  The effects of gain and loss on the band structures of a bulk topological
dielectric photonic crystal (PC) with $C_{6v}$ symmetry and the PC-air-PC
interface are studied based on first-principle calculation. To illustrate the
importance of parity-time (PT) symmetry, three systems are considered, namely
the PT-symmetric, PT-asymmetric, and lossy systems. We find that the system
with gain and loss distributed in a PT symmetric manner exhibits a phase
transition from a PT exact phase to a PT broken phase as the strength of the
gain and loss increases, while for the PT-asymmetric and lossy systems, no such
phase transition occurs. Furthermore, based on the Wilson loop calculation, the
topology of the PT-symmetric system in the PT exact phase is demonstrated to
keep unchanged as the Hermitian system. At last, different kinds of edge states
in Hermitian systems under the influences of gain and loss are studied and we
find that while the eigenfrequencies of nontrivial edge states become complex
conjugate pairs, they keep real for the trivial defect states.
