1. 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

2. OUTLINE 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

3. THANKS 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)‏

4. 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

5. 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

6. Nuclear Shell Model – realistic testing ground
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 (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

7. 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?

8. PROPERTY of RANDOM MATRICES ?
GROUND STATE ENERGY OF RANDOM MATRICES
EXPONENTIAL CONVERGENCE
PROPERTY of RANDOM MATRICES ?

9. ENERGY CONVERGENCE in SIMPLE MODELS
Tight binding model Shifted harmonic oscillator

10. REALISTIC
SHELL Cr
MODEL
Excited state
J=2, T=0
EXPONENTIAL
CONVERGENCE !
E(n) = E + exp(-an)
n ~ 4/N

11. REALISTIC
SHELL
MODEL
EXCITED STATES
51Sc
1/2-, /2-
Faster convergence:
E(n) = E + exp(-an)
a ~ 6/N

12. Partition structure in the shell model28 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

13. Energy dispersion for individual states is nearly constant
(result of geometric chaoticity!)

14. EXPONENTIAL DISTRIBUTION :
Nuclei (various shell model versions), atoms, IBM

15. From turbulent to laminar level dynamics

16. NEAREST LEVEL SPACING DISTRIBUTION
at interaction strength 0.2 of the realistic value
WIGNER-DYSON distribution
(the weakest signature of quantum chaos)

17. Level curvature distribution
for different interaction strengths

18. SPECTRAL RIGIDITY Regular spectra = L/15 (universal for small L)
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
= a log L +b for L>>1
R. Haq et al. 1982
SPECTRAL RIGIDITY

19. Spectral rigidity (calculations for 40Ca in the region of ISGQR)
[Aiba et al. 2003]
Critical dependence on interaction between 2p-2h states

20. Purity ? Missing levels ? Data agree with f=(7/16)=0.44 and
235U, I=3 or 4,
960 lowest levels
f=0.44
0, 4% and 10% missing
D. Mulhall, Z. Huard, V.Z.,
PRC 76, (2007).

22. COMPLEXITY of QUANTUM STATES RELATIVE!
Typical eigenstate:
GOE:
Porter-Thomas distribution for weights:
(1 channel)
Neutron width of neutron resonances as an analyzer

23. 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

24. Reduced neutron widths
(energy dependence
eliminated)
Distribution for
strengths of E2 transitions
in (sd)-shell model for
6 particles
(using local average)
Porter – Thomas distribution for single channels

25. 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

26. 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”

27. 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

28. N - scaling Single – particle matrix element
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

29. STATISTICAL ENHANCEMENT Parity nonconservation in scattering of slow
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

30. Transmission coefficients for two helicity states
235 U
Los Alamos data
E=63.5 eV
10.2 eV (0.08)%
(0.37)
(0.40) *
(0.86)
(0.11)
(1.30)
(0.86)
Transmission coefficients for two helicity states
(longitudinally polarized neutrons)

31. 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)

32. Parity non-conservation in fission by polarized neutrons –
Parity violating asymmetry
Parity preserving asymmetry
[Grenoble]
A. Alexandrovich et al
Parity non-conservation in fission by polarized neutrons –
on the level up to 0.001

33. ILL experiment
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

35. AVERAGE STRENGTH FUNCTION
Breit-Wigner fit (dashed)
Gaussian fit (solid)
Exponential tails

37. 52 Cr
Ground and excited states 56 Ni 56
Superdeformed headband

38. 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:

39. 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

40. EXPONENTIAL
CONVERGENCE
OF SINGLE-PARTICLE
OCCUPANCIES
(first excited state J=0) 52 Cr
Orbitals f5/2 and f7/2

41. 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

43. CONVERGENCE REGIMES
Fast
convergence
Exponential
convergence
Power law
Divergence

44. 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

45. CLOSED MESOSCOPIC SYSTEM
at high level density
Two languages: individual wave functions
thermal excitation
* Mutually exclusive ?
* Complementary ?
* Equivalent ?
Answer depends on thermometer

46. Temperature T(E)‏
T(s.p.) and T(inf) =
for individual states !

47. Single – particle occupation numbers
J= J= J=9
Single – particle occupation numbers
Thermodynamic behavior
identical in all symmetry classes
FERMI-LIQUID PICTURE

48. J=0
Artificially strong interaction (factor of 10)‏
Single-particle thermometer cannot resolve
spectral evolution

49. 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

50. EFFECTIVE TEMPERATURE of 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)‏

51. Information entropy is basis-dependent
- special role of mean field

52. INFORMATION ENTROPY AT WEAK INTERACTION

53. INFORMATION ENTROPY of EIGENSTATES
(a) function of energy; (b) function of ordinal number
ORDERING of EIGENSTATES of GIVEN SYMMETRY
SHANNON ENTROPY AS THERMODYNAMIC VARIABLE

