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-Fock Skyrmion WF Symmetric Gauge LLL Wavefunctions Classical # of reversed spins

Skyrmions Size n=0 LL Fertig et al. PRB ’94 ‘97 K g

Fertig et al. PRB ’94 Barrett & Tycko PRL ‘95 Skyrmions in Ground State

Skyrmion Crystal Brey et al. PRL ‘95

Cote et al. PRL ‘97 Skyrmion Crystal Collective Excitations k 1.5 ~ phonons k 1 ~ 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/ B= 45 Tesla

SU(4) Ferromagnetism Arovas/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 Interaction Energy Scale

Zero Field Conductivity   V gate 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   1.5 Columbia   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/

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 k 1.5 & 2 k 1 & k 2

Next Time: Skyrmions QHF Anistropy – Ising or XY QHF in Graphene Quantum Hall Superfluids & Pseudo-spin Transfer Torques