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

The chiral Hamiltonian for twisted graphene bilayers is analyzed in terms of its squared Hamiltonian which removes the particle-hole symmetry and thus one bipartite 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 twisted systems: kinetic energy, confinement potential and an interlayer interaction operator which is divided in two parts: a non-Abelian interlayer operator and an operator which contains an interaction energy between layers. Here, each of these components is analyzed as a function of the angle of rotation, as well as in terms of the wave-function localization properties. In particular, it is proved that the non-Abelian operator represents interlayer currents between each layer triangular sublattices, i.e., a second-neighbor interlayer current between bipartite sublattices. A crossover is seen between such contributions and thus the first magic angle is different from other higher order magic angles. Such angles are determined by a balance between the negative energy contribution from interlayer currents and the positive contributions from the kinetic and confinement energies. A perturbative analysis performed around the first magic angle allows to explore analytically the details of such energy balance.