WKB estimate of bilayer graphene's magic twist angles

Abstract

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.

Chiral Approximation to Twisted Bilayer Graphene: Exact Intra-Valley Inversion Symmetry, Nodal Structure and Implications for Higher Magic Angles

“…Our identification of zero mode wavefunctions with two quantum Hall wavefunctions may also shed light on the non-Abelian quantization condition [80][81][82] , where a semi-classical analysis was done recently in Ref. (83).…”

Spectral characterization of magic angles in twisted bilayer graphene

“…This calls to question a recently suggested WKB approach [28]: the explanation proposed there gives the asymptotic spacing of magic angles as…”

“…1.47. In addition, the approach proposed in [28] would seemingly apply to all tunnelling potentials satisfying (4) in the chiral model. However, the equidistant asymptotic distribution of magic angles is not stable under such perturbations, as we discuss in our next paragraph.…”