  Recent experiments in graphene heterostructures have observed Chern
insulators - integer and fractional Quantum Hall states made possible by a
periodic substrate potential. Here we study theoretically the competition
between different Chern insulators, which can be tuned by the amplitude of the
periodic potential, leads to a new family of quantum critical points described
by QED$_3$-Chern-Simons theory. At these critical points, $N_f$ flavors of
Dirac fermions interact through an emergent U$(1)$ gauge theory at Chern-Simons
level $K$, and remarkably, the entire family (with any $N_f$ or $K$) can be
realized at special values of the external magnetic field. Transitions between
particle-hole conjugate Jain states realize "pure" QED$_3$ in which multiple
flavors of Dirac fermion interact with a Maxwell U$(1)$ gauge field. The
multi-flavor nature of the critical point leads to an emergent SU$(N_f)$
symmetry. Specifically, at the transition from a $\nu=$1/3 to 2/3 quantum Hall
state, the emergent SU(3) symmetry predicts an octet of charge density waves
with enhanced susceptibilities, which is verified by DMRG numerical simulations
on microscopic models applicable to graphene heterostructures. We propose
experiments on Chern insulators that could resolve open questions in the study
of 2+1 dimensional conformal field theories and test recent duality inspired
conjectures.
