  In this work, we study the synergistic correlated states in two distinct
types of interacting electronic systems coupled by interlayer Coulomb
interactions. We propose that this scenario can be realized in a type of
Coulomb-coupled graphene-insulator heterostructures with gate tunable band
alignment. We find that, by virtue of the interlayer Coulomb coupling between
the interacting electrons in the two layers, electronic states that cannot be
revealed in either individual layer would emerge in a cooperative and
synergistic manner. Specifically, as a result of the band alignment, charge
carriers can be transferred between graphene and the substrate under the
control of gate voltages, which can yield a long-wavelength electronic crystal
at the surface of the substrate. This electronic crystal exerts a superlattice
Coulomb potential on the Dirac electrons in graphene, which generates subbands
with reduced non-interacting Fermi velocity. As a result, $e$-$e$ Coulomb
interactions within graphene would play a more important role, giving rise to a
gapped Dirac state at the charge neutrality point, accompanied by
interaction-enhanced Fermi velocity. Moreover, the superlattice potential can
give rise to topologically nontrivial subband structures which are tunable by
superlattice's constant and anisotropy. Reciprocally, the electronic crystal
formed in the substrate can be substantially stabilized in such coupled bilayer
heterostructure by virtue of the cooperative interlayer Coulomb coupling. We
further perform high-throughput first principles calculations to identify a
number of promising insulating materials as candidate substrates for graphene
to demonstrate these effects.
