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

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

5. I

6. 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)

7. Two-Dimensional Electron Gas

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

9. Integer Quantum Hall Effect  xy /(h/e 2 )  xx

10. Cyclotron Orbits

11. Landau Levels

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

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

14. Fractional Quantum Hall Effect

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

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

17. Fractionally Charged Quasiparticles

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

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

20. 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

21. II

22. 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)

23. Quantum Hall Edge States Skipping Orbits

24. Edge States X = k l 2 k F1 k F0  i = k F /2π

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

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

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

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

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

30. First Quantization Bosonization

31. Bosonization by Example

32. Luttinger Liquids 3D E k 1D

33. Density of States Anomaly

34. Spin-Charge Separation Alexi Tsvelik

35. Tunneling DOS Calculation Fermi Golden Rule

36. Tunneling DOS Calculation

37. Tunneling into Edge Tunneling Grayson, Chang et al. PRL 1998,2001

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

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

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

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

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

43. 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

44. Line Junction Systems – Split Gate

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

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

47. Interaction Parameter Theory Hartree-Fock Energy Functional

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

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

50. Quantum Hall Domain Walls  Baking Bread

51. Sine-Gordon Model Kang et al. Nature 2002 Sine-Gordon Model

52. Fun with 2D Electrostatics Co-Planar appox. Conformal transformation

53. Smooth Edge Model

54. III

55. III – Bilayer Condensates Reference J.P. Eisenstein and A.H. MacDonald Nature 432, 7018 (2004).

56. superfluid helium superconductor Bose-Einstein Condensates (BECs) BEC of sodium atoms Durfee & Ketterle, Optics Express 2, 299 (1998)

57. References Eisenstein and AHM - cond-mat/0404 Nature Dec (2004) Abolfath, Radzihovsky & AHM – PRB (2004)

58. History of Superconductivity Kammerlingh Onnes 1911 Bardeen-Cooper-Schrieffer (BCS) 1957 Brian Josephson 1962 Bednorz and Mueller 1986 T ρ

59. Electrons polarize nearby ions creating surplus of positive charge Attractive e-e Interactions

60. Pairs of electrons behave like bosons  coherent many- body wavefunction Order Parameter is Classical Energy Barriers are Large

61. Electron-Electron Pairs Cooper PairsOrder Parameter Superflow

62. 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

63. Ga Al Si Bilayers

64. III QHE for =1/2 + 1/2 Spontaneous Phase Coherence

65. QH Bilayers Easy-Plane Ferromagnets Excitonic BECs (Josephson Junctions)

66. Excitons – Elementary Excitations of Intrinsic Semiconductors e - h h e

67. … also Keldysh JETP 1968

68. Electron-Hole Pairs (n’,n)=( ,  ) =Ferromagnetism (n’,n)=(c,v) = Excitonic BECs (n’,n)=(TopLayer,BottomLayer) Order Parameter Counterflow Superflow

69. Excitonic BEC and Superfluidity?

70. 3D E c + E V 2D Bilayer E c + E V 2D Bilayer in Field Exciton Condensation in Semiconductors Keldysh 1964 Lezovik 1975 Kuramoto 1978

71. BCS Nambu-Gorkov & PHT Attractive Interactions Repulsive Interactions

72. E c + E V 2D eh Bilayer E c + E V 2D eh Bilayer in Field Exciton Condensation in Bilayers Lezovik 1975 Kuramoto 1978 Bilayer QH 1991 E c + E V 2D ee Bilayer in Field

73. WHAT? Spontaneous Interlayer Coherence WHY? Gain in Interlayer Correlation Energy exceeds loss in Intralayer Correlation Energy TT Disordered Ordered BB Top Layer Electron Bottom Layer Electron Cloud Mean-Field Theory Description

74. Electrons and Holes in the QH Regime Add magnetic field Particle-hole transformation Assemble Bilayer

75. How to detect an excitonic BEC No Odd Channel Resistivity 1996

76. e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - Independent Contacts

77. Interlayer Voltage 0 Tunneling rate 0 Weak to Strong Coupling Transition

78. I I e - voltmeter 0 0

79. Electron-hole Pair Current

80. Superflow in Electron-Electron Bilayers Kellogg and Eisenstein cond-mat/0401521

81. Superflow in Electron-Electron Bilayers Kellogg and Eisenstein cond-mat/0401521

82. Topological Charge = Electric Charge

83. Vortex-Flow Dissipation

84. Collective Dynamics & Dissipation Ferromagnets vs. Josephson Junctions vs. Bilayers J.J. Dynamics Thin Film Ferromagnet Dynamics

85. Joglekar TDHFA+SCBA Joglekar PRL (2002)

86. Collective Dynamics Ferromagnets vs. Josephson Junctions vs. Bilayers J.J. Dynamics Thin Film Ferromagnet Dynamics +I st

87. =1, Ql B =0.838, V 0 /(e 2 /  l B )=1.5, N  =36, Symmetric Disorder local density super-current:  pseudo-spin d/l B =0.5

88. Collective Spin Transport I Easy Plane Free Magnet Perpendicular Easy- Axis Pinned Magnet Konig AHM et al. PRB (2003); PRL (2001)

89. Easy-Plane Current-Driven Dynamics Easy-Plane Fero = Superconductor = Quantum Hall Bilayer LL Dynamics Uniaxial Anisotropy Current Driven Micromagnetic Exchange

90. Spin Supercurrent I Konig AHM et al. PRB (2003); PRL (2001) Super Spin Current

91. Nunez+AHM, cond-mat/0403710 Spin-Transfer Theory

92. Transport Orbitals eV Condensate Orbitals K = X l 2 Coherent Edge Transport E mpl  10 -6 eV Δ QP  10 -3 eV G  e 2 /h Δ t  10 -9 eV V *  10 -6 Volts

93. Excitonic BEC does occurs in Bilayer QH Systems Excitonic BEC does lead to dramatic collective transport Challenges for Theory Height, width and field-dependence of zero-bias tunneling peak?? Hall and Longitudinal Resistivity at Finite T??

94. Outline I 2DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractional 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