Constructing low-energy models for the electronic structure of twisted bilayer graphene

The recent, unexpected discovery of insulating and superconducting behaviour in magic angle twisted bilayer graphene (tBLG) has generated tremendous interest in theoretical communities to rationalise these observations. To study the angle dependent electronic properties, we have developed a tight binding model for tBLG. Importantly, the developed tight binding model accounts for atomic corrugation, and therefore, different interactions between AA and AB regions. Such asymmetry has been attributed to the isolation of bands near the Fermi energy, which has been experimentally observed. From the tight binding model, the Wannier orbitals and interaction matrix of these orbitals can be obtained. These calculations permit the construction of simplified Hamiltonians with no free parameters. Such downfolded Hamiltonians can be used in state-of-the-art functional renormalisation group methods to calculate the electronic phase diagram. The developed methodology naturally lends itself to systematic improvements and to other bilayer materials.