Special exceptional point acting as Dirac point in one dimensional -symmetric photonic crystal

We reveal that a special exceptional point (EP) can act as a Dirac point in a one-dimensional $\mathcal{P}\mathcal{T}$ -symmetric photonic crystal and prove it in detail using our extended first-principle theory. This theory was developed by applying biorthogonal bases of the non-Hermitian Hamiltonian to the $\vec{k}\cdot \vec{p}$ method to study the dispersion relations of non-Hermitian systems. By using this theory, we demonstrate that two linear dispersions can cross at a critical touching point, which is a special EP, and the corresponding effective Hamiltonian can be cast into a massless Dirac Hamiltonian under the biorthogonal bases of the non-Hermitian Hamiltonian; therefore, this point can be called the Dirac point, and the linear slope can also be predicted. In contrast to the Dirac point in a Hermitian system, which is induced by the degeneracy of two different eigenstates, the Dirac point here is induced by the degeneracy of parallel eigenstates in a non-Hermitian system. In addition, after increasing the non-Hermiticity, the Dirac point evolves into a pair of EPs, and their positions in the band structure can be predicted well by our theory. Our findings and theory are important for further understanding the physics of the Dirac point in non-Hermitian wave systems.