Theory of correlated Chern insulators in twisted bilayer graphene

Magic-angle twisted bilayer graphene is the best studied physical platform featuring moire potential induced narrow bands with non-trivial topology and strong electronic correlations. Despite their significance, the Chern insulating states observed at a finite magnetic field -- and extrapolating to a band filling, $s$, at zero field -- remain poorly understood. Unraveling their nature is among the most important open problems in the province of moir\'e materials. Here we present the first comprehensive study of interacting electrons in finite magnetic field while varying the electron density, twist angle and heterostrain. Within a panoply of correlated Chern phases emerging at a range of twist angles, we uncover a unified description for the ubiquitous sequence of states with the Chern number $t$ for $(s,t)=\pm (0,4), \pm(1,3),\pm(2,2)$ and $\pm(3,1)$. We also find correlated Chern insulators at unconventional sequences with $s+t\neq \pm 4$, as well as with fractional $s$, and elucidate their nature.