Download Quantum Hall effects

Quantum Hall effects
- an introduction M. Fleischhauer
AvH workshop, Vilnius, 03.09.2006
Kaiserslautern, April 2006
quantum Hall history
discovery: 1980
IQHE
Nobel prize: 1985
K. v. Klitzing
FQHE
discovery: 1982
Nobel prize: 1998
D. Tsui
H. Störmer R. Laughlin
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classical Hall effect (1880 E.H. Hall)
Lorentz-force on electron:
stationary current:
Hall resistance:
2
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
radial vector of
cyclotron motion cyclotron motion
electron
R
X
commutation relations:
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2D electrons in magnetic fields: Landau levels
mapping to oscillator:
H = hc R² / 2 l²m = h c ( a † a + ½ )
Landau levels
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2D electrons in magnetic fields: Landau levels
typical scales:
B
B
• 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 n´th 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
Kaiserslautern, April 2006
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
3D:no projected area in (xy)
2D always projected area in (xy)
A
A
B
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
0
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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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)
|0>
γ
Δ
Ωp
|1>
adiabatic eigenstates:
Ωs
|2>
+
Ω
γ
D
-
γ
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:
|0>
Ωp
R. Dum & M. Olshanii,
PRL 76, 1788 (1996)
|1>
Ωs
|2>
transformation to local adiabatic basis:
 gauge potential A + scalar potential
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(i) magnetic fields
Ωp
Ωs
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
B
external trap
G. Juzeliūnas 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
Veff
x
y
z
B
 =   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
Ω
D1
D2
2 x 2 vector matrix
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magnetic monopole field
singularity lines
Ω1
Ω3
Ω2
 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
• vortex light beams
• displaced beams (non-cylindrical geometry, currents)
• degenerate dark states
 non-Abelian magnetic fields (monopoles,...)
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quantum gases as many-body model systems
• lattice models:
Bose-Hubbard model;
Bose-Fermi-H. model;
spin models
• quantum-Hall physics:
rotating traps
vortices, vortex lattices;
lowest Landau level
• BCS – BEC
crossover:
Feshbach resonances;
fermionic superfluidity
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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
Veff
external trap
B
ratio of fields
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ultra-cold atoms
& molecules
many-body &
solid-state physics
instruments of
quantum optics &
coherent control
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quantum-Hall physics
filling factor
NФ # flux quanta ~ (R / lm )
N # atoms
2
• hydrodynamics:  >> 1
• quantum effects:  ~ 1
 = 0
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