Noncoplanar multiple- Q spin textures by itinerant frustration: Effects of single-ion anisotropy and bond-dependent anisotropy

Abstract

We theoretically investigate multiple-$Q$ spin textures, which are composed of superpositions of spin density waves with different wave numbers, for an effective spin model of centrosymmetric itinerant magnets. Our focus is on the interplay between biquadratic interactions arising from the spin-charge coupling and magnetic anisotropy caused by the spin-orbit coupling. Taking into account two types of the magnetic anisotropy, single-ion anisotropy and bond-dependent anisotropy, we elucidate magnetic phase diagrams for an archetypal triangular-lattice system in the absence and presence of an external magnetic field. In the case of the single-ion anisotropy, we find a plethora of multiple-$Q$ instabilities depending on the strength and the sign of the anisotropy (easy plane or easy axis), including a noncoplanar triple-$Q$ state regarded as a skyrmion crystal with topological number of two, and coplanar and noncoplanar double-$Q$ states. In an external magnetic field, we find that another noncoplanar triple-$Q$ state, a skyrmion crystal with topological number of one, is stabilized by the in-plane (out-of-plane) magnetic field under the easy-plane (easy-axis) anisotropy. A part of the results, especially for the relatively large biquadratic interaction, qualitatively reproduce those in the Kondo lattice model, which explicitly includes itinerant electrons [S. Hayami and Y. Motome, Phys. Rev. B 99, 094420 (2019)]. We also examine the stability of the field-induced skyrmion crystal by rotating the field direction. As a by-product, we show that a different triple-$Q$ state with nonzero chirality appears in the presence of the biquadratic interaction and the easy-axis anisotropy. Meanwhile, we find that the bond-dependent anisotropy also stabilizes both types of skyrmion crystals. We show that, however, for the skyrmion crystal with topological number of one, Bloch- and N\'eel-type skyrmion crystals are selectively realized depending on the sign of the bond-dependent anisotropy, since this anisotropy selects a particular set of the helicity and vorticity. Moreover, we find yet another multiple-$Q$ states with nonzero spin scalar chirality, including two types of meron crystals with the skyrmion numbers of one and two. The systematic investigation of multiple-$Q$ instabilities in triangular itinerant magnets will provide a reference to complex magnetic textures in centrosymmetric magnetic metals.