Ren Yafei, Gao Qiang, MacDonald A H, Niu Qian
Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA.
Phys Rev Lett. 2021 Jan 8;126(1):016404. doi: 10.1103/PhysRevLett.126.016404.
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.
双层石墨烯在一系列离散的神奇扭转角处呈现零能量平带。在没有亚晶格层间跳跃的情况下，零能量态满足一个带有非阿贝尔SU(2)规范势的狄拉克方程，该规范势无法全局对角化。我们通过引入一个与扭转角成比例的无量纲普朗克常数，围绕AB和BA转折点求解线性化狄拉克方程，并通过体WKB波函数连接艾里函数解，为该狄拉克方程开发了一种半经典WKB近似方案。我们在无量纲普朗克常数的一组离散值处找到了零能量解，这些值是我们通过解析得到的。我们解析得到的平带扭转角与先前工作中通过数值确定的扭转角非常接近。
