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Atomic nucleus, Fundamental Symmetries, and Quantum Chaos Vladimir Zelevinsky

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Fundamental Symmetries,

and

Quantum Chaos

Vladimir Zelevinsky

NSCL/ Michigan State University

FUSTIPEN, Caen

June 3, 2014

- Naftali Auerbach (Tel Aviv)
- B. Alex Brown (NSCL, MSU)
- Mihai Horoi (Central Michigan University)
- Victor Flambaum (Sydney)
- Declan Mulhall (Scranton University)
- Roman Sen’kov (CMU)
- Alexander Volya (Florida State University)

* Symmetries

* Mesoscopic physics

* From classical to quantum chaos

* Chaos as useful practical tool

* Nuclear level density

* Chaotic enhancement

* Parity violation

* Nuclear structure and EDM

PHYSICS of ATOMIC NUCLEI in XXI CENTURY

- Limits of stability - drip lines, superheavy…
- Nucleosynthesis in the Universe; charge asymmetry; dark matter…
- Structure of exotic nuclei
- Magic numbers
- Collective effects – superfluidity, shape transformations, …
- Mesoscopic physics – chaos, thermalization, level and width statistics, …
- ^ random matrix ensembles
- ^ physics of open and marginally stable systems
- ^ enhancement of weak perturbations
- ^ quantum signal transmission
- Neutron matter
- Applied physics – isotopes, isomers, reactor technology, …
- Fundamental physics and violation of symmetries:
- ^ parity
- ^ electric dipole moment (parity and time reversal)
- ^ anapole moment (parity)
- ^ temporal and spatial variation of fundamental constants

Uniformity of space = momentum conservation P

Uniformity of time = energy conservation E

Isotropy of space = angular momentum conservation L

Relativistic invariance

Indistinguishability of identical particles

Relation between spin and statistics

Bose – Einstein (integer spin 0,1, …)

Fermi – Dirac (half-integer spin 1/2, 3/2, …)

Coordinate inversion P

vectors and pseudovectors, scalars and pseudoscalars

Time reversal T

microscopic reversibility, macroscopic irreversibility

Charge conjugation C

excess of matter in our Universe

Conserved in strong and electromagnetic interactions

C and P violated in weak interactions

T violated in some special meson decays (Universe?)

C P T - strictly valid

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

* Condensed matter effects

MICRO ----- MESO ----- MACRO

- Complex nuclei
- Complex atoms
- Complex molecules (including biological)
- Cold atoms in traps
- Micro- and nano- devices of condensed matter
- --------
- Future quantum computers

Common features: quantum bricks, interaction, complexity;

quantum chaos, statistical regularities;

at the same time – individual quantum states

Symmetry preserves unfolded momentum

Stadium billiard – no symmetries

A single trajectory fills in phase space

CHAOS

- Constants of motion destroyed
- Trajectories labeled by

initial conditions

- Close trajectories exponentially

diverge

- Round-off errors amplified
- Unpredictability = chaos

MANY-BODY QUANTUM CHAOS AS AN INSTRUMENT

SPECTRAL STATISTICS – signature of chaos

- missing levels

- purity of quantum numbers *

- calculation of level density (given spin-parity) *

- 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 *

- development of computational tools *

- new approximations in many-body problem

MANY-BODY QUANTUM CHAOS AS AN INSTRUMENT

SPECTRAL STATISTICS – signature of chaos

- missing levels

- purity of quantum numbers *

- calculation of level density (given spin-parity) *

- 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 *

- development of computational tools *

- new approximations in many-body problem

different spectra

50 levels, rescaled

(a), (b), (c) – exact symmetries

(e), (f) – mixed symmetries

Arrows: s < (1/4) D

- Neutron resonances in 167Er, I=1/2
- Proton resonances in 49V, I=1/2
- I=2,T=0 shell model states in 24Mg
- Poisson spectrum P(s)=exp(-s)
- Neutron resonances in 182Ta, I=3 or 4
- Shell model states in 63Cu, I=1/2,…,19/2

SPECTRAL STATISTICS

Nearest level spacing distribution

(simplest signature of chaos)

Disordered spectrum P(s) = exp(-s)

= Poisson distribution

Regular system

“Aperiodic crystal” = Wigner P(s)

Chaotic system

Wigner distribution

- universality classes
- all states of similar complexity
- local spectral properties
- uncorrelated independent matrix elements

Gaussian Orthogonal Ensemble (GOE) – real symmetric

Gaussian Unitary Ensemble (GUE) –Hermitian complex

Extreme mathematical limit of quantum chaos!

Many other ensembles: GSE, BRM, TBRM, …

(shell model of 24Mg

as a typical example)

Fraction (%) of realistic strength

From turbulent to laminar level dynamics

Chaos due to particle interactions at high level density

different spectra

50 levels, rescaled

(a), (b), (c) – exact symmetries

(e), (f) – mixed symmetries

Arrows: s < (1/4) D

- Neutron resonances in 167Er, I=1/2
- Proton resonances in 49V, I=1/2
- I=2,T=0 shell model states in 24Mg
- Poisson spectrum P(s)=exp(-s)
- Neutron resonances in 182Ta, I=3 or 4
- Shell model states in 63Cu, I=1/2,…,19/2

Nearest level spacing distributions for the same cases

(all available levels)

NEAREST LEVEL SPACING DISTRIBUTION

at interaction strength 0.2 of the realistic value

WIGNER-DYSON distribution

(the weakest signature of quantum chaos)

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

Missing levels ?

