Aoki Phases in the Lattice Gross - Neveu Model with Flavored Mass terms

We investigate the parity-broken phase structure for staggered and naive fermions in the Gross-Neveu model as a toy model of QCD. We consider a generalized staggered Gross-Neveu model including two types of four-point interactions. We use generalized mass terms to split the doublers for both staggered and naive fermions. The phase boundaries derived from the gap equations show that the mass splitting of tastes results in an Aoki phase both in the staggered and naive cases. We also discuss the continuum limit of these models and explore taking the chirally-symmetric limit by fine-tuning a mass parameter and two coupling constants. This supports the idea that in lattice QCD we can derive oneor two-flavor staggered fermions by tuning the mass parameter, which are likely to be less expensive than Wilson fermions in QCD simulation.