strongly correlated coulomb systems a path integral monte carlo …bonitz/talks/gnv03... ·...
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Strongly correlated Coulomb Strongly correlated Coulomb systemssystems––a a path path integral Monte Carlo integral Monte Carlo approach approach
Michael Bonitz
Physics Department, University of Floridaon leave from Fachbereich Physik, Universität Rostock
Gainesville, 22 May 2003©Michael Bonitz, Universität Rostock, 2003
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Students/co-workers:
V. Golubnychiy, D. Semkat, A. Filinov, P. Ludwig (Rostock),V.S. Filinov (Moscow)
Cooperations:
J. Dufty (Gainesville), W. Ebeling (Berlin),V. Fortov, P. Levashov, Yu. Lozovik (Moscow),D. Kremp, W.D. Kraeft (Rostock)S.W. Koch, P. Thomas, W. Hoyer (Marburg)F. Peeters (Antwerp),M. Schlanges (Greifswald)
©Michael Bonitz, Universität Rostock, 2003
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OutlineOutline
1. Introduction: correlated Coulomb systems
2. Journey to the center of Jupiter
5. Summary and outlook
3. Path integral Monte Carlo: idea and applications
4. Mesoscopic Coulomb systems. „Artificial atoms“Wigner crystallization in Quantum dots and heterostructures.Single-electron control of collective behavior
©Michael Bonitz, Universität Rostock, 2003
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Coulomb Coulomb SystemsSystems
D. Hoffmann, GSI Darmstadt
Electron density, 1/ccm
Tem
pera
ture
, eV
KeV 4101 ≅
JupiterPlanet coresLightning
Lightning
Magnetic FusionMagnet fusion
SunSun core
Inertial FusionICF
MetalsSemiconductors Brown
dwarfs
Dusty Dusty !!PlasmasPlasmas
Plasmas Plasmas in trapsin traps
©Michael Bonitz, Universität Rostock, 2003
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CorrelationCorrelation andand Quantum Quantum effectseffectsCoulomb Interaction: reerU baab /)( =
TkU B/⟩⟨≡Γ BFs arEUr // ∝⟩⟨≡
- Fermi EnergyFE Ba - Bohr Radius
StrongCoulomb
correlations
1=ΓDeBroglie
wave length
Tmkh Bπλ 2/=
r
r=λOverlap
of wave functions, Spin effects
Quantum effects
l
13 == λχ n
1=sr
©Michael Bonitz, Universität Rostock, 2003
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Internal Energy Internal Energy of of Coulomb Coulomb MatterMatter
Energy isotherm of hydrogen
intEEE ideal +=
Trigger, Filinov, Ebeling, Fortov, Bonitz,JETP (2003)
©Michael Bonitz, Universität Rostock, 2003
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Equilibrium phase diagram Equilibrium phase diagram of of Coulomb Coulomb systemssystems
M. Bonitz, Physik Journal 7/8 2002, p.69
Universal scaling
RB Eascalesenergyandlength
,:
Hydrogensemiconductors
eVA 6.135.0
eVA 3105100 −⋅
THEORYWeak U: perturbation theory, Strong U: first-principle simulations
©Michael Bonitz, Universität Rostock, 2003
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OutlineOutline
1. Introduction: correlated Coulomb systems
2. Journey to the center of Jupiter
5. Summary and outlook
3. Path integral Monte Carlo: idea and applications
4. Mesoscopic Coulomb systems. „Artificial atoms“Wigner crystallization in Quantum dots and heterostructures.Single-electron control of collective behavior
©Michael Bonitz, Universität Rostock, 2003
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Mysterious Hydrogen Mysterious Hydrogen
Filinov, Fortov, Bonitz, Levashov, JETP Letters 74, 384 (2001)
Insulator-metal transition, Anomalous compressibility (?), plasma phase transition (?)
Hydrogen conductivity(experiment) in cm/ohm
410
210
010
210−
210− 010410− 410
Nellis et al.
+ Fortov et al.
T=3,000-10,000K
©Michael Bonitz, Universität Rostock, 2003
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((SomeSome of) of) the animationsthe animations,,shown shown at at this place this place in in the the talktalk
can be viewed can be viewed at at my my web web pagepage::http://elde.mpg.unihttp://elde.mpg.uni--rostock.de/mbrostock.de/mb
An An introduction introduction toto partpart of of the results is giventhe results is given inintwo recent reviewstwo recent reviews ((pdfpdf--filesfiles areare on on mymy web web pagepage):):M. M. BonitzBonitz, Physik Journal 7/8 2002,69 (in German), Physik Journal 7/8 2002,69 (in German)
M. M. Bonitz Bonitz et al., J. Phys. A: Math. Gen. 36, 5921 (2003)et al., J. Phys. A: Math. Gen. 36, 5921 (2003)
©Michael Bonitz, Universität Rostock, 2003
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Correlated electron-hole-plasma in optically excited semiconductors
Quasi-equilibriumExample: 2-dimensional quantum well
5.010:,2.0
→==
s
RB
rnincreaseETk
©Michael Bonitz, Universität Rostock, 2003
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Rs=8.6V. Filinov, W. Hoyer, S.W. Koch, and M. Bonitz 2001
Rigoros Path integral Monte Carlo simulations
Full account of - Coulomb interaction, - Quantum effects- Spin of electrons and holes
Dots: Electron- (hole-)Position and extension
(Wave function) in 2D Quantum well
Bs arr /=
©Michael Bonitz, Universität Rostock, 2003
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Rs=4.2
ExcitonsBiexcitons
Trions,Cluster,
...