Data agree with

f=(7/16)=0.44

and

4% missing levels

235U, I=3 or 4,

960 lowest levels

f=0.44

D. Mulhall, Z. Huard, V.Z.,

PRC 76, 064611 (2007).

0, 4% and 10% missing

RELATIVE!

Typical eigenstate:

GOE:

Porter-Thomas distribution for weights:

(1 channel)

Neutron width of neutron resonances as an analyzer

in the region of

giant quadrupole

resonance

Resolution:

(p,p’) 40 keV

(e,e’) 50 keV

Unresolved fine structure

D = (2-3) keV

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”

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?

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

in condensed matter physics

CONVERGENCE

OF SINGLE-PARTICLE

OCCUPANCIES

(first excited state J=0)

52

Cr

Orbitals f5/2 and f7/2

Shell Model and Nuclear Level Density

M. Horoi, J. Kaiser, and V. Zelevinsky, Phys. Rev. C 67, 054309 (2003).

M. Horoi, M. Ghita, and V. Zelevinsky, Phys. Rev. C 69, 041307(R) (2004).

M. Horoi, M. Ghita, and V. Zelevinsky, Nucl. Phys. A785, 142c (2005).

M. Scott and M. Horoi, EPL 91, 52001 (2010).

R.A. Sen’kov and M. Horoi, Phys. Rev. C 82, 024304 (2010).

R.A. Sen’kov, M. Horoi, and V. Zelevinsky, Phys. Lett. B702, 413 (2011).

R. Sen’kov, M. Horoi, and V. Zelevinsky, Computer Physics Communications

184, 215 (2013).

Statistical Spectroscopy:

S. S. M. Wong, Nuclear Statistical Spectroscopy (Oxford, University Press, 1986).

V.K.B. Kota and R.U. Haq, eds., Spectral Distributions in Nuclei and

Statistical Spectroscopy (World Scientific,

Singapore, 2010).

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!)

Also in multiconfigurational method (hybrid of shell model and

density functional)

at high level density

Two languages: individual wave functions

thermal excitation

* Mutually exclusive ?

* Complementary ?

* Equivalent ?

Answer depends on thermometer

Single – particle occupation numbers

28 Si

Thermodynamic behavior identical

in all symmetry classes

FERMI-LIQUID PICTURE

Artificially strong interaction (factor of 10)

Single-particle thermometer cannot resolve

spectral evolution

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)

Is there a pairing phase transition in mesoscopic system?

Invariant entropy

- Invariant entropy is basis independent
- Indicates the sensitivity of
- eigenstate to parameter G
- in interval [G,G+ G]

Contour plot of invariant correlational entropy showing a phase diagram as a function

of T=1 pairing (λT=1) and T=0 pairing (λT=0); three plots indicate phase diagram as

a function of non-pairing matrix elements (λnp) . Realistic case is λT=1=λT=0 =λnp=1

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

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

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 preserving asymmetry

[Grenoble]

A. Alexandrovich et al . 1994

Parity non-conservation in fissionby polarized neutrons –

on the level up to 0.001

233 U

by cold

polarized

neutrons,

(Grenoble)

A. Koetzle

et al. 2000

Asymmetry

determined

at the “hot”

chaotic stage

- STATISTICAL MECHANICS
- PHASE TRANSITIONS
- COMPLEXITY
- INFORMATICS
- CRYPTOGRAPHY
- LARGE FACILITIES
- LIVING ORGANISMS
- HUMAN BRAIN
- ECONOPHYSICS
- FUNDAMENTAL SYMMETRIES
- PARTICLE PHYSICS
- COSMOLOGY

… 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.

Dipole moment and violation of P- and T-symmetries

spin

Observation of the dipole moment is an indication of parity and time-reversal violation

T-reversal

Observation of the dipole moment is an indication of parity and time-reversal violation

d

d

spin

-29

e.cm

d(199Hg)<3.1x10

spin

Limits on EDM for the electron

Experiment: < 8.7 x 10-29 e.cm

Standard model ~ 10-38 e.cm

Physics beyond SM ~10-28 e.cm

NeutronEDM < 2.9 x 10

P-reversal

d

d

spin

-26

|-

Parity-doubletParity conservation:

Mixture by weak interaction W

Small parity violating interaction W

Perturbed ground state

Non-zero Schiff

moment

Nuclear ENHANCEMENTS

* Chaotic (statistical)

* Kinematic

* Structural

*accidental

VERY HARD TIME-CONSUMING

EXPERIMENTS…

- Many-body quantum chaos as universal phenomenon at high level density
- Experimental, theoretical and computational tool
- Role of incoherent interactions not fully understood
- Chaotic paradigm of statistical thermodynamics
- Nuclear structure mechanisms for enhancement of tiny effects, chaoric and regular

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