  Skyrmions are spatially localized magnetic swirls which carry a nonzero
integer topological charge. We study crystals of skyrmions in a two-dimensional
ferromagnet model with chiral interactions induced by the presence of broken
inversion symmetry. We show that non-linear quartic mode-coupling terms allowed
by symmetry enhances the Zeeman-field range over which the skyrmion-crystal
phase remains stable. Furthermore, it leads to a significant dependence of the
lattice constant of this spin crystal over this wide field range. Conduction
electrons coupled to such a tunable spin crystal are shown to experience a
Berry-flux density which varies with the Zeeman field. Such tunable skyrmion
crystals provide a distinct realization of a Berry-Hofstadter butterfly,
resulting in a phenomenon we term "topological quantum oscillations" in the
electronic density of states and associated observables.