Partially ionized e-h-plasma
©Michael Bonitz, Universität Rostock, 2003
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Rs=2.1 ©Michael Bonitz, Universität Rostock, 2003
Electron-hole-droplets
partiallydelocalized phase
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Rs=0.63
Correlateddelocalized phase
Fermi liquid
©Michael Bonitz, Universität Rostock, 2003
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OutlineOutline
1. Introduction: correlated Coulomb systems
2. Journey to the center of Jupiter
5. Summary and outlook
3. Path integral Monte Carlo: idea and applications
4. Mesoscopic Coulomb systems. „Artificial atoms“Wigner crystallization in Quantum dots and heterostructures.Single-electron control of collective behavior
©Michael Bonitz, Universität Rostock, 2003
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Path integral quantum Monte Carlo
UKHe TkH B ˆˆˆ,ˆ /ˆ+== −ρN-particle-density operator (canonical ensemble):
Equilibrium: Minimize total Energy ZTkF B ln−=
ρ̂TrZ =Partition function " yields all thermodynamic quantities
Problem: ρ̂ known only for limiting cases, where KU ˆˆ <<
Feynman: [ ]MTMkHTkH BB ee )/(ˆ/ˆ ⋅−− ≡)/(ˆ TMkH Be ⋅−" Use known result for
©Michael Bonitz, Universität Rostock, 2003
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FirstFirst--principle thermodynamics principle thermodynamics of of correlated quantumcorrelated quantum systemssystems
)exp( Hβρ −=N-particle density operator:
M. Bonitz (Ed.), Introduction to Computational Methods, Rinton Press, Princeton 2003
{ }Niiii rrrr rrr ,...,, 21≡
©Michael Bonitz, Universität Rostock, 2003
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Illustration: Illustration: SnapshotsSnapshots of of closed electronclosed electron „„pathspaths““
„Spin-less“ particles Exchange included
5 Electrons in 2D simulation box, Path ends labeled by thick dots
©Michael Bonitz, Universität Rostock, 2003
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Applications of Path integral Monte Carlo
" first-principle calculations of equilibrium properties:
a) many-body systems: high-order correlations, partial ionization,bound state renormalization, Bose condensation etc.
b) few-body systems: binding energies of atoms, molecules/excitons, bi-excitons, trions and larger complexes follow non-perturbatively
c) „Exact“ effective quantum pair potentials: input for DFT simulationsor semiclassical molecular dynamics
d) mesoscopic systems (N=10...50): Fermi liquid behavior, Wigner crystallization etc.
Rs=4.2 ©Michael Bonitz, Universität Rostock, 2003
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Electron-hole bound states in quantum wells- Quasi-2D e-h-plasma, finite well width z
- renormalized in-plane (well) interaction, e-h-wave function overlap varies (non-monotonically) with z
Largest e-h overlap" largest exciton binding energy
©Michael Bonitz, Universität Rostock, 2003Binding energies
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Binding energies
Exciton and biexciton binding energies as function of quantum
well width L
"PIMC simulations very accurate, avoid any basis expansion,
"applicable to finite temperature,"geometry of minor importance
A.Filinov, MB, and Yu.Lozovik,phys. stat. sol., accepted (2003)
©Michael Bonitz, Universität Rostock, 2003
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EffectiveEffective Quantum pair Quantum pair potentialspotentials
Result:Drastic improvement of previous potentials (Kelbg, Deutsch etc.)" applicable to strong coupling
including bound statesA.Filinov, M. Bonitz, W. Ebeling, J. Phys. A, 2003 (accepted)
