  We derive Lorentz-invariant four-fermion interactions, including
Nambu-Jona-Lasinio type and superconducting type, which are widely studied in
high-energy physics, from the honeycomb lattice Hamiltonian with Hubbard
interaction. We investigate the phase transitions induced by these two
interactions and consider the effects of the chemical potential and magnetic
flux (Haldane mass term) on these phase transitions. We find that the
charge-density-wave and superconductivity generated by the attractive
interactions are mainly controlled by the chemical potential, while the
magnetic flux delimits the domain of phase transition. Our analysis underscores
the influence of the initial topological state on the phase transitions, a
facet largely overlooked in prior studies. We present experimental protocols
using cold atoms to verify our theoretical results.
