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Quantum Hall Effect: Incompressible & Edge States, Line Junctions, and Superfluidity

This article discusses the current topics and research in the Quantum Hall Effect, including incompressible states, edge states, line junctions, and counterflow superfluidity. The article also covers related topics such as the Integer and Fractional Quantum Hall Effect, Laughlin wavefunction, composite fermions, and thermodynamic stability. It further explores experimental findings, bilayer mean field theory, and analogies such as the easy-plane ferromagnet and Josephson-junction analogy. The article concludes with discussions on line junction systems, split gate line junctions, and corner line junctions.

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Quantum Hall Effect: Incompressible & Edge States, Line Junctions, and Superfluidity

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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

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