1. Exact pair potential from exact 2-particle density matrix (numer.)
),(ln)( rrTkrU abBpairab ρ−=
2. Derive analytical potential(1 fit parameter)
T=30,000K
U, units of Ryd
©Michael Bonitz, Universität Rostock, 2003
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((SomeSome of) of) the animationsthe animations,,shown shown at at this place this place in in the the talktalk
can be viewed can be viewed at at my my web web pagepage::http://elde.mpg.unihttp://elde.mpg.uni--rostock.de/mbrostock.de/mb
An An introduction introduction toto partpart of of the results is giventhe results is given inintwo recent reviewstwo recent reviews ((pdfpdf--filesfiles areare on on mymy web web pagepage):):M. M. BonitzBonitz, Physik Journal 7/8 2002,69 (in German), Physik Journal 7/8 2002,69 (in German)
M. M. Bonitz Bonitz et al., J. Phys. A: Math. Gen. 36, 5921 (2003)et al., J. Phys. A: Math. Gen. 36, 5921 (2003)
©Michael Bonitz, Universität Rostock, 2003
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OutlineOutline
1. Introduction: correlated Coulomb systems
2. Journey to the center of Jupiter
5. Summary and outlook
3. Path integral Monte Carlo: idea and applications
4. Mesoscopic Coulomb systems. „Artificial atoms“Wigner crystallization in Quantum dots and heterostructures.Single-electron control of collective behavior
©Michael Bonitz, Universität Rostock, 2003
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Mesoscopic Electron clustersin Quantum dots („artificial atoms“)
Model: N=1...100 Electrons in spherical harmonic „trap“
quasi-2-dim
confinement by external Fields or Heterostructures
-
- Shell structure, hexagonaland spherical Symmetry
Classical ground state: 0=kinE
- strong N-dependence
- strong Coulomb interaction,- Quantum and spin effects
At high density:
" Challenge for Theory!
Path integral Monte CarloBedanov/Peeters©Michael Bonitz, Universität Rostock, 2003
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Coulomb Coulomb ((WignerWigner) ) crystal crystal Ground state of the electron gas in metals
E. Wigner, Physical Review 46, 1002 (1934):
" exchange and correlation energy of the electron gas
„If the electrons had no kinetic energy, they would settlein configurations which correspond to the absoluteminima of the potential energy. These are close-packedlattice configurations, with energies very near to that ofthe body-centered lattice....“
©Michael Bonitz, Universität Rostock, 2003
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Experimentally observed Experimentally observed Coulomb Coulomb ((WignerWigner) ) crystals crystals
- electrons on helium droplets,- dusty plasmas,- ions in traps or storage rings- predicted: in White dwarf stars"All classical systems (B=0)
Ca+ ions in Paul trapG. Werth, Uni Mainz
©Michael Bonitz, Universität Rostock, 2003
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Wigner crystallization of quantum electron clustersVariation of temperature or density (confinement)
40/ ≥KinCoulomb EUWigner crystal expected for: ),( srParameterCoupling Γ
Result: two crystal-“phases“: Intra-shell- and Inter-shell order [Lozovik, Bedanov/Peeters]
Extension and shape of highly correlated 19-e-Wave function
Probability:rot=0 " pink=max
Increase density: " growing overlap of electrons:OO-crystal " RO-crystal " „liquid“
©Michael Bonitz, Universität Rostock, 2003
A. Filinov, M. Bonitz, and Yu. Lozovik, Phys. Rev. Lett. 86, 3851 (2001), PR Focus April 2001
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Phase Phase diagramdiagram of of the mesoscopic the mesoscopic Wigner crystalWigner crystal
RM - Radial Melting, OM – Rotational Melting
Particle Number
Quantum Liquid(„Wigner-Molecule“)
Confinement Strength
Classical
Liquid
Tem
pera
ture
TkU B/⟩⟨≡Γ
BFs arEUr // ∝⟩⟨≡
Wigner Crystal strongest correlations
©Michael Bonitz, Universität Rostock, 2003
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Coulomb BilayersCoulomb BilayersTwo mesoscopic 2D clusters at fixed distance d
-
New effect: influence of inter-layercorrelations on ground state, crystal
1921 == NN0,0 == BT
d in units of interparticle distance
d
" Change of crystal symmetry with d
Peeters et al., Kalman et al., A. Filinov, MB, Yu. Lozovik, Contrib. Plasma Phys. 41, 357 (2001)
©Michael Bonitz, Universität Rostock, 2003
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Mesoscopic Mesoscopic ee--h h bilayersbilayers
2== he NN
0,0 == BTd x=d/a
12221
21
21
0 −++=x5
2
2
5 2
2
0
3),(
),(451),(),(
ωµω
ωωω
mda
daddada
D
D
=
−≈
Ground state exciton distance in trap:
Reduction of d: change from Coulombto dipole interaction at small x:
243
62
3
2
1),(
)(2),(),(
xedxd
xOde
axdxaUcor
−=
+−=
µ
µ
"Interaction of two excitons"Binding energy/interaction tuneable"Exciton crystals possible
