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We numerically investigate the stabilization mechanisms of skyrmion crystalsunder thermal fluctuations and external magnetic field in itinerantcentrosymmetric tetragonal magnets. By adopting an efficient steepest descentmethod with a small computational cost, we systematically construct themagnetic field-temperature phase diagrams of the effective spin model derivedfrom the itinerant electron model on a two-dimensional square lattice. As aresult, we find that a square-type skyrmion crystal is stabilized by either orboth of the high-harmonic wave-vector interaction and the biquadraticinteraction under an external magnetic field. Especially, we discover that theformer high-harmonic wave-vector interaction can stabilize the skyrmion crystalonly at finite temperatures when its magnitude is small. In addition to theskyrmion crystal, we also find other stable multiple-$Q$ states in the phasediagram. Lastly, we discuss the correspondence of the phase diagrams betweenthe effective spin model and the skyrmion-hosting material GdRu$_2$Si$_2$. Thepresent results suggest a variety of multiple-$Q$ states could be driven bythermal fluctuations and external magnetic fields in centrosymmetric itinerantmagnets.

Widely-sweeping magnetic field-temperature phase diagrams for skyrmion-hosting centrosymmetric tetragonal magnets