Charge ordering and spontaneous topological Hall effect in bilayer skyrmion crystals

Magnetic skyrmion crystals with zero net skyrmion charge and zero topological Hall response are interesting candidate phases which can occur at a vanishing magnetic field in centrosymmetric systems. We study a minimal bilayer model of skyrmion crystals having opposite chirality and topological charge in the two layers, and show that it can host nearly flat electronic bands with quasiuniform Berry curvature and quantum metric. Using Hartree-Fock theory, we show that weak to moderate short-range electron interactions induce two distinct types of symmetry breaking patterns depending on the band dispersion: An intra-unit-cell charge density modulation from Chern band mixing or a layer-imbalanced phase with a nonzero ferroelectric polarization. Both phases break inversion symmetry leading to a spontaneous and large net topological Hall effect, with the phase diagram tunable by external electric fields. Our results may be relevant to centrosymmetric skyrmion materials such as ${\mathrm{Gd}}_{2}{\mathrm{PdSi}}_{3}$ and ${\mathrm{Gd}}_{3}{\mathrm{Ru}}_{4}{\mathrm{Al}}_{12}$ as well as artificially engineered heterostructures. We also discuss its relation to recent work on twisted transition metal dichalcogenide bilayers.