Title: Hybrid skyrmion and anti-skyrmion phases in polar $C_{\rm 4v}$ systems

Abstract: We investigate the stability of the skyrmion crystal phase in a tetragonal polar system with the Dzyaloshinskii-Moriya interaction by focusing on the symmetry of ordering wave vectors forming the skyrmion crystal. Our analysis is based on numerical simulations for an effective spin model, which is derived from the weak-coupling regime in the Kondo lattice model on a polar square lattice. We show that a hybrid square skyrmion crystal consisting of Bloch and N\'eel spin textures emerges even under polar $C_{\rm 4v}$ symmetries when the ordering wave vectors correspond to low-symmetric wave vectors in momentum space, which is in contrast to the expectation from the Lifshitz invariants. We also show the instability toward the anti-skyrmion crystal and rhombic skyrmion crystal depending on the direction of the Dzyaloshinskii-Moriya vector in momentum space. Furthermore, we show that the regions of the skyrmion crystal phases are affected by taking into account the symmetric anisotropic exchange interaction. Our results open the potential direction of engineering the hybrid skyrmion crystal and anti-skyrmion crystal phases in polar magnets.