  Strongly correlated quantum matter exhibits a rich variety of remarkable
properties, but the organizing principles that underlie the behavior remain to
be established. Graphene heterostructures, which can host narrow moire electron
bands that amplify the correlation effect, represent a new setting to make
progress on this overarching issue. In such correlated moire systems, an
insulating state is a prominent feature of the phase diagram and may hold the
key to understanding the basic physics. Here we advance the notion of a fragile
insulator, a correlation-driven insulating state that is on the verge of a
delocalization transition into a bad metal. Using a realistic multiorbital
Hubbard model as a prototype for narrow band moire systems, we realize such a
fragile insulator and demonstrate a nematic order in this state as well as in
the nearby bad metal regime. Our results are consistent with the observed
electronic anisotropy in the graphene moire systems and provide a natural
understanding of what happens when the insulator is tuned into a bad metal. We
propose the fragile insulator and the accompanying bad metal as competing
states at integer fillings that analogously anchor the overall phase diagram of
the correlated moire systems and beyond.
