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Two monolayers of graphene twisted by a small `magic' angle exhibit nearlyflat bands leading to correlated electronic states and superconductivity, whoseprecise nature including possible broken symmetries, remain under debate. Herewe theoretically study a related but different system with reduced symmetry -twisted {\em double} bilayer graphene (TDBLG), consisting of {\em two} Bernalstacked bilayer graphene sheets, twisted with respect to one another. Unlikethe monolayer case, we show that isolated flat bands only appear on applicationof a vertical displacement field $D$. We construct a phase diagram as afunction of twist angle and $D$, incorporating interactions via a Hartree-Fockapproximation. At half filling, ferromagnetic insulators are stabilized,typically with valley Chern number $C_v=2$. Ferromagnetic fluctuations in themetallic state are argued to lead to spin triplet superconductivity frompairing between electrons in opposite valleys. Response of these states to amagnetic field applied either perpendicular or parallel to the graphene sheetsis obtained, and found to compare favorably with a recent experiment. Wehighlight a novel orbital effect arising from in-plane fields that can exceedthe Zeeman effect and plays an important role in interpreting experiments.

Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene