  Graphene bilayers exhibit zero-energy flat bands at a discrete series of
magic twist angles. In the absence of intra-sublattice inter-layer hopping,
zero-energy states satisfy a Dirac equation with a non-abelian SU(2) gauge
potential that cannot be diagonalized globally. We develop a semiclassical WKB
approximation scheme for this Dirac equation by introducing a dimensionless
Planck's constant proportional to the twist angle, solving the linearized Dirac
equation around AB and BA turning points, and connecting Airy function
solutions via bulk WKB wavefunctions. We find zero energy solutions at a
discrete set of values of the dimensionless Planck's constant, which we obtain
analytically. Our analytic flat band twist angles correspond closely to those
determined numerically in previous work.

Mesoscale and Nanoscale Physics;Materials Science