Double- Q and quadruple- Q instabilities at low-symmetry ordering wave vectors under tetragonal symmetry

Multiple-Q states are expressed as a superposition of spin density waves at multiple ordering wave vectors, which results in unconventional complicated spin textures, such as skyrmions, hedgehogs, and vortexes. We investigate the multiple-Q instability by focusing on the low-symmetry ordering wave vectors in momentum space. By systematically performing the simulated annealing for effective spin models with various ordering wave vectors on a two-dimensional square lattice, we classify the magnetic phase diagram into four types according to the position of the ordering wave vectors. Three out of four cases lead to a plethora of isotropic multiple-Q instabilities yielding collinear, coplanar, and noncoplanar double-Q and quadruple-Q magnetic phases, while the remaining case leads to an anisotropic double-Q instability when the multiple-spin interaction is introduced. Our results indicate that exotic multiple-Q phases distinct from the skyrmion crystal phase are expected when the ordering wave vectors lie on the low-symmetry positions in the Brillouin zone.