Topological features and gauge fields in graphene superlattices. Detecting Topological Currents in Graphene Superlattices

The quest for crystals whose electronic bands show non trivial topological features has not ceased since the discovery of the Quantum Hall Effect. The amazing current quantization in QHE devices can be traced back to edge states which emerge from non trivial electronic structures[1, 2]. Topological properties are typically associated to integer numbers, so that they are quite robust against perturbations. The search for such systems culminated in the development of the concept of the topological insulator[3, 4] (TI).A two dimensional TI can be viewed as two copies of a Quantum Hall system, related by time reversal symmetry (TRS). In the absence of perturbations which break TRS, each copy should show quantized electronic transport, as in the Integer Quantum Hall Effect. The topology of a given (isolated) electronic band in a two dimensional system is described by an integral over the Brillouin Zone[5]. The integrand is determined by the momentum dependence of the wavefunction. The value of this integral must be an integer, the Chern number. The simplest bands with non trivial Chern numbers are the Landau levels induced by a constant magnetic field in a two dimensional electron gas. Alternatively, some relatively simple singularities in band structures, like gapped Dirac points in triangular lattices, lead to non trivial integrands, which, in turn, may give rise to non zero Chern numbers[2, 3]. Bands associated to a single atomic orbital in the unit cell cannot have a non trivial topology. Graphene is a material with Dirac cones precisely at the Fermi level. It combines interesting fundamental properties, amazing resilience, and little disorder. In fact, the band structure of graphene is behind some of the proposals described above[2, 3]. Including spin and valley degeneracy, graphene has four Dirac cones. The region in momentum space near the apex of each cone contributes the the integrated Chern number of the valence and conduction bands with ±1/2. In order to turn graphene into a system with non trivial topology i) the Dirac cones need to be gapped, and ii) the contribution from each cone needs to have the right sign, so that no cancellations occur. The realization of these premises is a tall order: ordinary gaps in graphene require the absence of spatial inversion symmetry, and TRS implies that the sum of the contributions from all cones vanishes. The way out of this dilemma proposed in[3] was the formation of gaps by the spin-orbit interaction. Each spin band becomes topologically non trivial, and graphene turns into a TI. Unfortunately, carbon is a