Angle dependence of the Landau level spectrum in twisted bilayer graphene

Abstract

In the context of the low-energy effective theory, the exact Landau level spectrum of quasiparticles in twisted bilayer graphene with small twist angle is analytically obtained by spheroidal eigenvalues. We analyze the dependence of the Landau levels on the twist angle to find the points where the twofold degeneracy for twist angles is lifted in the nonzero modes and below (above) which massive (massless) fermion pictures become valid. In the perpendicular magnetic field of 10 T, the degeneracy is removed at ${\ensuremath{\theta}}_{\mathrm{deg}}\ensuremath{\sim}{3}^{\ensuremath{\circ}}$ for a few low levels, specifically ${\ensuremath{\theta}}_{\mathrm{deg}}\ensuremath{\simeq}2.{56}^{\ensuremath{\circ}}$ for the first pair of nonzero levels and ${\ensuremath{\theta}}_{\mathrm{deg}}\ensuremath{\simeq}3.{50}^{\ensuremath{\circ}}$ for the next pair. Massive quasiparticle appears at $\ensuremath{\theta}<{\ensuremath{\theta}}_{\mathrm{c}}\ensuremath{\simeq}1.{17}^{\ensuremath{\circ}}$ in 10 T, which match perfectly with the recent experimental results. Since our analysis is applicable to the cases of arbitrary constant magnetic fields, we make predictions for the same experiment performed in arbitrary constant magnetic fields, for example, for $B=40$ T we get ${\ensuremath{\theta}}_{\mathrm{c}}\ensuremath{\simeq}2.{34}^{\ensuremath{\circ}}$ and the sequence of angles ${\ensuremath{\theta}}_{\mathrm{deg}}=5.11,7.01,8.42,\ensuremath{\cdots}$ for the pairs of nonzero energy levels. The symmetry restoration mechanism behind the massive (massless) transition is conjectured to be a tunneling (instanton) in momentum space.