All Magic Angles in Twisted Bilayer Graphene are Topological

We show that the electronic structure of the low-energy bands in the small angle-twisted bilayer graphene consists of a series of semimetallic and topological phases. In particular, we are able to prove, using an approximate low-energy particle-hole symmetry, that the gapped set of bands that exist around all magic angles have a nontrivial topology stabilized by a magnetic symmetry, provided band gaps appear at fillings of $\pm 4$ electrons per moiré unit cell. The topological index is given as the winding number (a $\mathbb{Z}$ number) of the Wilson loop in the moiré Brillouin zone. Furthermore, we also claim that, when the gapped bands are allowed to couple with higher-energy bands, the $\mathbb{Z}$ index collapses to a stable ${\mathbb{Z}}_2$ index. The approximate, emergent particle-hole symmetry is essential to the topology of graphene: When strongly broken, nontopological phases can appear. Our Letter underpins topology as the crucial ingredient to the description of low-energy graphene. We provide a four-band short-range tight-binding model whose two lower bands have the same topology, symmetry, and flatness as those of the twisted bilayer graphene and which can be used as an effective low-energy model. We then perform large-scale (11000 atoms per unit cell, 40 days per $\mathbf{k}$-point computing time) ab initio calculations of a series of small angles, from 3° to 1°, which show a more complex and somewhat different evolution of the symmetry of the low-energy bands than that of the theoretical moiré model but which confirm the topological nature of the system.

Figure 1

(a) Fermi velocity at the charge neutrality point of the MBM plotted as a function of the twist angle. The dimensionless parameter $\alpha =w/[2v_F|\mathbf{K}|\sin (\theta /2)]$ uniquely determines the band structure of the MBM (up to a scaling); $w$ is the interlayer coupling, $v_F$ is the Fermi velocity of single-layer graphene, $|\mathbf{K}|$ is the distance between $\mathrm{\Gamma }$ and $K$, and $\theta$ is the twist angle. (See Ref. [18] or Sec. II in Ref. [22] for more details.) In this plot, we set $w=110\mathrm{meV}$ and $v_F|\mathbf{K}|=19.81\mathrm{eV}$. The gapped regions ${\mathrm{\Theta }}_a$, where the two bands near charge neutrality point are fully disconnected from all other bands, are shadowed with gray and labeled as $A–D$. The magic angles, where the Fermi velocity vanishes, are marked by dashed lines. By numerical calculation, we find that ${\alpha }_{m1}\approx 0.605$ (${\theta }_{m1}\approx 1.05°$), ${\alpha }_{m2}\approx 1.28$ (${\theta }_{m1}\approx 0.497°$), ${\alpha }_{m3}\approx 1.83$ (${\theta }_{m3}\approx 0.348°$), ${\alpha }_{m4}\approx 2.71$ (${\theta }_{m4}\approx 0.235°$), ${\alpha }_{m5}\approx 3.30$ (${\theta }_{m5}\approx 0.193°$), and ${\alpha }_{m6}\approx 3.82$ (${\theta }_{m6}\approx 0.167°$). (b) The Wilson loops of the four gapped phases have the same winding number, 1.

Figure 2

The four-band tight-binding model for one valley. The lower two bands have identical irreps and Wilson loop winding of the gapped middle two bands in the one-valley MBM. (a) The tight-binding model. The energy splitting between $s$ and $p_z$ is $\mathrm{\Delta }$. The hopping parameters $t_{s,p}$, $t_{s,p}^{\prime }$, and $\lambda$ are all, in general, complex numbers. (b) The band structure. (c) The Wilson loop of the lower two bands. (b) and (c) are calculated with the parameters $t_s=t_p=t$, $t_s^{\prime }=t_p^{\prime }=-\frac{1}{3}t$, $\lambda =(2/\sqrt{27})t$, and $\mathrm{\Delta }=0.15t$. As discussed in Sec. V in Ref. [22], such parameters are chosen to flatten the bandwidth.

Figure 3

Electronic band structures with different commensurate twist angles ${\theta }_i$. (a) The ab initio electronic band structures clearly show the breaking of particle-hole symmetry. The bands change nonmonotonically as decreasing $\theta$. The inset is the enlargement of the black box area. The two blue bands indicate two $2{\mathrm{\Gamma }}_3$ bands; the gray band is the $2{\mathrm{\Gamma }}_2$ band; the red band is the $2{\mathrm{\Gamma }}_1$ band. (b) The gap at $K$ at charge neutrality is a function of the twist angle.