General Theory of Flat Bands and Magic Angles in Twisted Dirac Materials

Twisted bilayer graphene has been a center of recent theoretical and experimental interest due to the emergence of magic angles and the interaction-driven insulating and superconducting phases that result from nearly-flat bands. The Bistritzer-MacDonald model provides a framework to understand the origin of the flat bands. Using perturbation theory, we generalize this analysis to any interface between two twisted 2D Dirac materials, one with N Dirac cones and one with M Dirac cones (where N and M may be odd, as surface states of topological insulators), under assumptions of time-reversal and rotational symmetry. We then use this framework to consider two special cases: a TI-on-TI interface and a graphene-on-TI interface. We find that spin-flipping interactions serve a vital role in forming magic angles in both systems.