Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky
description
Transcript of Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky
![Page 1: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/1.jpg)
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
![Page 2: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/2.jpg)
THANKS• B. Alex Brown (NSCL, MSU)
• Mihai Horoi (Central Michigan University)
• Declan Mulhall (Scranton University)
• Alexander Volya (Florida State University)
• Njema Frazier (NNSA)
![Page 3: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/3.jpg)
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
![Page 4: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/4.jpg)
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
![Page 5: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/5.jpg)
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
………
![Page 6: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/6.jpg)
GROUND STATE ENERGY OF RANDOM MATRICES
EXPONENTIAL CONVERGENCE
SPECIFIC PROPERTY of RANDOM MATRICES ?
Banded GOE Full GOE
![Page 7: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/7.jpg)
ENERGY CONVERGENCE in SIMPLE MODELS
Tight binding model Shifted harmonic oscillator
![Page 8: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/8.jpg)
REALISTICSHELLMODEL
EXCITED STATES 51Sc
1/2-, 3/2-
Faster convergence:E(n) = E + exp(-an) a ~ 6/N
![Page 9: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/9.jpg)
REALISTIC SHELL 48 CrMODEL
Excited stateJ=2, T=0
EXPONENTIALCONVERGENCE !
E(n) = E + exp(-an) n ~ 4/N
![Page 10: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/10.jpg)
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
![Page 11: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/11.jpg)
Energy dispersion for individual states is nearly constant (result of geometric chaoticity!)
28Si
![Page 12: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/12.jpg)
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, …
![Page 13: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/13.jpg)
From turbulent to laminar level dynamics
![Page 14: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/14.jpg)
NEAREST LEVEL SPACING DISTRIBUTION
at interaction strength 0.2 of the realistic value
WIGNER-DYSON distribution
(the weakest signature of quantum chaos)
![Page 15: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/15.jpg)
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
![Page 16: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/16.jpg)
Spectral rigidity (calculations for 40Ca in the region of ISGQR) [Aiba et al. 2003]
Critical dependence on interaction between 2p-2h states
![Page 17: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/17.jpg)
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
![Page 18: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/18.jpg)
Level curvature distribution for different interaction strengths
Shell Model 28Si
![Page 19: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/19.jpg)
EXPONENTIAL DISTRIBUTION :
Nuclei (various shell model versions), atoms, IBM
![Page 20: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/20.jpg)
![Page 21: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/21.jpg)
Information entropy is basis-dependent - special role of mean field
![Page 22: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/22.jpg)
INFORMATION ENTROPY AT WEAK INTERACTION
![Page 23: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/23.jpg)
INFORMATION ENTROPY of EIGENSTATES (a) function of energy; (b) function of ordinal number
ORDERING of EIGENSTATES of GIVEN SYMMETRY SHANNON ENTROPY AS THERMODYNAMIC VARIABLE
![Page 24: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/24.jpg)
Smart information entropy (separation of center-of-mass excitationsof lower complexity shifted up in energy)
12C
CROSS-SHELL MIXING WITH SPURIOUS STATES
1183 states
![Page 25: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/25.jpg)
NUMBER of PRINCIPAL COMPONENTS
1.44
![Page 26: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/26.jpg)
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
![Page 27: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/27.jpg)
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
![Page 28: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/28.jpg)
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
![Page 29: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/29.jpg)
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)
![Page 30: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/30.jpg)
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)
![Page 31: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/31.jpg)
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
![Page 32: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/32.jpg)
Fission of233 Uby coldpolarized neutrons,(Grenoble)
A. Koetzle et al. 2000
Asymmetry determined at the “hot”chaotic stage
![Page 33: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/33.jpg)
![Page 34: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/34.jpg)
AVERAGE STRENGTH FUNCTIONBreit-Wigner fit (solid)Gaussian fit (dashed) Exponential tails
![Page 35: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/35.jpg)
![Page 36: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/36.jpg)
52 Cr
Ground and excited states
56 Ni
Superdeformed headband
56
![Page 37: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/37.jpg)
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:
![Page 38: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/38.jpg)
EXPONENTIALCONVERGENCEOF SINGLE-PARTICLEOCCUPANCIES
(first excited state J=0)
52 Cr
Orbitals f5/2 and f7/2
![Page 39: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/39.jpg)
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
![Page 40: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/40.jpg)
![Page 41: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/41.jpg)
CONVERGENCE REGIMES
Fastconvergence
Exponentialconvergence
Power law
Divergence
![Page 42: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/42.jpg)
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
![Page 43: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/43.jpg)
CLOSED MESOSCOPIC SYSTEM
at high level density
Two languages: individual wave functions thermal excitation
* Mutually exclusive ?* Complementary ?* Equivalent ?
Answer depends on thermometer
![Page 44: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/44.jpg)
J=0 J=2 J=9
Single – particle occupation numbers
Thermodynamic behavior identical in all symmetry classes
FERMI-LIQUID PICTURE
![Page 45: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/45.jpg)
J=0
Artificially strong interaction (factor of 10)
Single-particle thermometer cannot resolve spectral evolution
![Page 46: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/46.jpg)
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
![Page 47: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/47.jpg)
Temperature T(E)
T(s.p.) and T(inf) =for individual states !
![Page 48: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/48.jpg)
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)
![Page 49: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/49.jpg)
Interaction: 0.1 1 10
Exp (S)Various measures
Level density
Information Entropy inunits of S(GOE)
Single-particle entropyof Fermi-gas
![Page 50: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/50.jpg)
* 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
![Page 51: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/51.jpg)
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
![Page 52: Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky](https://reader036.fdocuments.net/reader036/viewer/2022070411/568148ad550346895db5c083/html5/thumbnails/52.jpg)
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 :