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The chiral Hamiltonian for twisted graphene bilayers is analyzed in terms ofits squared Hamiltonian which removes the particle-hole symmetry and thus onebipartite lattice, allowing to write the Hamiltonian in terms of a $2\times 2$matrix. This brings to the front the three main physical actors of twistedsystems: kinetic energy, confinement potential and an interlayer interactionoperator which is divided in two parts: a non-Abelian interlayer operator andan operator which contains an interaction energy between layers. Here, each ofthese components is analyzed as a function of the angle of rotation, as well asin terms of the wave-function localization properties. In particular, it isproved that the non-Abelian operator represents interlayer currents betweeneach layer triangular sublattices, i.e., a second-neighbor interlayer currentbetween bipartite sublattices. A crossover is seen between such contributionsand thus the first magic angle is different from other higher order magicangles. Such angles are determined by a balance between the negative energycontribution from interlayer currents and the positive contributions from thekinetic and confinement energies. A perturbative analysis performed around thefirst magic angle allows to explore analytically the details of such energybalance.

physical review. b./physical review. b

Why the first magic-angle is different from others in twisted graphene  bilayers: interlayer currents, kinetic and confinement energy and  wavefunction localization