  We formulate new two dimensional fermions breaking $\gamma_{5}$hermiticity,
based on the minimal doubling fermion. We investigate their properties: (I)
Symmetries, (II) eigenvalue distributions, and (III)the number of poles. As a
simple application of the fermions, the Gross-Neveu model in two dimensions is
studied using the fermion. We obtain the parity broken phase diagram called
Aoki phase and the chiral broken phase diagram of the model with an imaginary
chemical potential.
