Humanity makes great use of the electric field effect: charging and discharging capacitors in low density semiconductors systems is the underpinning of the analog and digital electronics that define our age. At the same time, we know quantum matter to include far more than just electrical conductors and insulators. I will describe the physics of crystalline graphite multilayers with rhombohedral stacking, where the competition between electron hopping within- and between- the graphene planes leads to a flat dispersion characterized by high electronic density of states and Berry curvature, which can be tuned by a perpendicular electric field. Using electrostatic gates to tune both this interlayer potential and the total carrier density, I will show that a dizzying variety of magnetic and superconducting states can be realized, often within the same device. The exceptional experimental reproducibility of these structurally simple systems allows us to investigate a variety of effects in a controlled environment, including the role of spin orbit coupling or a moire potential, providing insight into the mechanisms of magnetism and superconductivity. Most strikingly, quantized Hall effects and superconductivity can be realized in the same field-effect transistor for only slightly different values of a gate voltage, providing a versatile platform both to both study the mechanisms underlying these phases as well as build highly controllable interfaces between these paradigmatic phases of quantum matter.
