WKB estimate of bilayer graphene's magic twist angles

TL;DR

This paper develops a semiclassical WKB method to analytically estimate the magic twist angles in bilayer graphene, matching previous numerical results and providing insight into the zero-energy flat bands.

Contribution

It introduces a novel WKB approximation scheme for the Dirac equation in bilayer graphene, enabling analytical determination of flat band twist angles.

Findings

Analytical flat band twist angles closely match numerical results.

Zero-energy solutions occur at discrete values of the dimensionless Planck's constant.

The method provides a new approach to understanding flat bands in twisted bilayer graphene.

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.