Moiré Landau levels of a $C_4$-symmetric twisted bilayer system in the absence of a magnetic field

Abstract: It is widely known that the twisted bilayer graphene (TBG) shows flat bands at magic angles, which can be well described by the effective continuum model derived by Bistritzer and MacDonald (BM). We propose in this paper a similar twisted bilayer system but defined on the square lattice with $\pi$-flux per plaquette, and study its spectrum using the BM Hamiltonian with a mass term which is originated from the staggered potential. The basic difference between the TBG and the present model is simply rotational symmetry, $C_3$ vs $C_4$, as well as a mass term. Nevertheless, the feature of the flat bands is quite different: Those of the TBG appear at magic angles only, while the present model shows many flat bands, which are reminiscent of Landau levels, quite stably at any angles even in the absence of a magnetic field other than $\pi$-flux which keeps time reversal (TR) symmetry. Moreover, flat bands emerge in the mass gap of the Dirac spectrum, and each state composing these flat bands is well-localized at the position forming the moir\'e lattice. It turns out that the moir\'e potential serves as a periodic magnetic field, which can give energies smaller that the gap around moir\'e lattice positions. We derive a local Hamiltonian valid around the moir\'e lattice sites and show that it indeed reproduces the energies of the flat bands within the mass gap. Since these mid-gap states are localized at the moir\'e lattice, they form degenerate levels, which may be referred to as moir\'e Landau levels, although the mechanism of degeneracies are different from the conventional Landau levels. Interestingly, doubled fermions of the BH Hamiltonian associated with two layers have opposite charges when they couple with the effective moir\'e magnetic filed, which concern TR symmetry.