x=d/a
ae
h
attractive repulsive
Interaction P. Ludwig, A. Filinov, M. Bonitz (2002)©Michael Bonitz, Universität Rostock, 2003
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Wigner crystallization Wigner crystallization in in quantum quantum ee--h h bilayersbilayersResult: two crystal phases: with/without inter-layer orderingn=0.07
Exciton liquid
n=0.25
Decoupled e/h liquids
Exciton crystal
n=0.15
Decoupled e/h crystals
n=0.2
3 electrons and 3 holes, fixed layer distance d
probability to find individual electrons/holes (one e is fixed): red=0 " pink=max
A. Filinov, M. Bonitz, Yu. Lozovik, J. Phys. A, accepted©Michael Bonitz, Universität Rostock, 2003
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Phase Phase diagram diagram of of mesoscopicmesoscopic ee--h h bilayers bilayers
Ne=Nh=8, d in units of exciton Bohr radius, Ha=2 Ryd
A. Filinov, P. Ludwig, V. Golubnychiy, M. Bonitz, Yu. Lozovik, phys. stat. sol. (b) 2003, accepted©Michael Bonitz, Universität Rostock, 2003
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ParticleParticle numbernumber dependencedependence of of meltingmelting parameters parameters in in single single dotdot
37137Bulk
518320
6415419„Magic“
N
∞
srrrΓ0Γ
11104.3 ⋅
330
−
sor
11100.3 ⋅
400
−
©Michael Bonitz, Universität Rostock, 2003
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((SomeSome of) of) the animationsthe animations,,shown shown at at this place this place in in the the talktalk
can be viewed can be viewed at at my my web web pagepage::http://elde.mpg.unihttp://elde.mpg.uni--rostock.de/mbrostock.de/mb
An An introduction introduction toto partpart of of the results is giventhe results is given inintwo recent reviewstwo recent reviews ((pdfpdf--filesfiles areare on on mymy web web pagepage):):M. M. BonitzBonitz, Physik Journal 7/8 2002,69 (in German), Physik Journal 7/8 2002,69 (in German)
M. M. Bonitz Bonitz et al., J. Phys. A: Math. Gen. 36, 5921 (2003)et al., J. Phys. A: Math. Gen. 36, 5921 (2003)
©Michael Bonitz, Universität Rostock, 2003
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„Dusty Plasmas“Alternative road to strong correlations: " very high particle charge Q
Experiments of A. Piel, A. Melzer and co-workers (Univ. Kiel)
Web page http://www.ieap.uni-kiel.de/plasma/ag-piel
Dust particles: Q=(5000...10000) ed ~ 0.0095mm
Melamin-Formaldehyd-spheres, (electron-Microscop picture)
Charging in HF-dischargein plasma chamber
Vertical E-field compensatesgravitation " dust particles „float“
©Michael Bonitz, Universität Rostock, 2003
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Response to external excitation
Web page: http://www.ieap.uni-kiel.de/plasma/ag-piel
Experiments of A. Piel, A. Melzer and co-workers (Univ. Kiel)
Two-dimensional mesoscopic dusty plasma crystal
OMOM1920 Γ>Γ>>Γ
Tangential excitation by 2 laser pulses
©Michael Bonitz, Universität Rostock, 2003
N=20N=19
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Potential Potential ApplicationsApplications
2/1 srn ∝Γ= /1T
„switch“ between insulator (crystal) and conductorSingle electron- „transistor“
Crystallization/melting without change of density and temperature: " by addition/removal of a single electron
MB, V. Golubnichyi, A.Filinov, Yu.Lozovik , Microelectronic Engineering 63, 141(2002)©Michael Bonitz, Universität Rostock, 2003
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((SomeSome of) of) the animationsthe animations,,shown shown at at this place this place in in the the talktalk
can be viewed can be viewed at at my my web web pagepage::http://elde.mpg.unihttp://elde.mpg.uni--rostock.de/mbrostock.de/mb
An An introduction introduction toto partpart of of the results is giventhe results is given inintwo recent reviewstwo recent reviews ((pdfpdf--filesfiles areare on on mymy web web pagepage):):M. M. BonitzBonitz, Physik Journal 7/8 2002,69 (in German), Physik Journal 7/8 2002,69 (in German)
M. M. Bonitz Bonitz et al., J. Phys. A: Math. Gen. 36, 5921 et al., J. Phys. A: Math. Gen. 36, 5921 (2003)(2003)
©Michael Bonitz, Universität Rostock, 2003
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Summary and Outlook
I. Coulomb systems: fascinating variety of structures"Planets/stars, atoms, molecules, excitons, Wigner crystal..."Coulomb interaction important for many fields. Analogies, Overlap...
II. Quantum effects and strong correlations:
-+++++++QMD
+++++++++MD
++++++++++QKinetics
--+++++PIMC
Fast processes
Dynamics,Transport
Quantumeffects
Correla-tions
Development,Combination
of all approaches
Perspective
" there is no universal theory/computational method!http://elde.mpg.
uni-rostock.de/mb©Michael Bonitz, Universität Rostock, 2003