  Higher-order exceptional points (EPs), which appear as multifold degeneracies
in the spectra of non-Hermitian systems, are garnering extensive attention in
various multidisciplinary fields. However, constructing higher-order EPs still
remains as a challenge due to the strict requirement of the system symmetries.
Here we demonstrate that higher-order EPs can be judiciously fabricated in PT
-symmetric staggered rhombic lattices by introducing not only on-site gain/loss
but also nonHermitian couplings. Zero-energy flatbands persist and
symmetry-protected third-order EPs (EP3) arise in these systems owing to the
non-Hermitian chiral/sublattice symmetry, but distinct phase transitions and
propagation dynamics occur. Specifically, the EP3 arises at the Brillouin zone
(BZ) boundary in the presence of on-site gain/loss. The single-site excitations
display an exponential power increase in the PT -broken phase. Meanwhile, a
nearly flatband sustains when a small lattice perturbation is applied. For the
lattices with non-Hermitian couplings, however, the EP3 appears at the BZ
center. Quite remarkably, our analysis unveils a dynamical
delocalization-localization transition for the excitation of the dispersive
bands and a quartic power increase beyond the EP3. Our scheme provides a new
platform towards the investigation of the higher-order EPs, and can be further
extended to the study of topological phase transitions or nonlinear processes
associated with higher-order EPs.
