WKB Estimate of Bilayer Graphene’s Magic Twist Angles

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