Non-$γ_{5}$hermiticity minimal doubling fermion

Abstract: 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.