strongly correlated coulomb systems a path integral monte carlo …bonitz/talks/gnv03... ·...

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

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

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


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


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


Internal Energy Internal Energy of of Coulomb Coulomb MatterMatter

Energy isotherm of hydrogen

intEEE ideal +=

Trigger, Filinov, Ebeling, Fortov, Bonitz,JETP (2003)


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


OutlineOutline







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


((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)


http://elde.mpg.uni-rostock.de/mb

Correlated electron-hole-plasma in optically excited semiconductors

Quasi-equilibriumExample: 2-dimensional quantum well

5.010:,2.0

→==

s

RB

rnincreaseETk


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 /=


Rs=4.2

ExcitonsBiexcitons

Trions,Cluster,

...

Partially ionized e-h-plasma


Rs=2.1 ©Michael Bonitz, Universität Rostock, 2003

Electron-hole-droplets

partiallydelocalized phase

Rs=0.63

Correlateddelocalized phase

Fermi liquid


OutlineOutline







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


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≡


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


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

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

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)


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


OutlineOutline







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

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....“


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


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“


A. Filinov, M. Bonitz, and Yu. Lozovik, Phys. Rev. Lett. 86, 3851 (2001), PR Focus April 2001

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


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)


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

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

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

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

−


„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“


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


N=20N=19

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




M. M. Bonitz Bonitz et al., J. Phys. A: Math. Gen. 36, 5921 et al., J. Phys. A: Math. Gen. 36, 5921 (2003)(2003)



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

strongly correlated coulomb systems a path integral monte carlo …bonitz/talks/gnv03... ·...

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