  When the crystalline symmetries that protect a higher-order topological phase
are not preserved at the boundaries of the sample, gapless hinge modes or
in-gap corner states cannot be stabilized. Therefore, careful engineering of
the sample termination is required. Similarly, magnetic textures, whose quantum
fluctuations determine the supported magnonic excitations, tend to relax to new
configurations that may also break crystalline symmetries when boundaries are
introduced. Here we uncover that antiskyrmion crystals provide an
experimentally accessible platform to realize a magnonic topological quadrupole
insulator, whose hallmark signature are robust magnonic corner states.
Furthermore, we show that tuning an applied magnetic field can trigger the
self-assembly of antiskyrmions carrying a fractional topological charge along
the sample edges. Crucially, these fractional antiskyrmions restore the
symmetries needed to enforce the emergence of the magnonic corner states. Using
the machinery of nested Wilson loops, adapted to magnonic systems supported by
noncollinear magnetic textures, we demonstrate the quantization of the bulk
quadrupole moment, edge dipole moments, and corner charges.
