Large Anomalous Hall Effect in Topologically Trivial Double-$Q$ Magnets

Abstract

Multi-$Q$ magnets consist of superposed spin density waves with distinct magnetic modulation vectors, enabling a wide range of magnetic orders depending on their combination. Among them, topologically nontrivial spin textures, such as a magnetic skyrmion, has been extensively studied owing to the emergence of topological Hall effects induced by real-space scalar spin chirality. Contrary to this expectation, we theoretically investigate another route to enhancing the Hall response under a topologically \textit{trivial} double-$Q$ spin textures. Despite the cancellation of the scalar spin chirality, the double-$Q$ magnetism exhibits a pronounced Hall response with a nonmonotonic dependence on the uniform magnetization, which is in stark contrast to a ferromagnetic state and a single-$Q$ spiral state. Analyzing the multi-orbital Kondo lattice model, we show that orbital hybridization induced by the double-$Q$ superstructure enhances the Berry curvature in $\mathbf{k}$-space, leading to a large anomalous Hall effect. This mechanism accounts for the observed giant anomalous Hall effect in GdRu$_2$Si$_2$ and GdRu$_2$Ge$_2$, thereby highlighting topologically trivial double-$Q$ spin textures as promising spintronic materials.

Figures (3)

• Figure 1: Panels (a) and (b) illustrate the 1$Q$ and 2$Q$ spin configurations, respectively. Panels (c) and (d) show the anomalous Hall conductivity $\sigma_{xy}$ for the 1$Q$ and 2$Q$ states as functions of the chemical potential $\mu$ and the out-of-plane canting angle $\zeta$ of the magnetic moments. $\zeta = 0^{\circ}$ corresponds to coplanar spin configurations, while $\zeta = 90^{\circ}$ represents a fully polarized configuration. The same color scale is used for panels (c) and (d). For $-1.5 < \mu < 0$, the maximum values of $|\sigma_{xy}|$ are 0.647 for the 2$Q$ state and 0.145 for the 1$Q$ state. The same figure, plotted over the chemical-potential range $0.5 < \mu < 2.5$, is shown in Fig. 1 of the Supplemental Materials Suppl.
• Figure 2: Berry curvature of the energy eigenstates at the $\Gamma$ point for the 2$Q$ magnetic configuration. The vertical axis denotes the eigenenergy, and the horizontal axis represents the trajectory in the model-parameter space $(J,\;\zeta,\;\lambda)$. Along the parameter path, only one parameter is varied at a time while the others are held fixed: $(J,\;\zeta,\;\lambda): (0,\;0^{\circ},\;10^{-9}) \rightarrow (0.1,\;0^{\circ},\;10^{-9}) \rightarrow (0.1,\;10^{\circ},\;10^{-9}) \rightarrow (0.1,\;10^{\circ},\;0.1)$. Violet shading in the region for $0<J<0.1$ and $0^{\circ}<\zeta< 10^{\circ}$ indicates the absolute value in degenerate regions where contributions cancel, while the blue-to-red colormap is used in the region for $10^{-9} < \lambda < 0.1$ where degeneracy is lifted.
• Figure 3: Anomalous Hall conductivity $\sigma_{xy}$ as a function of the parameter $\zeta$. The three line plots correspond to the fully polarized (black dash-dotted), 1$Q$ (blue solid), and 2$Q$ (red bold) magnetic states, respectively. In all cases, the electron filling ratio $\nu$ is fixed at 0.84, correspnding to $\mu \simeq 1.77$ for $\zeta=0^{\circ}$ (See Fig. 1 of the Supplemental Material Suppl). The sampling interval for the canting angle $\zeta$ is $\Delta\zeta = 1^\circ$ in all cases, except for the 2$Q$ configuration in the range $0^\circ < \zeta < 10^\circ$, for which a finer resolution of $\Delta\zeta = 0.1^\circ$ is employed to capture the rapid variation in $\sigma_{xy}$.