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