Triangular antiferromagnetism on the honeycomb lattice of twisted bilayer graphene

We present the electronic band structures of states with the same symmetry as the three-sublattice planar antiferromagnetic order of the triangular lattice. Such states can also be defined on the honeycomb lattice provided the spin density waves lie on the bonds. We identify cases which are consistent with observations on twisted bilayer graphene: a correlated insulator with an energy gap, yielding a single doubly degenerate Fermi surface upon hole doping. We also discuss extensions to metallic states which preserve spin-rotation invariance, with fluctuating spin density waves and bulk ${\mathbb{Z}}_2$ topological order.

Figure 1

Moiré superlattice resulting from two graphene sheets, indicated in red and blue, twisted by angle $\theta$ relative to each other. Here we focus on commensurate twist angles and assume that the rotation axis goes through the AA site [11, 12]. The regions of local AA, AB, and BA stacking each form a triangular lattice. The generators of the point group $D_3$ of the system are illustrated in the inset, where the twist angle $\theta$ has been chosen larger to make the geometry more clearly visible.

Figure 2

(a) Schematic of the triangular antiferromagnetism on the honeycomb lattice. The red, green, and blue (r, g, and b) hexagons label the three spin orientations indicated by the arrows at the hexagon centers: ${\mathit{S}}_{\mathrm{red}}=(-1/2,\sqrt{3}/2,0,0)$, ${\mathit{S}}_{\mathrm{green}}=(-1/2,-\sqrt{3}/2,0)$, ${\mathit{S}}_{\mathrm{blue}}=(1,0,0)$. The spins on the red, green, or blue bonds have the same orientation as the spin of that color. All types of bonds considered are shown emanating from a single site. (b) Triangular lattice sites are shown as squares. The dual honeycomb-lattice sites are indicated by turquoise and orange circles, corresponding to the $A$ and $B$ sublattices, respectively. The primitive vectors, ${\mathit{e}}_1$ and ${\mathit{e}}_2$, are drawn in navy, and ${\mathbf{\tau }}_{A,B}$ indicate which honeycomb-lattice site is associated to each site of the triangular lattice: ${\mathit{r}}_{A,B}=\mathit{r}+{\mathbf{\tau }}_{A,B}.$

Figure 3

The spectrum of the triangular lattice model with ${120}^{\circ }$ coplanar spin order, $H_t^{▵}+H_{\text{mag}}^{▵}$, along the one-dimensional momentum cut indicated in (a) is shown in (b). The Brillouin zone (BZ) of the moiré lattice is the large hexagon depicted in black. It contains the reduced BZ, colored in blue, which is $1/3$ the size of the full BZ. (c) and (d) show the Fermi surfaces (red and orange solid lines) in the magnetic BZ along with the momentum dependence of the band that crosses the Fermi energy [indicated in red and orange in part (b)] upon hole and electron doping of the half-filled state, respectively. In all plots, we have chosen $P_0/t=2.5$ leading to a full gap at half-filling of the triangular lattice, which corresponds to quarter-filling of the flat bands.

Figure 4

(a), (b) $P_2=0.75$, $t_3=-0.15$. (a) The band structure of the Hamiltonian $H_t^{⬡}+H_{\mathrm{mag}}^{⬡}$ along the momentum cut in Fig. 3. As a result of a residual antiunitary symmetry, all bands plotted are twofold degenerate. (b) Band structure of the band immediately below the gap at one quarter-filling; this corresponds to the band drawn in red in (a). The Fermi surface which is obtained upon hole doping is also drawn in red. The magnetic BZ is outlined in black. (c), (d) $P_2=0.9$, $t_3=0$. (c) and (d) show the same information as (a) and (b) but without third-nearest-neighbur hopping.

Figure 5

(a), (b), (c), and (e) plot the band structure along the cut in Fig. 3 for various values of $P_1$ and $V$. The orders present are shown in the inset on the top right. (d) and (f) show the Fermi surfaces which are obtained by hole-doping the models of (c) and (e), respectively. These are superposed on a color plot of the upper band (colored orange). (a) $V=0.0$, $P_1=0.2$. (b) $V=2.0$, $P_1=0.0$. Here, all bands have a twofold spin degeneracy. (c), (d) $V=2.0$, $P_1=2.0$. The red and orange bands contribute the red and orange Fermi surfaces in (d). (e), (f) $V=2.0$, $P_1=0.9$. The same as shown in (c) and (d), but with a larger magnetic moment.