Emergent momentum scale, localization, and van Hove singularities in the graphene twist bilayer

We identify an angle-dependent momentum scale as the fundamental property of a bilayer composed of mutually rotated graphene layers. The interlayer scattering processes at these characteristic momentum values define an effective Brillouin zone (a Jones zone) which, in general, differs from the Brillouin zone generated by the real-space lattice, which is physically irrelevant. From this we develop a numerical method that increases, for the twist bilayer, the efficiency of the standard tight-binding method by a factor of ≈10³ at no loss of accuracy. The efficiency of the method is based on (i) the fact that the twist Hamiltonian is exceptionally sparse in a basis of single-layer graphene (SLG) states, (ii) a solution of trivial Diophantine problem (B\'ezout's identity) allows one to know in advance which matrix elements take nonzero values, and (iii) to access the electronic structure in a few electron volts about the Dirac point a truncated SLG basis consisting only of states in a somewhat larger energy window are required, leading to a much reduced size of the Hamiltonian. This allows a complete survey of the system which reveals (i) an angle-dependent series of van Hove singularities, (ii) an increasing mixing of SLG states as the twist angle is reduced leading to the appearance of localization of the twist bilayer wave functions at all energies in the small-angle limit, and (iii) a zero-energy peak in the density of states in an approximately self-similar small-angle regime.