SCZ Memorial Workshop

Magic Angle TBG: FQHE Reprise

Rafi Bistrizer Jung Jung Su Fengcheng WuMing Xie Pawel Potasz

University of Texas at Austin

Intrinsic Spin Hall Effect

Murakami, Nagaosa, and SCZ Science (2003)Sinova et al. PRL (2004)

Bilayer Two-Dimensional Electron Systems

< "#$ "% > ≠ 0

Spatially IndirectExciton Condensate

PRB (1994,95,96)

Electrons and Holes in the QH Regime

Add magnetic field

Particle-hole transformation

Assemble Bilayer

4

Kinetic EnergyQuenching !

!" + !$ = 1

Bilayer Two-Dimensional Electron Systems

Merons !

Charge e/2

Exciton Superflow – in pictures

Drag Counterflow

Su & AHM Nature Phys. (2008)Eisenstein von Klitzing

TL

BL BR

TRe e

e

e

h

Sublattice-Pseudospins

A

B

2D Massless Dirac Equation

vD ~ 106 m/s

2D wavevectorSublatticepseudospin

Bilayer AB stacking

A1 B1

A2 B2

Novoselov et al. – Nature Phys. (2006)

McCann & Falko – PRL (2006)

Bilayer Graphene N=0 FQHE

J.I.A. Li et al. Science (2017)

Fengcheng Wu et al. PRB (2014)

SO(5) Symmetry in Quantum Hall Effect

! = #°

%&

%'

Twisted vdW Heterojunctions

Periodic Tunneling Hamiltonian

TAB(r) AA

BA

AB

Rafi Bistritzer & AHM, PNAS (2011)

Fourier X-form of π-bandinterlayer hopping amplitude

tq decreases very rapidly with q

�1 parameter model

Magic Angles !

Bistritzer & AHM PNAS (2011) Morrell et al. PRB (2010)

43210-2-4

Magic Angle Flat Bands

Insulator Superconductor

-3 -1

T

Ong State

Gaps without Breaking C2T symmetry!Ong State ! = 0

Magic Angle Phase Diagram TAA/TAB=0.8

Continuum Model and Moiré Bands

φ = 2π/3

Rafi Bistritzer & AHM, PNAS (2011)

MATBG Mass Operator

("#, "%, "&)= "(1 + 2+, 1 − +, 1 − +)

+

.

Fock self-energy from remote valence bands

Sensitive Band Topology

Broken Flavor SymmetriesK↑ valence band empty " = −% (non-Ong State)

Xie & AHM arXiv:1812.04213

Insulators at All Quarters – Some Chern

Efetov Group – ICFO (Barcelona)

0 2 4 6 8-10

0

10

20

30

40

E [m

eV]

Nxky+kx

Finite-Size Exact Diagonalization

QP Bands

IncludingRemote-BandSelf-Energy

Ferromagnetic Insulators

Electrons

Holes

Band Shuffling

SCZ Memorial Workshop

Magic Angle TBG: FQHE Reprise

Rafi Bistrizer Fengcheng WuMing Xie Pawel Potasz

University of Texas at Austin