A comparative study is made on the metal-insulator transition of Dirac fermions in the honeycomb and pi-flux Hubbard models at half filling by means of the variational cluster approximation and cluster dynamical impurity approximation. Paying particu
lar attention to the choice of the geometry of solver clusters and the inclusion of particle-bath sites, we show that the direct transition from the Dirac semimetallic state to the antiferromagnetic Mott insulator state occurs in these models, and therefore, the spin liquid phase is absent in the intermediate region, in agreement with recent quantum-Monte-Carlo--based calculations.
Excitonic density-wave states realized by the quantum condensation of electron-hole pairs (or excitons) are studied in the two-band Hubbard model with Hunds rule coupling and the pair hopping term. Using the variational cluster approximation, we calc
ulate the grand potential of the system and demonstrate that Hunds rule coupling always stabilizes the excitonic spin-density-wave state and destabilizes the excitonic charge-density-wave state and that the pair hopping term enhances these effects. The characteristics of these excitonic density-wave states are discussed using the calculated single-particle spectral function, density of states, condensation amplitude, and pair coherence length. Implications of our results in the materials aspects are also discussed.
