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
OUTLINE
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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
THANKS
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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)
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
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
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 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
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?
GROUND STATE ENERGY OF RANDOM MATRICES
EXPONENTIAL CONVERGENCE
PROPERTY of RANDOM MATRICES ?
ENERGY CONVERGENCE in SIMPLE MODELS
Tight binding model
Shifted harmonic oscillator
REALISTIC
SHELL
MODEL
48 Cr
Excited state
J=2, T=0
EXPONENTIAL
CONVERGENCE !
E(n) = E + exp(-an)
n ~ 4/N
REALISTIC
SHELL
MODEL
EXCITED STATES
51Sc
1/2-,
3/2-
Faster convergence:
E(n) = E + exp(-an)
a ~ 6/N
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
Energy dispersion for individual states is nearly constant
(result of geometric chaoticity!)
EXPONENTIAL DISTRIBUTION :
Nuclei (various shell model versions), atoms, IBM
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
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
R. Haq et al. 1982
SPECTRAL RIGIDITY
= 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 ?
Missing levels ?
235U, I=3 or 4,
960 lowest levels
f=0.44
Data agree with
f=(7/16)=0.44
and
4% missing levels
0, 4% and 10% missing
D. Mulhall, Z. Huard, V.Z.,
PRC 76, 064611 (2007).
COMPLEXITY of QUANTUM STATES
RELATIVE!
Typical eigenstate:
GOE:
Porter-Thomas distribution for weights:
(1 channel)
Neutron width of neutron resonances as an analyzer
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
Reduced neutron widths
(energy dependence
eliminated)
Distribution for 20 000
strengths of E2 transitions
in (sd)-shell model for
6 particles
(using local average)
Porter – Thomas distribution for single channels
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
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”
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
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
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
ILL experiment
A. 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
AVERAGE STRENGTH FUNCTION
Breit-Wigner fit (dashed)
Exponential tails
Gaussian fit (solid)
52
Cr
Ground and excited states
56
Ni
Superdeformed headband
OTHER OBSERVABLES ?
Occupation numbers
Add a new partition of dimension d
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Corrections to wave functions
where
Occupation numbers are diagonal in a new partition
The same exponential convergence:
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
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
Temperature T(E)
T(s.p.) and T(inf) =
for individual states !
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
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
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)
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
NUMBER of PRINCIPAL COMPONENTS
Exp (S)
Various
measures
Level density
Information
Entropy in
units of S(GOE)
Single-particle
entropy
of Fermi-gas
Interaction: 0.1
1
10
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
(seniority 0)
Model of two levels with
pair transfer
Capacity 16 + 16, N=16
Critical value 0.3
(in BCS ¼)
Averaging interval 0.01
First excited state
“pair vibration”
No instability in
the exact solution
Softening at the same point 0.3
Shell model 48Ca
Ground state
invariant entropy;
phase transition
depends on
non-pairing
interactions
Occupancy of
f7/2 shell
Correlation energy
~ 2 MeV
48 Ca
(a) Invariant entropy and the line of phase transitions
(b) Occupancy of the f7/2 orbital
(c) Effective number of T=1 pairs
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/
PAIR CORRELATOR
(b) Only pairing
(d) Non-pairing
interactions
(f) All interactions
PAIRING
PHASE
TRANSITION
PAIR CORRELATOR as a THERMODYNAMIC FUNCTION
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
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
J=0, T=0 states in 24 Mg
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
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
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
Superradiance, collectivization by
decay Analog in a complex system
Dicke coherent state
N identical two-level atoms
coupled via common radiation
Volume ¿ 3
Interaction via continuum
Trapped states ) self-organization
 ~ D and few channels
• Nuclei far from stability
• High level density (states of
same symmetry)
• Channel thresholds
Single-particle decay in a many-body system
Evolution of complex energies =E-i /2
as a function of 
•System 8 s.p. levels, 3 particles
•One s.p. level in continuum e= –i/2
Total states 8!/(3! 5!)=56; states that decay fast 7!/(2! 5!)=21
Quasistationary states are determined by continuum
Ingredients
• Intrinsic states: P-space
– States of fixed symmetry
– Unperturbed energies 1; some 1>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
EFFECTIVE HAMILTONIAN
Specific features of the
continuum shell model
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Remnants of traditional shell model
Non-Hermitian Hamiltonian
Energy-dependent Hamiltonian
Decay chains
New effective interaction – unknown…
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
Pygmy resonance
Orthogonal:
GDR not seen
A model of 20 equally
distant levels is used
Parallel:
Most effective excitation
of GDR from continuum
At angle:
excitation of GDR
and pigmy
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
Typical Example
N=1000
=0 and v=1
Critical decay strength  about 2
Decay width as a function of energy
Location of particle
Disordered problem
Example: disorder +
N=1000
localization
=random number and v=1
Critical decay
strength  about 2
Location of a particle
Distribution of widths
as a function of decay strength
Weak decay: Random Distribution
Transitional region:
Formation of superradiant states
Strong decay: Superradiance
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
QUESTIONS and PROBLEMS
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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
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
Spectra are chaotic:
Gaussian level density,
Wigner-Dyson level spacing distribution,
Exponential distribution of
off-diagonal many-body matrix elements
(average over many realizations)
Distribution of ground state spins
6 particles, j=11/2
Fraction of ground states of
spin J=0 and J=J(max)
(single j model)
GROUND STATE SPIN DISTRIBUTION (6 particles, j=21/2)
• Natural multiplicity
(b) Boson approximation
(c) Uniform V(L) from –1 to +1 (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
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
Widths of level distributions in the J-class for a single-j model
(6 particles)
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
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)
Dotted lines – statistical predictions for the state M=J
Predictions for energy of individual states with J=0 and J=J(max)
compared to exact diagonalization
(6 particles, j = 21/2)
Collectivity of low-lying states
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:
Collectivity out of chaos:
Johnson, Dean, Bertsch 1998
V.Z., Volya
2004
Johson, Nam
2007
Horoi, V.Z.
2009
Predominance of prolate deformations :
Teller, Wheeler 1938 – alpha-carcass
Bohr, Wheeler 1939 - liquid drop
Lemmer
1960 - extra kinetic energy of large orbital momenta
Castel, Goeke 1976 - the same in terms of collective energy
Castel, Rowe, Zamick 1990 - adding self-consistency
Frisk
1990 - single-particle level density
Arita et al.
1998 periodic orbits and their bifurcations
Deleplanque et al. 2004 –
Hamamoto, Mottelson 1991 - metallic clusters
2009 – surface properties of deformed field
“The nature of the parameters responsible for the prolate dominance
has not yet been adequately understood”
ALAGA RATIO
(take sequences
J=0, J=2)
Distribution of Alaga ratio
0.5
4.0
Selection by E(4)/E(2)
[3.0, 3.6]
4.0
All cases
J=0, J=2
N= 10 000
4 neutrons +
4 protons
0f7/2 + 1p3/2
Interaction:
(a) weak
(b) strong
Here A(rot) = 4.10
Selection N(rot):
A between 3.90 and 4.30
Selection N(prolate):
Q(2)<0
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)
Strong interaction 4.0
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Matrix elements
1 9-12: pf mixing,
2 16: quadrupole pair transfer,
20-24: quadrupole-quadrupole forces
in particle-hole channel = formation of the mean field