Recent years have witnessed an explosively quick development in the field of van der Waals materials, leading to discoveries such as correlated insulators at unexpected densities, superconductors stabilized with the application of a magnetic field, and both integer and, recently, fractional Chern insulators. This talk will present a phenomenological model that captures many aspects of these systems and will attempt to indicate the remaining open fundamental questions.

In the first part, we introduce and analyze a model suitable for describing twisted bilayer graphene. I will show how, using a variational mean-field theory, we can determine the normal-state phase diagram of our model as a function of the band filling. The model clarifies how the intricate form of the interactions and the particle-hole asymmetry of the electronic spectrum determine the phase diagram. It also demonstrates how subtle device differences may lead to experimental observation of different behaviors. If time permits, I will describe what proper model modifications are suitable for describing other systems, such as untwisted and twisted trilayer graphene and transition metal dichalcogenide structures.