Low-energy theory for the graphene twist bilayer

The graphene twist bilayer represents the prototypical system for investigating the stacking degree of freedom in few-layer graphenes. The electronic structure of this system changes qualitatively as a function of angle, from a large-angle limit in which the two layers are essentially decoupled---with the exception of a 28-atom commensuration unit cell for which the layers are coupled on an energy scale of $\ensuremath{\approx}8\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$---to a small-angle strong-coupling limit. Despite sustained investigation, a fully satisfactory theory of the twist bilayer remains elusive. The outstanding problems are (i) to find a theoretically unified description of the large- and small-angle limits, and (ii) to demonstrate agreement between the low-energy effective Hamiltonian and, for instance, ab initio or tight-binding calculations. In this article, we develop a low-energy theory that in the large-angle limit reproduces the symmetry-derived Hamiltonians of Mele [Phys. Rev. B 81, 161405 (2010)], and in the small-angle limit shows almost perfect agreement with tight-binding calculations. The small-angle effective Hamiltonian is that of Bistritzer and MacDonald [Proc. Natl. Acad. Sci. (U.S.A.) 108, 12233 (2011)], but with the momentum scale $\mathrm{\ensuremath{\Delta}}K$, the difference of the momenta of the unrotated and rotated special points, replaced by a coupling momentum scale ${g}^{(c)}=\frac{8\ensuremath{\pi}}{\sqrt{3}a}sin\frac{\ensuremath{\theta}}{2}$. Using this small-angle Hamiltonian, we are able to determine the complete behavior as a function of angle, finding a complex small-angle clustering of van Hove singularities in the density of states (DOS) that after a ``zero-mode'' peak regime between $0.{90}^{\ensuremath{\circ}}<\ensuremath{\theta}<0.{15}^{\ensuremath{\circ}}$ limits $\ensuremath{\theta}<0.{05}^{\ensuremath{\circ}}$ to a DOS that is essentially that of a superposition DOS of all bilayer stacking possibilities. In this regime, the Dirac spectrum is entirely destroyed by hybridization for $\ensuremath{-}0.25