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Capri Spring School

Capri Spring School

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Capri Spring School

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  1. Capri Spring School Current Topics in Quantum Hall Research Allan MacDonald University of Texas at Austin

  2. QHE – Incompressible States • QHE – Edge States & Line Junctions • 3.QHE – Bilayer Spontaneous Coherence • & Counterflow Superfluidity

  3. I

  4. I – References on QHE cond-mat/9410047 Introduction to the Physics of the Quantum Hall Regime (figures available by e-mail request The Quantum Hall Effect (Richard Prange and Steven Girvin)

  5. Two-Dimensional Electron Gas

  6. Molecular Beam Epitaxy heated cells Al Ga As ultra high vacuum high quality GaAs substrate

  7. Integer Quantum Hall Effect xx xy/(h/e2)

  8. Cyclotron Orbits

  9. Landau Levels

  10. Lowest Landau Level Bottom of Ladder Analytic Wavefunctions Orbit Center Ladder Operator

  11. Incompressible States & Streda Formula Edge Current Conductance and LL degeneracy Compressibility

  12. Fractional Quantum Hall Effect

  13. Haldane Pseudopotentials 2-particle states Haldane Pseudopotentials Center of Mass & Relative Details Hardly Matter!

  14. Laughlin Wavefunction COM & Relative for each pair Hard-core model E=0 Eigenstates Laughlin Wavefunction FQHE Hamiltonian LLL Wavefunctions

  15. Fractionally Charged Quasiparticles

  16. Composite Fermions Flux Attachment  =1/3   = 1 Fractionally Charged Quasiparticles  = 2/5

  17. Thermodynamic Stability? Hard Core Model Chemical Potential vs. Density

  18. Outline I 2DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractionally Charge Quasiparticles, Composite Fermions, Thermodynamic Instability II Edge States,Chiral Luttinger Liquids, Line Junctions, Experiment Canonical Line Junction Models, Sine-Gordon Models III Experiment, Bilayer Mean Field Theory, Easy-Plane Ferromagnet Analogy, Josephson-Junction Analogy Counterflow Superfluidity, Interlayer Tunneling

  19. II

  20. II – Quantum Hall Edge State References Review: A.M. Chang, Rev. Mod. Phys. 74, 1449 (2003) Original Chiral Luttinger Liquid Paper: X.G. Wen, Phys. Rev. B 41, 12838 (1990)

  21. Quantum Hall Edge States Skipping Orbits

  22. Edge States X = k l2 i = kF/2π kF0 kF1

  23. Field Theory of QH Edge Hamiltonian More on V later

  24. Free Chiral Bosons Field Theory of QH Edge Filling Factor  Creation & Annihilation

  25. Local Fermi Wavevector Chiral Density Wave  Field Theory of QH Edge  Conjugate Variable

  26. Edge Magnetoplasmons Frequency Domain:Wassermeier et al. PRB (1990)

  27. Magnetoplasmons in time Domain ns Time Domain: Ashoori et al. PRB (1992)

  28. First Quantization Bosonization

  29. Bosonization by Example

  30. Luttinger Liquids 3D 1D E k

  31. Density of States Anomaly

  32. Spin-Charge Separation Alexi Tsvelik

  33. Tunneling DOS Calculation Fermi Golden Rule

  34. Tunneling DOS Calculation

  35. TunnelingintoEdge Tunneling Grayson, Chang et al. PRL 1998,2001

  36. Edge State Measurements Noise: Glattli et al. PRL (1997); Heiblum et al. (1997)

  37. But … what’s this?? 0 voltmeter 2DEG Hall Bar Roddaro et al. (Pisa) PRL 2003, 2004

  38. and … what’s this?? Roddaro et al. (Pisa) PRL 2003, 2004, 2005

  39. X=3L/4 X=0 Quantum Hall Condensate Quantum Hall Condensate X=L/2 X=L/4 Quantum Hall Line Junction

  40. Magnetoplasmons in Line Junction Systems Safi Schulz PRB 1995,1999

  41. Outline I 2DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractionally Charged Quasiparticles, Composite Fermions, Thermodynamic Instability II Edge States,Chiral Luttinger Liquids, Line Junctions, Experiment Canonical Line Junction Models, Sine-Gordon Models III Experiment, Bilayer Mean Field Theory, Easy-Plane Ferromagnet Analogy, Josephson-Junction Analogy Counterflow Superfluidity, Interlayer Tunneling

  42. Line Junction Systems – Split Gate

  43. Line Junction Systems – CEO Kang et al. Nature 2002

  44. Corner Line Junctions Grayson et al. 2004, 2005

  45. Hartree-Fock Energy Functional Interaction Parameter Theory

  46. ’   = δk/2π Interaction Parameter Theory Simple Chiral Edge ε(k) X = k l2

  47. ε(k) Attraction to NeutralizingBackground EMP Velocity ’  X = k l2  = δk/2π Interaction Parameter Theory Simple Chiral Edge

  48. Baking Bread  Quantum Hall Domain Walls