Angle Dependence of Landau Level Spectrum in Twisted Bilayer Graphene

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 two-fold 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 $\theta_{{\rm deg}}\sim 3^\circ$ %angles around 3 degrees for a few low levels, specifically, $\theta_{\rm deg}\simeq 2.56^\circ$ for the first pair of nonzero levels and $\theta_{\rm deg}\simeq 3.50^\circ$ for the next pair. Massive quasiparticle appears at $\theta<\theta_{{\rm c}}\simeq 1.17^\circ$ in 10\,T, %angles less than 1.17 degrees. 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, e.g., for B=40\,T we get $\theta_{\rm c}\simeq 2.34^\circ$ and the sequence of angles $\theta_{\rm deg} = 5.11, 7.01, 8.42,...$ 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.