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 approximate to 8 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 Delta K, the difference of the momenta of the unrotated and rotated special points, replaced by a coupling momentum scale g((c)) = 8 pi/root 3a sin 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 degrees < theta < 0.15 degrees limits theta < 0.05 degrees 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 -0.25 < E < 0.25 eV with an average band velocity approximate to 0.3(vF)((SLG)) (where SLG denotes single-layer graphene). We study the fermiology of the twist bilayer in this limit, finding remarkably structured constant energy surfaces with multiple Lifshitz transitions between K-and Gamma-centered Fermi sheets and a rich pseudospin texture.