  Recent acoustic and electrical-circuit experiments have reported the
third-order (or octupole) topological insulating phase, while its counterpart
in classical magnetic systems is yet to be realized. Here we explore the
collective dynamics of magnetic vortices in three-dimensional breathing
cuboids, and find that the vortex lattice can support zero-dimensional corner
states, one-dimensional hinge states, two-dimensional surface states, and
three-dimensional bulk states, when the ratio of alternating intralayer and
interlayer bond lengths goes beyond a critical value. We show that only the
corner states are stable against external frustrations because of the
topological protection. Full micromagnetic simulations verify our theoretical
predictions with good agreement.
