Topological Mott Insulator in Semiconductor
Moire Heterostructure

Liang Fu

Yang Zhang

Trithep Devakul

Margarita DavydovaCornell:

Jie ShanKin Fai Mak

Two Types of Insulators
Band insulators• even integer fillings• energy gap due to Pauli exclusion

Mott insulators• odd integer fillings• interaction induced gap

Normal Band Insulators• energy gap remains finite as the crystal is taken apart• topologically equivalent to atomic limit
EnergyInteratomic separation

Band insulator: NaCl = Na1+ & Cl1-

Topological Insulators• cannot be taken apart smoothly• rely on the wave nature of electron
TopologicalEk

Atomic Insulator

Atomic versus Topological Insulators

Dimmock et al(1966)• PbTe = Pb2+ & Te2• SnTe ≠ Sn2+ & Te2-Hsieh et al(2012)

Continuous Phase Transition
between normal and topological band insulators
Dirac field theory: 𝐻 = −𝑖 ∇ ⋅ Γ + 𝑚Γ0

Ando et al (2012)• change of topology = sign change of Dirac mass
• boundary (= domain wall) hosts massless Weyl fermion• interaction is irrelevant at critical point

Mott Insulators
NiO = Ni2+ & O2-• electrons bound to individual atoms• electron motion prohibited by local repulsion 𝑈𝑛𝑖↑ 𝑛𝑖↓

MottTopology

Part 1: Inverting Mott Insulators

Continuous phase transition from 120∘ AFM Mott insulator
to Chern insulator with spin chirality
Zhang, Devakul & LF, PNAS (2021)
Devakul & LF, PRX (2022)

Moire Superlattices
Twisted bilayer graphene

Semiconductor heterostructureWSe2/MoSe2

𝑎𝐾𝐾′𝒑2
𝐻=+ 𝑉(𝒓 )2𝑚𝑉(𝒓) : superlattice potential

Tight-binding regime at large moire period: ℏ2 /𝑚𝑎2 ≪ 𝑉=> triangular lattice of “quantum dots” & Hubbard model𝑎𝑀 ∼ 8nm𝑛:12Mott insulator at 𝑛 = 1 (half band filling)

1 2
3 3

Wigner crystals at 𝑛 = , …

WC & MIT: Chenhao Jin, Cenke Xu, Senthil, Kim, Das Sarma, Philips, MacDonald …

Local Moment and AFM Interaction𝜒∝Tang et al, Nature (2020)

1𝑇−𝜃Charge-Transfer Mott Insulator

Mott-Hubbard insulator

𝑈𝑈 > Δ:
Charge transfer insulator(similar to CuO2)

UHLHUHΔMXXMLH• moire potential may have two minima in a unit cell• doped charges at n>1 occupy secondary minima to avoid 𝑈.
• insulating gap at n=1 set by Δ
Zhang, Yuan & LF, PRB (2020)

Intralayer Charge Transfer in WSe2/WS2

Xiaodong Xu’s group (submitted)
See also Xu et al, arXiv:2202.02055

𝑛=1

𝑛=2

charge-transfer
Mott insulator

honeycomb lattice
Mott-Hubbard

𝐸MoTe2/WSe2• Electric field tunes interlayer charge transfer

Moire Bands in MoTe2/WSe2
13 x13 MoTe2

𝑎𝑀 = 4.6nm

MoTe2

WSe2

Δ

• Δ = 0.13eV at zero E field• majority layer: MoTe2
minority layer: WSe2
• moire band due to lattice corrugation
with bandwidth ~ 50 meV

14 x14 WSe2

E Field Tunes Band Inversion• electric field inverts minibands on two layers• band inversion + p-wave interlayer tunneling => valley Chern number

Prediction: E field induced quantum spin Hall insulator at n=2
Zhang, Devakul & LF, PNAS (2021)

