Spiral-induced Anomalous Hall Effect from Odd-parity Spin-nodal Lines

Abstract

Spin spirals represent a fundamental class of noncollinear yet coplanar magnetic structures that give rise to diverse emergent phenomena reflecting spin chirality. We investigate metallic systems hosting commensurate spin spirals and uncover an unconventional anomalous Hall effect (AHE) induced by spiral magnetism. The spin spiral introduces odd-parity spin splitting with polarization perpendicular to the helical plane, forming spin-nodal lines in the electronic structure. In the presence of spin-orbit coupling, we find that these nodal lines become gapped by finite magnetization, concentrating the Berry curvature near the gap and generating a distinctive AHE. We identify the interplay among the spin-orbit coupling, helical plane orientation, and magnetization direction as the key ingredient for this spiral-induced AHE, which is expected to occur across a wide range of materials hosting commensurate spin spirals.

Figures (5)

• Figure 1: Schematic picture of the spiral-induced AHE. The red arrow illustrates electron flow on a spin spiral order with the $yz$ helical plane and spiral pitch $L = 4$. The blue arrows and circles on the square lattice denote localized spins and their helical planes, respectively. The green and purple arrows represent the net magnetization $M_z$ and the spiral propagation vector $\bold{Q}$ along the $x$ axis, respectively.
• Figure 2: Low-energy electronic band structures for the $yz$ spiral state at (a) $J = 0.8$ and (b) $J = 2.0$. The left panels show the band dispersions for $\lambda=m=0$; the lowest-energy bands are projected onto the bottom plane. The right panels plot the energy dispersions along the $\bar{\mathrm{X}}\Gamma\mathrm{X}$ line at $k_y = 0$. The dashed lines show the results with SOC ($\nu=y$, $\lambda = 0.01$) and magnetization [$\mu=z$, (a) $m = 0.03$, (b) $m = 0.10$]. The horizontal lines represent the Fermi levels $\varepsilon_{\mathrm{F}}$ used for the AHC calculations in Fig. \ref{['f3']}(a) and the Fermi surfaces in Figs. \ref{['f3']}(b) and \ref{['f3']}(c). In both (a) and (b), the color plots visualize the expectation value of the spin polarization along the $x$ direction, $\langle s^x_{\bold{k}}\rangle$.
• Figure 3: (a) Net magnetization $M_z$ dependence of the AHC $\sigma_{xy}$ in the $yz$ spiral state for several parameter sets of $J$ and $n_\mathrm{e}$. (b, c) Color plots of the Berry curvature $\Omega^z_{\bold{k}}$ of the lowest (left) and the second-lowest (right) energy bands for $J = 2.0$ at (b) $m = 0.1$ ($M_z \sim 0.05$) and (c) $m = 0.5$ ($M_z \sim 0.3$). The black rectangles denote the first BZ. The orange and green lines represent the Fermi surfaces at $n_\mathrm{e} = 0.014$ and $0.050$, respectively.
• Figure 4: (a) Full electronic band structure along the representative symmetric lines in the first BZ, colored by the spin polarization $\langle s^z_{\bold{k}}\rangle$, at $J=2.0$, $\lambda=0.01$, and $M_z \sim 0.05$ ($m = 0.1$). The high-symmetry points are denoted in Fig. \ref{['f3']}(b). (b) Corresponding $\sigma_{xy}$ in the $yz$ spiral state as a function of $\varepsilon_\mathrm{F}$ The horizontal dashed lines represent the energy levels where $\sigma_{xy}$ has peaks, as guides for the eyes.
• Figure 5: Similar plots to Fig. \ref{['f4']} in the main text for $L = 5$ at $J = 2.0$, $\lambda = 0.01$, and $m = 0.1$.