Band structure of twisted bilayer graphene: Emergent symmetries, commensurate approximants, and Wannier obstructions

A remarkable feature of the band structure of bilayer graphene at small twist angle is the appearance of isolated bands near neutrality, whose bandwidth can be reduced at certain magic angles (e.g., $\theta \sim 1.{05}^{\circ }$). In this regime, correlated insulating states and superconductivity have been experimentally observed. A microscopic description of these phenomena requires an understanding of universal aspects of the band structure, which we discuss here. First, we point out the importance of emergent symmetries, such as valley conservation, which are excellent symmetries in the limit of small twist angles and dictate qualitative features of the band structure. These have sometimes been overlooked when discussing commensurate approximants to the band structure, which we also review here, and solidify their connection with the continuum theory which incorporates all emergent symmetries. Finally, we discuss obstructions to writing tight-binding models of just the isolated bands, and in particular a symmetry-based diagnostic of these obstructions, as well as relations to band topology and strategies for resolving the obstruction. Especially, we construct a four-band model where the two lower isolated bands realize all identified Wannier obstructions of the single-valley nearly flat bands of twisted bilayer graphene.

Figure 1

Pattern of momentum mapping for the two different types of twist angles. Here, $K$, $K_{\theta }$, and $K_{\mathrm{m}}$, respectively, denote the $K$ point of the unrotated, rotated, and moiré Brillouin zones. $K^{\prime }$, $K_{\theta }^{\prime }$, and $K_{\mathrm{m}}^{\prime }$ denote their respective time-reversal partners.

Figure 2

Examples of commensurate lattices. Here, we twist the top layer counterclockwise by $\theta /2$ and the bottom by $-\theta /2$. The radius of the disk in each panel is chosen to be $3a/(2sin(\theta /2))$. Two relatively large twist angles ${\theta }_{\mathrm{I}}\equiv \theta (5,1)\sim 6.0^{\circ }$ and ${\theta }_{\mathrm{II}}\equiv \theta (16,3)\simeq 5.7^{\circ }$ are considered. If smaller twist angles were used, the differences between the lattices would be even harder to discern. As indicated by the subscripts, they are respectively of types I and II, which lead to different primitive lattice vectors (black arrows). The twisting centers are chosen to be either an aligned hexagon center or an aligned carbon site, and we consider both a normal twist by $\theta$, as well as a “conjugated” twist by $\pi /3-\theta$. Note that conjugation changes the angle type (Appendix pp1), so panels (a), (c), (f), and (h) are type I lattices whereas (b), (d), (e), and (g) are of type II. Although the exact lattice constant for lattices (e)–(h) is larger, one can see visually that their approximate lattice constant is the same as that of lattices (a)–(d), which is given by $L^{\prime }\left(\theta \right)=a/(2sin(\theta /2))$. Except for lattices (c) and (h), the exact point group of all the lattices shown is $D_6$.

Figure 3

(a) Tight-binding model of valley-resolved isolated moiré bands from a two-orbital model on the honeycomb lattice. (b) Schematic band structure of this model. Two connected sets of bands separated by a band-gap result. The lower one, for example, captures the band structure of the nearly flat bands of twisted bilayer graphene, in a single valley while preserving all symmetry ($D_6$ point group). (c) Upon adding further symmetry-allowed perturbations, one can attempt to reproduce the band structure of twisted bilayer graphene. Dispersion shown is for parameters ${\widehat{t}}_1=0.4{\mu }_0+0.6{\mu }_z,{\widehat{t}}_2=0.1i{\mu }_x$.