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Skyrmions are spatially localized magnetic swirls which carry a nonzerointeger topological charge. We study crystals of skyrmions in a two-dimensionalferromagnet model with chiral interactions induced by the presence of brokeninversion symmetry. We show that non-linear quartic mode-coupling terms allowedby symmetry enhances the Zeeman-field range over which the skyrmion-crystalphase remains stable. Furthermore, it leads to a significant dependence of thelattice constant of this spin crystal over this wide field range. Conductionelectrons coupled to such a tunable spin crystal are shown to experience aBerry-flux density which varies with the Zeeman field. Such tunable skyrmioncrystals provide a distinct realization of a Berry-Hofstadter butterfly,resulting in a phenomenon we term "topological quantum oscillations" in theelectronic density of states and associated observables.

Tunable skyrmion crystals and topological quantum oscillations in magnetic metals