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Magnetic multiple-$Q$ states consisting of multiple spin density waves are asource of unconventional topological spin textures, such as skyrmion andhedgehog. We theoretically investigate a topologically nontrivial double-$Q$state with a net spin scalar chirality on a two-dimensional square lattice. Wefind that a double-$Q$ spiral superposition of the ordering wave vectorslocated at the Brillouin zone boundary gives rise to unconventional noncoplanarspin textures distinct from the skyrmion crystal. We show that such adouble-$Q$ state is stabilized by the interplay among the easy-axis anisotropicinteraction, high-harmonic wave-vector interaction, and external magneticfield. Furthermore, the obtained double-$Q$ state becomes a Chern insulatingstate with a quantum Hall conductivity when the Fermi level is located in theband gaps. Our present results provide another platform to realize topologicalmagnetic states other than skyrmion crystals by focusing on the symmetry ofconstituent ordering wave vectors in momentum space.

Chern insulating state with double-$Q$ ordering wave vectors at the Brillouin zone boundary