Quantum Hall effects - an introduction -

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Quantum Hall effects- an introduction -
M. Fleischhauer
AvH workshop, Vilnius, 03.09.2006
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quantum Hall history
discovery 1980
IQHE
Nobel prize 1985
K. v. Klitzing
FQHE
discovery 1982
Nobel prize 1998
H. Störmer
R. Laughlin
D. Tsui
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classical Hall effect (1880 E.H. Hall)
Lorentz-force on electron
stationary current
Hall resistance
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Dirac flux quantum
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Landau levels
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2D electrons in magnetic fields Landau levels
Hamiltonian
coordinate transformation
center of cyclotron motion
radial vector of cyclotron motion
electron
R
X
commutation relations
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2D electrons in magnetic fields Landau levels
mapping to oscillator
H h? R² / 2 l² h? ( a a ½ )

c
c
m
Landau levels
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2D electrons in magnetic fields Landau levels
typical scales

• length

magnetic length

• energy

cyclotron frequency
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2D electrons in magnetic fields Landau levels
degeneracy of Landau levels
center of cyclotron motion (X,Y) arbitrary ?
degeneracy

• 2D density of states (DOS)

one state per area of cyclotron orbit

• filling factor

atoms / flux quanta
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2D electrons in magnetic fields Landau levels
wavefunction of lowest Landau level (LLL) in
symmetric gauge
symmetric gauge
Landau gauge
introduce complex coordinate
b
LLL
analytic
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2D electrons in magnetic fields Landau levels
angular momentum of Landau levels
eigenstates of nth Landau level
angular momentum states of LLL
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2D electrons in magnetic fields Landau levels
wavefunction
j
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Integer Quantum Hall effect
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Integer Quantum Hall effect
spinless (for simplicity) and noninteracting
electrons Pauli principle
Slater determinant
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Integer Quantum Hall effect
compressibility
at integer fillings
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Integer Quantum Hall effect
Hall current
Heisenberg drift equations of cycoltron center
no plateaus ?!
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Integer Quantum Hall effect
Hall plateaus impurities
gap !

• impurities pin electrons to localized states
• electrons in impurity states do not contribute
  to current
• gap
• ? impurity states fill first

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Fractional Quantum Hall effect
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Fractional Quantum Hall effect
Laughlin state

• take e-e interaction into account

• generic wavefunction

• requirements

• wave function anstisymmetric
• eigenstate of angular momentum
• Coulomb repulsion ? Jastrow-type of wave function

Laughlin wave function
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Fractional Quantum Hall effect
angular momentum of Laughlin wave function and
filling factor
maximum single-particle angular momentum
filling factor of Laughlin state
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Fractional Quantum Hall effect
fractional Hall plateaus
fractional Hall states are gapped
? 1
? 1/3
? 1/5
? 1/7
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composite particle picture of FQHE
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composite particle picture of FQHE
composite particle electron m magnetic flux
quanta
? composite fermion
? composite boson
effective magnetic field
composite particle are anyons (fractional
statistics) exist only in 2D
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composite particle picture of FQHE
some remarks about anyons

• two-particle wave function

• exchange particles

• exchange particles a second time

? in 3D
Boson
Fermion
3Dno projected area in (xy)
2D always projected area in (xy)
A
B
A
B
particles can pick up e.g. Aharanov-Bohm phase
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composite particle picture of FQHE
? 1 / m
FQE
(A) electron flux quanta
form composite boson
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Bose condensation of composite bosons
(B) electron flux quanta
form composite fermion
?
IQHE for composite fermions
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composite particle picture of FQHE
Jain hierarchy

• experiment FQHE also for

composite fermion picture
since
?
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FQHE for interacting bosons
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FQHE for interacting bosons
exact diagonalization ? FQH effect for
Laughlin state for point interaction
composite fermions
boson single flux quantum


IQHE for composite fermions
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Thanks!
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effective magnetic fields in rotating traps
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atoms in dark states
for dark states see e.g. E. Arimondo,
Progress in Optics XXXV (1996)
adiabatic eigenstates

?
O
D
-
?
dark state (no fluoresence)
p
s
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center of mass motion of atoms in dark states

• space-dependent dark states atomic motion

0gt
R. Dum M. Olshanii, PRL 76, 1788 (1996)
O
O
p
s
1gt
2gt
transformation to local adiabatic basis
? gauge potential A scalar potential
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(i) magnetic fields
O
O
p
s
effective vector potential magnetic field

relative momentum vector
difference of center of mass of light beams
relative orbital angular momentum needed !
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magnetic fields (a) vortex light beams
V
eff
ratio of fields
external trap
B
G. Juzeliunas and P.Öhberg, PRL 93, 033602
(2004) P. Öhberg, J. Ruseckas, G. Juzeliunas,
M.F. PRA 73, 025602 (2006)
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magnetic fields (b) shifted light beams
x
V
eff
?x
B
z
y
? ? ?? ?x

• Quantum-Hall effect in non-cylindrical systems
• non-stationary situation possible (current in z)

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(ii) non-Abelian gauge fields
J. Ruseckas, G. Juzeliunas, P. Öhberg, M.F.
Phys.Rev.Lett 95 010404 (2005)

• more than one relevant adiabatic state !
  TRIPOD scheme

2 x 2 vector matrix
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magnetic monopole field
singularity lines
O
O
2
1
O
3
? point singularity at the origin
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summary

• motion of atom in space-dependent dark states
• ? gauge potential A
• light beams with relative OAM
• ? magnetic field B
• degenerate dark states
• ? non-Abelian magnetic fields (monopoles,...)

• vortex light beams
• displaced beams (non-cylindrical geometry,
  currents)

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quantum gases as many-body model systems

• lattice models

Bose-Hubbard model Bose-Fermi-H. model spin
models

• BCS BEC
• crossover

Feshbach resonances fermionic superfluidity

• quantum-Hall physics
• rotating traps
• vortices, vortex lattices
• lowest Landau level

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quantum gases as many-body model systems

• quantum-Hall physics
• rotating traps
• vortices, vortex lattices
• lowest Landau level

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magnetic fields (a) vortex light beams
V
eff
external trap
B
ratio of fields
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many-body solid-state physics
ultra-cold atoms molecules
instruments of quantum optics coherent control
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quantum-Hall physics
filling factor
2
(R / l )
N flux quanta N atoms
?
m

• hydrodynamics ? gtgt 1

• quantum effects ? 1 ? ?

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