Dirac points for twisted bilayer graphene with in-plane magnetic field

Abstract

We study Dirac points of the chiral model of twisted bilayer graphene (TBG) with constant in-plane magnetic field. The striking feature of the chiral model is the presence of perfectly flat bands at magic angles of twisting. The Dirac points for zero magnetic field and non-magic angles of twisting are fixed at high symmetry points $K$ and $K^{\prime }$ in the Brillouin zone, with $\Gamma$ denoting the remaining high symmetry point. For a fixed small constant in-plane magnetic field, we show that as the angle of twisting varies between magic angles, the Dirac points move between $K$, $K^{\prime }$ points and the $\Gamma$ point. In particular, near magic angles, the Dirac points are located near the $\Gamma$ point. For special directions of the magnetic field, we show that the Dirac points move, as the twisting angle varies, along straight lines and bifurcate orthogonally at distinguished points. At the bifurcation points, the linear dispersion relation of the merging Dirac points disappears and exhibit a quadratic band crossing point (QBCP). The results are illustrated by links to animations suggesting interesting additional structure.