Correlated insulating phases of twisted bilayer graphene at commensurate filling fractions: A Hartree-Fock study

Yi Zhang, Kun Jiang, Ziqiang Wang, and Fuchun Zhang

Phys. Rev. B 102, 035136 – Published 20 July 2020

Abstract

Motivated by the recently observed insulating states in twisted bilayer graphene, we study the nature of the correlated insulating phases of the twisted bilayer graphene at commensurate filling fractions. We use the continuum model and project the Coulomb interaction onto the flat bands to study the ground states by using a Hartree-Fock approximation. In the absence of the hexagonal boron nitride substrate, the ground states are the intervalley coherence states at charge neutrality (filling $ν = 0$ , or four electrons per moiré cell) and at $ν = - 1 / 4$ and $- 1 / 2$ (three and two electrons per cell, respectively) and the $C_{2} T$ symmetry-broken state at $ν = - 3 / 4$ (one electron per cell). The hexagonal boron nitride substrate drives the ground states at all $ν$ into $C_{2} T$ symmetry broken-states. Our results provide good reference points for further study of the rich correlated physics in the twisted bilayer graphene.

Received 15 January 2020
Revised 7 June 2020
Accepted 1 July 2020

DOI:https://doi.org/10.1103/PhysRevB.102.035136

Physics Subject Headings (PhySH)

Electronic structure

Bilayer films Graphene Unconventional superconductors

Bloch wave theory Hartree-Fock methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yi Zhang ¹, Kun Jiang^2,3, Ziqiang Wang³, and Fuchun Zhang^1,4

¹Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China
²Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
³Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
⁴Chinese Academy of Sciences Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China

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Issue

Vol. 102, Iss. 3 — 15 July 2020

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Images

Figure 1
(a) Schematic phase diagram of TBG in the plane spanned by the commensurate filling fraction and hBN potential $Δ_{hBN}$ . Red stands for the $C_{2} T$ -breaking insulator ( $C_{2} T I$ ), while cyan stands for the intervalley coherent (IVC) state. (b) TBG contains two sets of flat bands from valleys $τ = +$ and $τ = -$ . The IVC state is favored by the intervalley scattering (cyan arrow). $C_{2} T I$ is favored by the interband scattering in each valley (red arrow). (c) Atomic structure of the TBG with twisted angle $θ = 6 . 01^{\circ}$ , where the red and blue dots represent the A and B sublattices. The AA region corresponds to the region where the two red dots from the two layers lie on top of each other, while AB and BA correspond to the region where the red and blue dots overlap with each other. (d) Schematic plot of the Brillouin zone folding in TBG, where the small hexagon represents the moiré Brillouin zone reciprocal to the moiré superlattice structure in (c). $G_{1}$ and $G_{2}$ are the reciprocal lattice vectors of the moiré Brillouin zone.
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Figure 2
(a) Total energy per particle for the ferromagnetic insulator (FMI), $C_{2} T$ -breaking insulator ( $C_{2} T I$ ), and intervalley coherent (IVC) states at charge neutrality for various values of the intrasublattice interlayer coupling $u_{0}$ , where we choose $Δ E$ to be relative to the average energy of these three states to clarify the small energy difference. (b)–(d) Typical energy dispersions of states of these three groups of states, where $u_{0} = 0.08$ eV and the chemical potential is moved to the center of the gap for convenience. The noninteraction bands calculated from $H_{B M}$ are also plotted for comparison.
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Figure 3
(a) Total energy per particle for the ferromagnetic insulator (FMI), flavor-polarized $C_{2} T$ -breaking insulator ( $C_{2} T FMI$ ), and intervalley coherent (IVC) states at half filling for various values of the intrasublattice interlayer coupling $u_{0}$ , where we choose $Δ E$ to be relative to the average energy of these three states to clarify the small energy difference. (b)–(d) Typical energy dispersions of states of these three groups of states, where $u_{0} = 0.08$ eV and the chemical potential is moved to the center of the gap for convenience.
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Figure 4
Evolution of the intervalley coherent (IVC) state and $C_{2} T$ -breaking order parameters with the increase of the staggered potential $Δ_{hBN}$ for $u_{0} = 0.08$ eV at (a) half filling $ν = - 1 / 2$ and (b) quarter filling $ν = - 1 / 4$ .
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Figure 5
Total energy per particle for the flavor-polarized $C_{2} T$ -breaking insulator ( $C_{2} T FMI$ ) and mixed intervalley coherent (IVC) states at (a) $ν = - \frac{3}{4}$ and (b) $ν = - \frac{1}{4}$ for various values of the intrasublattice interlayer coupling $u_{0}$ , where we choose $Δ E$ to be relative to the average energy of these two states to clarify the small energy difference. (c) and (d) Typical energy dispersions of states of these two groups of states, where $u_{0} = 0.08$ eV and the chemical potential is moved to the center of the gap for convenience.
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Figure 6
Total energy per particle for the ferromagnetic insulator (FMI), flavor-polarized $C_{2} T$ -breaking insulator ( $C_{2} T FMI$ ), and intervalley coherent (IVC) states at half filling for various values of the gate distance $d_{s}$ . Inset: the relative energy $Δ E$ to the average energy of these three states to clarify the small energy difference.
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