  The notion of synthetic dimensions has expanded the realm of topological
physics to four dimensional (4D) space lately. In this work, non-Hermiticity is
used as a synthetic parameter in PT-symmetric photonic crystals to study the
topological physics in 4D non-Hermitian synthetic parameter space. We realize a
3D exceptional hypersurface (EHS) in such 4D parameter space, and the
degeneracy points emerge due to the symmetry of synthetic parameters. We
further demonstrate the existence of exceptional degenerate points (EDPs) on
the EHS that originates from the chirality of exceptional points (EPs), and the
exceptional surface near EDPs behaves like a Dirac cone. We further show that a
very narrow reflection plateau can be found near these EDPs, and their
sensitivity towards the PT-symmetry breaking environmental perturbation can
make these degeneracy points useful in optical sensing and many other nonlinear
and quantum optical applications.
