Nature of the Correlated Insulator States in Twisted Bilayer Graphene

Ming Xie and A. H. MacDonald

Phys. Rev. Lett. 124, 097601 – Published 2 March 2020

Abstract

We use self-consistent Hartree-Fock calculations performed in the full $π$ -band Hilbert space to assess the nature of the recently discovered correlated insulator states in magic-angle twisted bilayer graphene (TBG). We find that gaps between the flat conduction and valence bands open at neutrality over a wide range of twist angles, sometimes without breaking the system’s valley projected $C_{2} T$ symmetry. Broken spin-valley flavor symmetries then enable gapped states to form not only at neutrality, but also at total moiré band filling $n = \pm p / 4$ with integer $p = 1$ , 2, 3, when the twist angle is close to the magic value at which the flat bands are most narrow. Because the magic-angle flat band quasiparticles are isolated from remote band quasiparticles only for effective dielectric constants larger than $\sim 20$ , the gapped states do not necessarily break $C_{2} T$ symmetry and as a consequence the insulating states at $n = \pm 1 / 4$ and $n = \pm 3 / 4$ need not exhibit a quantized anomalous Hall effect.

Received 10 December 2018
Revised 1 January 2020
Accepted 5 February 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.097601

Physics Subject Headings (PhySH)

Quantum anomalous Hall effect Topological insulators

Graphene Topological materials

Electron-correlation calculations Hartree-Fock methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Ming Xie and A. H. MacDonald

Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA

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Vol. 124, Iss. 9 — 6 March 2020

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Images

Figure 1
Phase diagram of neutral bilayers as a function of Coulomb interaction strength, characterized by inverse dielectric constant $ε^{- 1}$ , and twist angle $θ$ . (a) Insulating states (blue regions) appear for ever weaker interactions as the narrow-band magic angle regime near $θ \sim 1.1 °$ is approached. Above the magic angle, the bilayer’s Dirac points are gapped when the effective fine structure constant $α^{*} = e^{2} / ε ℏ v_{D}^{*}$ exceeds $\sim 0.4$ (solid line), where $v_{D}^{*}$ is the Dirac velocity of the twisted bilayer,. ( $v_{D}^{*}$ goes to zero as the magic angle is approached [2].) For $ε^{- 1} ≳ 0.05$ remote band degrees of freedom play a role in determining the phase diagram details, enabling in particular insulating states that do not break $C_{2} T$ symmetry (hatched region of the phase diagram). (b) $C_{2} T$ -breaking order parameter $ρ_{B} - ρ_{A}$ (i.e., sublattice polarization) and (c) global energy gap as a function of interaction strength $ε^{- 1}$ for $θ = 1.1 °$ (blue solid line) and 1.5° (red dashed line).
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Figure 2
Properties of quasiparticles at $n = 0$ . Only valley $K$ states are shown. (a) Energy dispersion of the $C_{2} T$ breaking (blue) and $C_{2} T$ preserving (red) states at $θ = 1.1 °$ and $ε = 5$ . The dotted lines illustrate the noninteracting flat bands. The inset specifies the high symmetry lines in the moiré BZ (mBZ) along which the band energies have been plotted. (b) Berry curvature for lowest conduction band (upper) and the highest valence band (lower) of the broken $C_{2} T$ -symmetry solution. Features in these plots are related to avoided crossings between the flat conduction band and higher energy remote conduction bands. (c) Order parameters for $C_{2} T$ symmetry breaking as a function of the reciprocal lattice vector $G$ magnitude. The segments marked by vertical dashed lines group $G s$ with the same magnitude.
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Figure 3
Quasiparticle dispersion of the SCHF ground states for $θ = 1.1 °$ and $ε = 10$ at filling factors: (a) $n = - 3 / 4$ and (b) $n = - 1 / 2$ . The lowest conduction bands and the highest valence bands are plotted as solid lines while remote bands are plotted as dashed lines with flavor-dependent colors. The charge gaps (shaded gold) are $E_{g} = 2.86 meV$ for $n = - 3 / 4$ and $E_{g} = 3.15 meV$ for $n = - 1 / 2$ . The flavor dependent band occupation numbers $n_{K / K^{'}, ↑ / ↓}$ are measured from neutrality so that 0 means that the corresponding valence band is occupied and $- 1$ means that it is empty. (c) $C_{2} T$ order parameter and (d) global energy gap as a function of twist angle at $n = - 1 / 2$ in a state with one flat valence band occupied for each valley. The red (dashed) and blue (solid) lines correspond to $ε = 10$ and 25, respectively. The flavor polarized insulators are metastable, i.e., the assumed gap is self-consistent, in an interaction strength dependent interval (shaded regions) on the high-twist angle side of the magic angle.
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