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 :