  Topological phonon polaritons (TPhPs) are highly protected and localized edge
modes that are capable of achieving a strong confinement of electromagnetic
waves and immune to impurities and disorder. Here we realize TPhPs by
constructing one-dimensional dimerized silicon carbide nanoparticle chains,
which mimic the topological property of the well-known Su-Schrieffer-Heeger
(SSH) model. We analytically calculate the complex band structure of such
chains by taking all near-field and far-field dipole-dipole interactions into
account. For longitudinal modes, we demonstrate that, despite the
non-Hermiticity and breaking of the chiral symmetry, the band topology can be
still characterized by the complex Zak phase, which is quantized and indicates
a topological phase transition when the dimerization parameter $\beta$ changes
from less than 0.5 to larger than 0.5, like the conventional Hermitian SSH
model. By calculating the eigenmodes of a finite chain, we find such a
dimerized chain with $\beta>0.5$ supports nontrivial topological eigenmodes
localized over both of its edges, indicating the validity of the bulk-boundary
correspondence. On the other hand, for transverse modes, we discover a
topological phase transition by increasing the lattice constant, which is due
to the presence of strong long-range far-field transverse dipole-dipole
interactions decaying with the distance $r$ as $1/r$ for an infinitely long
chain. However, we surprisingly find the emergence of non-Hermitian skin effect
in a finite chain, which leads to the breakdown of the bulk-boundary
correspondence. Furthermore, by incorporating the effect of localized bulk
eigenmodes and proposing a modified complex Zak phase for a finite lattice, we
still recover the topological behavior of the conventional SSH model. We also
demonstrate the excitation of topological phonon polaritons and show their
enhancement to the photonic LDOS.
