Electronic structure of turbostratic graphene

We explore the rotational degree of freedom between graphene layers via the simple prototype of the graphene twist bilayer, i. e. , two layers rotated by some angle. It is shown that, due to the weak interaction between graphene layers, many features of this system can be understood by interference conditions between the quantum states of the two layers, mathematically expressed as Diophantine problems. Based on this general analysis we demonstrate that while the Dirac cones from each layer are always effectively degenerate, the Fermi velocity v₅ of the Dirac cones decreases as 0^; the form we derive for v₅ () agrees with that found via a continuum approximation in J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. Lett. 99, 256802 (2007). From tight-binding calculations for structures with 1. 47⁰ the Dirac point is fourfold degenerate, while at =0 one has the twofold degeneracy of the AB stacked bilayer. Interestingly, in this limit the electronic properties are in an essential way determined globally, in contrast to the ``nearsightedness'' W. Kohn, Phys. Rev. Lett. 76, 3168 (1996) of electronic structure generally found in condensed matter.