  We theoretically investigate plasmon polaritons in cubic lattices of
spherical metallic nanoparticles. The nanoparticles, each supporting
triply-degenerate localized surface plasmons, couple through the Coulomb
dipole-dipole interaction, giving rise to collective plasmons that extend over
the whole metamaterial. The latter hybridize with photons forming plasmon
polaritons, which are the hybrid light-matter eigenmodes of the system. We
derive general analytical expressions to evaluate both plasmon and
plasmon-polariton dispersions, and the corresponding eigenstates. These are
obtained within a Hamiltonian formalism, which takes into account retardation
effects in the dipolar interaction between the nanoparticles and considers the
dielectric properties of the nanoparticles as well as their surrounding. Within
this model we predict polaritonic splittings in the near-infrared to the
visible range of the electromagnetic spectrum that depend on polarization,
lattice symmetry and wavevector direction. Finally, we show that the
predictions of our model are in excellent quantitative agreement with
conventional finite-difference frequency-domain simulations, but with the
advantages of analytical insight and significantly reduced computational cost.
