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

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

I

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

Two-Dimensional Electron Gas

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

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

Cyclotron Orbits

Landau Levels

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

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

Fractional Quantum Hall Effect

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

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

Fractionally Charged Quasiparticles

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

Thermodynamic Stability? Hard Core Model Chemical Potential vs. Density

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

II

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, (1990)

Quantum Hall Edge States Skipping Orbits

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

Field Theory of QH Edge Hamiltonian More on V later

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

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

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

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

First Quantization Bosonization

Bosonization by Example

Luttinger Liquids 3D E k 1D

Density of States Anomaly

Spin-Charge Separation Alexi Tsvelik

Tunneling DOS Calculation Fermi Golden Rule

Tunneling DOS Calculation

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

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

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

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

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

Magnetoplasmons in Line Junction Systems Safi Schulz PRB 1995,1999

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

Line Junction Systems – Split Gate

Line Junction Systems – CEO Kang et al. Nature 2002

Corner Line Junctions Grayson et al. 2004, 2005

Interaction Parameter Theory Hartree-Fock Energy Functional

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

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

Quantum Hall Domain Walls  Baking Bread

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

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

Smooth Edge Model

III

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

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

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

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

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

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

Electron-Electron Pairs Cooper PairsOrder Parameter Superflow

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

Ga Al Si Bilayers

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

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

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

… also Keldysh JETP 1968

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

Excitonic BEC and Superfluidity?

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

BCS Nambu-Gorkov & PHT Attractive Interactions Repulsive Interactions

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

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

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

How to detect an excitonic BEC No Odd Channel Resistivity 1996

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

Interlayer Voltage 0 Tunneling rate 0 Weak to Strong Coupling Transition

I I e - voltmeter 0 0

Electron-hole Pair Current

Superflow in Electron-Electron Bilayers Kellogg and Eisenstein cond-mat/

Superflow in Electron-Electron Bilayers Kellogg and Eisenstein cond-mat/

Topological Charge = Electric Charge

Vortex-Flow Dissipation

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

Joglekar TDHFA+SCBA Joglekar PRL (2002)

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

=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

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

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

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

Nunez+AHM, cond-mat/ Spin-Transfer Theory

Transport Orbitals eV Condensate Orbitals K = X l 2 Coherent Edge Transport E mpl  eV Δ QP  eV G  e 2 /h Δ t  eV V *  Volts

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??

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