Quantum Hall Ferromagnets-II Ramin Abolfath Luis Brey Anton Burkov Rene Cote Jim Eisenstein Herb...
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Quantum Hall Ferromagnets-II
Ramin Abolfath Luis Brey Anton Burkov Rene Cote Jim Eisenstein Herb Fertig Steve Girvin Charles Hanna Tomas Jungwirth Kentaro
Nomura Leo Radzihovksy (Doug) Rajaraman Mark Rasolt Jairo Sinova Kun
Yang
Les Houches June 2006
University of Texas at Austin
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Skyrmions
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Skrymion in 2D O(3) Ferromagnet
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QHF Energy Functional
Landau Gauge
Energy Functional
Exchange Interaction
Form Factor
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Quantum Hall Skyrmions
Carry Charge !!
Sondhi et al. PRB ‘93
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Skyrmions in Ground State- II
Hartree-FockSkyrmion WF
Symmetric GaugeLLL
Wavefunctions
Classical # of reversed spins
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Skyrmions Size n=0 LL
Fertig et al. PRB ’94 ‘97
K
g
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Fertig et al. PRB ’94Barrett & Tycko PRL ‘95
Skyrmions in Ground State
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Skyrmion Crystal
Brey et al. PRL ‘95
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Cote et al. PRL ‘97
Skyrmion Crystal Collective Excitations
k1.5 ~ phonons
k1 ~ new
Larmour
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QHF Graphene
Nomura cond-mat/06 – Alicea cond-mat/06 – Yang cond-mat/06
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Graphene Landau Levels
Conduction
Valence
Semenoff Haldane Kane Mele
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Graphene Quantum Hall Effect
Novoselov et al. – Manchester – Nature (2005)
xx
xy
n
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Zhang et al. – Columbia – cond-mat/0602649
B= 45 Tesla
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SU(4) FerromagnetismArovas/Karlhede (1999) – Ezawa (2002)
A(E)
Landau Band
Exchange Integral
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QHF Stoner Criterion
Fogler & Shklovskii (1995); Maude et al. (2005)
AX > 1 QHF
Landau Band Width
InteractionEnergy Scale
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Zero Field Conductivity
Vgate
Geim Nature (2005)
Golden Rule
?
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Coulomb Scattering I
where
2-3
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Coulomb Scattering II
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Self-consistent Screening
Das Sarma Ando
Screened Exchange
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Phase Diagram
Manchester 1.5Columbia 5.0
Nomura cond-mat/0603 PRL
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Broken Symmetry
SU(N)/(SU(M) x SU(N-M)) coset space
2 (N-M) parameters
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Graphene Skyrmions - I Graphne Form Factor
Yang et al., cond-mat/0605666
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Graphene Skyrmions – II
SU(N)/(SU(M-1) x SU(N-M-1)) coset space
2 (N-M) + 2 (N-1) parameters
Skyrm Lattice has N Goldstone modes
k1.5 & 2 k1 & k2
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Next Time:
• Skyrmions
• QHF Anistropy – Ising or XY
• QHF in Graphene
• Quantum Hall Superfluids & Pseudo-spin Transfer Torques