Theories for the correlated insulating states and quantum anomalous Hall effect phenomena in twisted bilayer graphene

The experimentally observed correlated insulating states and quantum anomalous Hall (QAH) effect in twisted bilayer graphene (TBG) have drawn significant attention. However, up to date, the specific mechanisms of these intriguing phenomena are still open questions. Using an all-band Hartree-Fock variational method, we have explained the correlated insulating states and QAH effects at some integer fillings of the flat bands in TBG. Our results indicate that states breaking flavor (valley and spin) symmetries are energetically favored at all integer fillings. In particular, the correlated insulating states at $\pm 1/2$ filling and at the charge neutrality point are all valley polarized states which break $C_{2z}$ and time-reversal ($\mathcal{T}$) symmetries but preserve $C_{2z}\mathcal{T}$ symmetry. Such valley polarized states exhibit “moiré orbital antiferromagnetic ordering” on an emergent honeycomb lattice with compensating circulating current pattern in the moiré supercell. Within the same theoretical framework, our calculations indicate that the $C=\mp 1$ QAH states at $\pm 3/4$ fillings of the magic-angle TBG are spin and orbital ferromagnetic states, which emerge when a staggered sublattice potential is present. We find that the nonlocalness of the exchange interactions tends to enhance the bandwidth of the low-energy bands due to the exchange-hole effect, which reduces the gaps of the correlated insulator phases. The nonlocal exchange interactions also dramatically enhance the spin polarization of the system, which significantly stabilizes the orbital and spin ferromagnetic QAH state at $3/4$ filling of TBG aligned with hexagonal boron nitride (hBN). We also predict that, by virtue of the orbital ferromagnetic nature, the QAH effects at electron and hole fillings of hBN-aligned TBG would exhibit hysteresis loops with opposite chiralities.

Figure 1

(a) The lattice structure and Brillouin zone of the twisted bilayer graphene (TBG). The band structure of TBG at the magic angle $\theta =1.{05}^{\circ }$ are shown in (b) and (c): (b) without the hBN substrate and (c) aligned with the hBN substrate. See text for more details.

Figure 2

The two paths of spontaneous symmetry breakings in twist bilayer graphene: (a) without the alignment of hBN substrate and (b) with the alignment of hBN substrate. See text for more discussions.

Figure 3

The indirect gaps (in units of eV) of the Hartree-Fock ground states (with local-exchange approximation) of magic-angle TBG (without hBN alignment) at different fillings: (a) at 1/2 filling, (b) at 0 filling, (c) at 1/4 filling, (d) at 3/4 filling.

Figure 4

The real-space distributions of current densities and magnetic fields for the correlated insulating states in magic-angle TBG at 1/2 filling, with $\epsilon =3.5$, $\kappa =0.005{\text{Å}}^{-1}$: (a) in the $\beta$ phase [see Fig. 3] with the staggered sublattice potential $\mathrm{\Delta }=0$ and (b) in the ${\alpha }_M$ phase [see Fig. 6] with staggered sublattice potential (induced by the aligned hBN substrate) $\mathrm{\Delta }=15\mathrm{meV}$ exerted on the bottom graphene layer. The color coding indicates the strength of magnetic field in units of Gauss. The directions and amplitudes of the black arrows represent the directions and amplitudes of the current densities.

Figure 5

The band structures of the Hartree-Fock ground states of magic-angle TBG at different fillings: (a) 1/2 filling, $\epsilon =3.5$, $\kappa =0.005{\text{Å}}^{-1}$, (b) 0 filling, $\epsilon =5$, $\kappa =0.01{\text{Å}}^{-1}$, (c) 1/4 filling, $\epsilon =3.5$, $\kappa =0.005{\text{Å}}^{-1}$, (d) 3/4 filling, $\epsilon =5$, $\kappa =0.01{\text{Å}}^{-1}$. The blue and red lines in (a) and (c) represent the majority and minority spins. The gray dashed lines mark the chemical potentials.

Figure 6

The calculated anomalous Hall conductivities (in units of $e^2/h$) of the Hartree-Fock ground states for TBG aligned with hBN substrate with local-exchange approximation: (a) at $3/4$ filling, (b) at $-3/4$ filling, (c) at 1/2 filling, and (d) at $-\mathrm{1/2}$ filling. The insets schematically show the hysteresis loops of the QAH effects at different fillings.

Figure 7

The indirect gaps (in units of eV) of the Hartree-Fock ground states with nonlocal exchange interactions for magic-angle TBG (without hBN alignment): (a) at 1/2 filling, (b) at 0 filling, (c) at 1/4 filling, (d) at 3/4 filling.

Figure 8

The anomalous Hall conductivities (in units of $e^2/h$) and indirect gap (in units of eV) of the Hartree-Fock ground states with nonlocal exchange interactions for hBN-aligned TBG at the magic angle: (a) AHC at 3/4 filling, (b) AHC at 1/4 filling, and (c) indirect gap at 1/2 filling.

Figure 9

The dependence of the Fock order parameters $O_0^{F,zz}(\mathbf{k},\mathbf{Q})$ and $O_0^{F,xy}(\mathbf{k},\mathbf{Q})$ on atomic wave vector $\mathbf{k}$ and moiré reciprocal lattice vector $\mathbf{Q}$. (a) and (b): The $\mathbf{k}$ dependence of the $\left|\mathbf{Q}\right|=0$ Fock order parameters $O_0^{F,zz}(\mathbf{k},|\mathbf{Q}|=0)$ and $O_0^{F,xy}(\mathbf{k},|\mathbf{Q}|=0)$, in units of eV. $k_x$ and $k_y$ are in units of the moire reciprocal lattice constant $q_M=4\pi /\left(\sqrt{3}{L}_s\right)$. (c) and (d): $O_0^{F,zz}(\mathbf{k},|\mathbf{Q}|=0)$ and $O_0^{F,xy}(\mathbf{k},|\mathbf{Q}|=0)$ plotted along the vertical dashed line passing through the $M_s$ point shown in (a) and (b). The blue circles in (c) and (d) represent the actual calculated data, and the blue lines represent Gaussian fittings to the actual data. (e) and (f): The $\left|\mathbf{Q}\right|$ dependence of $O_0^{F,zz}(\mathbf{k},\mathbf{Q})$ and $O_0^{F,xy}(\mathbf{k},\mathbf{Q})$ at $\mathbf{k}=M_s$ $[O_0^{F,zz}(M_s,\mathbf{Q})$ and $O_0^{F,xy}(M_s,\mathbf{Q})]$, in units of eV.