Flat bands due to twisted bilayer graphene: A coordinate transformation approach

We consider two-layer graphene with one layer twisted by a small angle θ. This can be investigated using a coordinate transformation given by an angle θ with respect to the untwisted layer. For the untwisted layer two sublattices are a(R) and b(R + δ) and in the twisted layer we have A(R') and B(R' + δ). We consider hopping between the two layers −t_a,A and −t_a,B . As a result, we obtain four eigenvalues. Two eigenvalues have low energy with vanishing dispersion as a function of the angle. The solution is based on the replacement of the twisted coordinate in the momentum space by a spinor transformation B'_σ(k'(k)) = exp(iσ₃θ/2) B_σ(k). In the presence of interaction the coordinates are affected by the transformation and thus contain the phase exp(iσ₃θ/2).