
R. Sen’kov
………..
1978
-
2008
N. Auerbach
V. Spevak
G. Gribakin
M. Kozlov
J. Ginges
A. Lisetskiy
A. Volya
up to
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
10%
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 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 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)
Exponential tails
Gaussian fit (dashed)
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
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.
New approach to many-body theory for
mesoscopic systems –
instead of blunt diagonalization mean field out of chaos,
coherent modes plus
thermalized chaotic background
1.
Chaos-free scalable quantum computing
(internal and external chaos)
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.