Edge Transport in MoTe2/WSe2

Zhao et al, arXiv:2207.02312

Quantum Anomalous Hall Effect at Half Fillingband insulator

Mott insulator

topologicalband insulator

topologicalMott insulator

E Field Induced Mott-QAH Transition• absence of E field hysteresis• robust and reproducible

Mott to QAH Insulators
Mott-QAH transition by inverting charge transfer gap
QAHMottMetal𝑈↗

UHΔ↘

𝑈CTΔ𝑊

𝛿<0LH
Devakul & LF, PRX (2022)

𝛿~ Δ − 𝑊120∘ -AFM Mott Insulator
𝐸𝐸UH𝑈

↓↑

CT
Δ×2×1

LHDOS

𝑘 in magnetic BZ• Quasiparticle bands in magnetically ordered insulator aredifferent from noninteracting bands.
• Low-energy states: spin-polarized holes on majority layer &spin-degenerate electrons on minority layer

Interacting Field Theory𝐸(𝑘)ℋeff = න𝜓 † 𝐻eff 𝜓 𝑑𝒌 + 𝑔 න𝑛𝐵↑ 𝑟 𝑛𝐵↓ 𝑟 𝑑𝒓

𝑘2−2𝑚𝐴×2

𝛿

0×1

𝐻eff =""""

𝜓 = (𝜓𝐴 , 𝜓𝐵↑ , 𝜓𝐵↓ )𝜆(𝑘𝑥 +𝑖𝑘𝑦 ) 𝜆(𝑘𝑥 − 𝑖𝑘𝑦 )
𝑘2+𝛿
2𝑚𝐵

00𝑘2+𝛿
2𝑚𝐵𝑘•••

spin degeneracy at k=0 on minority layer protected by 𝐶3 & 𝑇𝑠𝑧
p-wave hybridization dictated by band symmetry
𝑔: electron repulsion on minority layer

Interacting Field TheoryQuasiparticle band at 𝑔 = 0𝛿0−4𝑚𝐵𝜆2𝛿After band inversion 𝛿 < 0, quadratic band touching appears at Fermi
level, which is unstable to repulsion 𝑔 Sun, Yao, Fradkin, Kivelson, PRL 2009

𝑔 > 0 changes from irrelevant to marginally relevant at band inversion!

QAH with non-coplanar magnetism

FM spinsplitting

𝛿<0|𝐶| = 1• chiral spin order: (canted) xy-AFM in MoTe2 & Ising FM in WSe2
• Ising FM opens Chern gap at quadratic band touching

Continuous Mott-Chern Transition

𝛿>0FM spinsplitting

𝛿<0𝐶=0|𝐶| = 1

Inverting charge transfer gap induces simultaneous change ofmagnetism & topology.
Devakul & LF, PRX (2022)

Topological Band & Mott Insulators
Inverting single-particle gap(even-integer filling)

Inverting many-body gap(odd-integer filling)
𝐸−+−+UH𝑈CTΔLH
2007

2022

AFM Mott insulators with negative charge transfer gap:
potential route to high-temperature QAH

Bridging Mott and Chern

Comparison with Other QAH Systems

Chang et al (2013)
Magnetically doped TI film
• FM of dopant opens Chern gap at surface Dirac point• even integer filling
Sharpe et al, Serlin et al (2019)
Magic-angle graphene• fully valley-polarized flat Chern band

𝐶𝐾 = +1

𝐶𝐾′ = −1

Zhang & Senthil, MacDonald,
Xie et al, Pan et al …

QAH in MoTe2/WSe2 differs fundamentally from flat band FM.

Comparison with ExperimentHartree-Fock Phase Diagram

MoTe2/WSe2

Prediction for Magnetism• Mott: zero spin 𝑆𝑧 polarization• QAH: finite but incomplete spin 𝑆𝑧 polarization
increasing with B field and E field• Intervalley XY magnetic order: gapless magnon

Evidence for Canted Spin Texture in QAH

Tao et al, arXiv: 2208.07452

Outlook• charge gap across Mott-Chern transition• spin superfluidity• critical exponents• inverting quantum spin liquid