Non-${\gamma }_5$hermiticity fermions in two dimensions

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.