WKB Estimate of Bilayer Graphene’s Magic Twist Angles

Graphene bilayers exhibit zero-energy flatbands at a discrete series of magic twist angles. In the absence of intrasublattice interlayer 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 wave functions. We find zero-energy solutions at a discrete set of values of the dimensionless Planck’s constant, which we obtain analytically. Our analytic flatband twist angles correspond closely to those determined numerically in previous work.

Figure 1

$A$ in the moiré unit cell. The real and imaginary parts are represented by the horizontal and vertical components of the vectors. The yellow, green, and red dots identify turning points, and the wavy lines identify the branch cut.

Figure 2

(a),(b) The color map of the real and the imaginary part of $S$. Dotted lines stand for the Stokes lines.

Figure 3

Stokes diagram in a moiré unit cell bounded by the red solid lines. The yellow, green, and red dots identify turning points. The branch cuts and Stokes lines are plotted by purple wavy and dotted lines, respectively. Turning points at the corners are labeled 1-6 while the bounded WKB solution regions are labeled by Roman numerals I-VI.

Figure 4

Validity regions of different solutions in a moiré unit cell. The WKB solutions are valid away from the turning points $AA$, $AB$, and $BA$ as indicated by the regions outside the black circles in solid lines. The solutions of the linearized equation are valid inside the dashed circle.

Figure 5

Comparison between numerical and WKB estimations of the magic angles via (a) the parameter $\alpha$ and (b) the magic angle difference ${\alpha }_{n+1}-{\alpha }_n$. Inset: relative error $\delta =({\alpha }_n^{\mathrm{WKB}}-{\alpha }_n^{\text{Num}})/{\alpha }_n^{\text{Num}}$. The numerical $\alpha$ values indicated by blue circles and connected with a solid line were extracted from Ref. [3].