54. 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

55. NUMBER of PRINCIPAL COMPONENTS

56. Exp (S)‏
Various
measures
Level density
Information
Entropy in
units of S(GOE)‏
Single-particle
entropy
of Fermi-gas
Interaction:

57. 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

58. Model of two levels with pair transfer Capacity 16 + 16, N=16
Critical value 0.3
(in BCS ¼)‏
Averaging interval 0.01
(seniority 0)‏
First excited state
“pair vibration”
No instability in
the exact solution
Softening at the same point 0.3

59. Shell model 48Ca
Ground state
invariant entropy;
phase transition
depends on
non-pairing
interactions
Occupancy of
f7/2 shell
Correlation energy
~ 2 MeV

60. 48 Ca
Invariant entropy and the line of phase transitions
Occupancy of the f7/2 orbital
Effective number of T=1 pairs

61. 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/

62. PAIR CORRELATOR
(b) Only pairing
(d) Non-pairing interactions
(f) All interactions

63. PAIRING
PHASE
TRANSITION
PAIR CORRELATOR as a THERMODYNAMIC FUNCTION

64. 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

65. 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

66. J=0, T=0 states in 24 Mg
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
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

67. 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

72. Ingredients Intrinsic states: P-space Continuum states: Q-space
States of fixed symmetry
Unperturbed energies e1; some e1>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

73. EFFECTIVE HAMILTONIAN

78. Specific features of the continuum shell model
Remnants of traditional shell model
Non-Hermitian Hamiltonian
Energy-dependent Hamiltonian
Decay chains
New effective interaction – unknown…

79. 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

82. Most effective excitation
Pygmy resonance
Orthogonal:
GDR not seen
Parallel:
Most effective excitation
of GDR from continuum
At angle:
excitation of GDR
and pigmy
A model of 20 equally
distant levels is used

83. 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

84. Typical Example N=1000 e=0 and v=1 Critical decay strength g about 2
Decay width as a function of energy
Location of particle

85. Disordered problem

86. Example: disorder + localization
e=random number and v=1
Critical decay
strength g about 2
Location of a particle

87. Distribution of widths as a function of decay strength
Weak decay: Random Distribution
Transitional region:
Formation of superradiant states
Strong decay: Superradiance

88. 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

89. QUESTIONS and PROBLEMS
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

90. STATISTICS of GROUND STATE SPINS ?
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

91. Spectra are chaotic:
Gaussian level density,
Wigner-Dyson level spacing distribution,
Exponential distribution of
off-diagonal many-body matrix elements
(average over many realizations)

92. Distribution of ground state spins
6 particles, j=11/2

93. Fraction of ground states of spin J=0 and J=J(max)
(single j model)‏

95. GROUND STATE SPIN DISTRIBUTION (6 particles, j=21/2)‏
Natural multiplicity (b) Boson approximation
(c) Uniform V(L) from –1 to (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

96. 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.

97. 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

98. Widths of level distributions in the J-class for a single-j model
(6 particles)‏

99. Statistical theory of parentage coefficients ?
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

101. 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)‏

102. Dotted lines – statistical predictions for the state M=J

103. Predictions for energy of individual states with J=0 and J=J(max)‏
compared to exact diagonalization
(6 particles, j = 21/2)‏

105. Collectivity of low-lying states

106. 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:

107. Collectivity out of chaos: Johnson, Dean, Bertsch 1998
V.Z., Volya
Johson, Nam
Horoi, V.Z
Predominance of prolate deformations :
Teller, Wheeler – alpha-carcass
Bohr, Wheeler liquid drop
Lemmer extra kinetic energy of large orbital momenta
Castel, Goeke the same in terms of collective energy
Castel, Rowe, Zamick adding self-consistency
Frisk single-particle level density
Arita et al
periodic orbits and their bifurcations
Deleplanque et al –
Hamamoto, Mottelson metallic clusters
2009 – surface properties of deformed field
“The nature of the parameters responsible for the prolate dominance
has not yet been adequately understood”

109. ALAGA RATIO
(take sequences J=0, J=2)

110. N=
Distribution of Alaga ratio
4 neutrons +
4 protons
0f7/2 + 1p3/2
4.0
0.5
4.0
Interaction:
weak
strong
Selection by E(4)/E(2)‏
[3.0, 3.6]
All cases
J=0, J=2
Here A(rot) = 4.10
Selection N(rot):
A between 3.90 and 4.30
Selection N(prolate):
Q(2)<0

111. 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)‏

112. Strong interaction 4.0
Matrix elements
9-12: pf mixing,
16: quadrupole pair transfer,
20-24: quadrupole-quadrupole forces
in particle-hole channel = formation of the mean field