  We construct 2D non-$\gamma_{5}$hermiticity fermions based on the minimal
doubling fermion. We investigate symmetries, reflection positivity, eigenvalue
distribution and the number of poles for our fermions. As simple tests for
application to the fermion, the Gross-Neveu model in two dimensions is studied
using the non-$\gamma_{5}$hermiticity fermion. We draw the parity broken phase
diagram, called Aoki phase and the chiral broken phase diagram for the model
with an imaginary chemical potential.
