  Non-Hermitian systems with parity-time (PT) symmetric complex potentials can
exhibit a phase transition when the degree of non-Hermiticity is increased. Two
eigenstates coalesce at a transition point, which is known as the exceptional
point (EP) for a discrete spectrum and spectral singularity for a continuous
spectrum. The existence of an EP is known to give rise to a great variety of
novel behaviors in various fields of physics. In this work, we study the
complex band structures of one-dimensional photonic crystals with PT symmetric
complex potentials by setting up a Hamiltonian using the Bloch states of the
photonic crystal without loss or gain as a basis. As a function of the degree
of non-Hermiticity, two types of PT symmetry transitions are found. One is that
a PT-broken phase can re-enter into a PT-exact phase at a higher degree of
non-Hermiticity. The other is that two spectral singularities, one originating
from the Brillouin zone center and the other from the Brillouin zone boundary,
can coalesce at some k-point in the interior of the Brillouin zone and create a
singularity of higher order. Furthermore, we can induce a band inversion by
tuning the filling ratio of the photonic crystal, and we find that the
geometric phases of the bands before and after the inversion are independent of
the amount of non-Hermiticity as long as the PT-exact phase is not broken. The
standard concept of topological transition can hence be extended to
non-Hermitian systems.
