Altermagnetic spin textures: Emergent electrodynamics, quantum geometry, and probes

Abstract

Emergent electrodynamics arising from spatially and temporally varying magnetic textures provides a framework for spin control in quantum materials. While this principle is established for ferromagnetic and antiferromagnetic textures, its consequences for altermagnets -- magnetic orders with vanishing net magnetization but finite spin splitting -- remain largely unexplored. In this work, we develop an effective low-energy theory of itinerant electrons coupled to smoothly varying altermagnetic spin textures. In the adiabatic regime, we show that altermagnetic textures generate additional emergent electromagnetic fields and quantum-geometric effects that are absent in conventional magnetic systems. These effects include emergent Zeeman fields that encode the structure of the altermagnetic order parameter, enabling local spin manipulation and a way to distinguish different altermagnetic orders. Moreover, we demonstrate a quantum-metric-induced, spin-dependent electron lensing effect that provides a mechanism for spin filtering, and discuss the local admixture of effective odd-parity magnetic components. Our results suggest that textured altermagnets could serve as a versatile resource for spintronics functionalities and a probe of altermagnetism.

Figures (5)

• Figure 1: Circular Néel domain wall separating two altermagnetic domains. (a) Real-space schematic of the Néel vector field, $\boldsymbol{n}(\boldsymbol{r})$, on the two sublattices ($\tau_z=\pm$; orange and blue). Arrows indicate the local in-plane spin orientation. The shaded annulus denotes the domain wall. The solid circle marks the domain wall radius, $R$. (b) Radial rotation profile, $\phi(r)$, describing the $\pi$-rotation of $\boldsymbol{n}$ across the wall. The wall center is at $r = R$. The illustrations are schematic and not to scale. In the paper, we assume a texture with $a \ll w \ll R$ ($a$ is the lattice constant and $w$ is the wall width).
• Figure 2: Multipolar emergent Zeeman fields from altermagnetic domain walls. (a) Emergent Zeeman field, $V_z(\boldsymbol{r})$ (normalized by its maximum), for a circular domain wall in a $d$-wave altermagnet, showing a quadrupolar pattern localized at the wall. (b) Corresponding result for a $g$-wave altermagnet, showing an octupolar pattern. (c,d) Angular cuts, $V_z(R,\varphi)$, at the wall radius, showing the $\cos(2\varphi)$ and $\sin(4\varphi)$ dependence. The multipolar form of $V_z$ provides a novel probe for altermagnetic order.
• Figure 3: Texture-induced effective metric near a circular domain wall. (a,b) Spatial map of $\lambda_{\max}-\lambda_{\min}$ for $\tau_z=\pm$, where $\lambda_{\max/\min}(\boldsymbol{r})$ are the eigenvalues of the metric tensor, $g_{\text{eff},\tau}(\boldsymbol{r})$, that enters the kinetic energy. At fixed $\boldsymbol{r}$, the local dispersion forms an ellipse in momentum space and $\lambda_{\max}-\lambda_{\min}$ quantifies its elongation. (c,d) Major axis of the ellipse (eigenvector of $\lambda_{\max}$) evaluated along the wall, which sets the orientation of the local dispersion.
• Figure 4: Altermagnetic domain wall as a sublattice-selective electron lens. Semiclassical trajectories incident from the left on a circular domain wall (gray disk), with $\tau=+1$ (blue) and $\tau=-1$ (red). Left: metric lensing from the texture-induced effective metric $g^{\mathrm{eff}}_{\tau,ij}(\boldsymbol{r})$. Right: potential lensing from the emergent scalar potential $V_\tau(\boldsymbol{r})$. Potential lensing produces opposite deflections of the two $\tau$ sectors (focusing/defocusing). In contrast, metric lensing generates a transmission/reflection asymmetry.
• Figure 5: Emergent spin-orbit coupling and odd-parity magnetism from altermagnetic textures. (a) Circular Néel domain wall. Faint arrows show the in-plane Néel vector field, $\boldsymbol{n}(\boldsymbol{r})$. The marked point lies at azimuthal angle $\varphi=\pi/4$, where the emergent Zeeman field, $V_z \propto \cos 2\varphi$, vanishes. (b) Local constant-energy contours, $E_\pm(\boldsymbol{p})=\mu$, evaluated at that point. Because $V_z=0$, the splitting arises purely from the metric-induced spin-orbit term, $\boldsymbol{\mathtt{g}}_{\boldsymbol{p}} \propto \boldsymbol{p}\cdot\nabla\phi$. The splitting vanishes for $\boldsymbol{p} \perp \nabla\phi$ (dashed line), producing a nodes. (c) Skyrmion texture. The color scale shows $n_z$. Arrows indicate the in-plane components of $\boldsymbol{n}(\boldsymbol{r})$. (d) Local constant-energy contours at the point in (c). Here, both $\nabla\theta$ and $\sin\theta\nabla\phi$ contribute to $\boldsymbol{\mathtt{g}}_{\boldsymbol{p}}$, which eliminates any nodes.