Phase Diagram of Massive Dirac Fermions with Tunable Interactions in High Magnetic Fields*

We study the strongly correlated states of massive fermions in two dimensions with Berry's phase $\pi$ and $2\pi$, in the limit of high magnetic fields. Due to the chiral band structure and massive carriers, the effective Coulomb interactions depend on the external magnetic field, and lead to a number of phases within a single low-lying Landau level. The tunability of the interactions allows the study of the transitions between phases in a more direct manner than in GaAs-based systems where the form of the interactions is independent of the magnetic field. We map the phase diagram at partial fillings $\nu = 1/3, 1/2, 3/5$ of the low-lying Landau levels, and find transitions between fractional quantum Hall states, compressible Fermi-liquid-like states, as well as charge-density-waves. We also find a new, broad regime of the effective interactions which favor the paired non-Abelian states. Our study identifies the strongly correlated phases expected in high-mobility graphene, bilayer graphene, topological insulators, and other materials with the non-trivial Berry phases, and provides a realistic method for studying the phase transitions between them.