Spin-Flux Skyrmions: Anomalous Electron Dynamics and Spin-Hall Currents

TL;DR

The paper addresses unexplained Hall-resistivity anomalies in skyrmion-hosting materials by formulating an explicit SU(2) gauge-field description that distinguishes conventional skyrmions from a new class called spin-flux skyrmions. This distinction stems from the doubly-connected SO(3) topology, yielding two topologically inequivalent rotation paths and, for spin-flux skyrmions, a monopolar $\sigma_x$ component in the emergent field that contributes an additional Hall term. The authors derive the emergent-field structure for both textures, show how the average emergent field depends on skyrmion density and core radius, and develop a spin-dependent Drude theory linking these fields to the Hall response, including a tunable in-plane spin-polarization effect. The work provides a mechanism to explain nonmonotonic Hall behavior with field and proposes measurable signatures (e.g., finite $B_{zx}^{av}$ and spin-flip scattering) that can identify spin-flux skyrmions, with potential implications for tunable spintronic devices.

Abstract

We introduce a topologically distinct skyrmion, termed a spin-flux skyrmion, which shares the same real-space magnetization profile as a conventional skyrmion but differs fundamentally in its underlying topological structure. This distinction originates from the path traced by its rotation matrices within the doubly connected SO(3) group manifold, leading to a nontrivial spinor phase of $e^{iπ}$ upon encircling the texture. Using an explicit SU(2) gauge field formalism, we derive the emergent magnetic field components generated by both conventional and spin-flux skyrmions. While conventional skyrmions exhibit a dominant $σ_z$ component with weak dipolar $σ_x, σ_y$ contributions, spin-flux skyrmions possess an additional monopolar $σ_x$ component that yields a finite average emergent field for a finite density of skyrmions. This nontrivial component introduces a nontrivial term in the Hall conductivity, enabling a direct explanation of experimental Hall resistivity anomalies that cannot be accounted for by conventional skyrmions alone. Moreover, we show that this additional term couples to the in-plane spin polarization of conduction electrons, providing a further tunable handle to control the transverse Hall response.

Figures (8)

• Figure 1: We illustrate the distinction between two skyrmions corresponding to magnetic moment rotation fields $U_1$ and $U_2=U_2^zU_2^y$, in terms of the SO(3) group manifold (panels (a) and (b)). In this picture, a solid ball of radius $\pi$ is used to represent all possible spin rotations: each point inside the ball corresponds to a unique rotation, the displacement from the origin indicates the rotation angle, and the direction specifies the rotation axis. The surface of the ball therefore represents a rotation angle of $\pi$. For the conventional $U_1$ skyrmion (panel (a)), we consider circular paths in coordinates space with radii $r=\rho_0$ (the skyrmion core radius) and $r=\infty$, which we label as $p_1$ and $p_1^{\prime}$, respectively. All rotation matrices describing this skyrmion configuration lie infinitesimally belong the surface of the SO(3) ball (shown in blue and red). Here, the rotation field corresponds to a fixed rotation angle of $\pi$ about the spatially varying axis unit vector $\hat{n}(\vec{r})$. As the skyrmion is encircled, the spin rotation trajectory in SO(3) forms a closed loop on the surface. Both paths, $p_1$ and $p_1^{\prime}$, can be continuously deformed to a single point and are considered topologically trivial within SO(3). For the $U_2$(spin-flux) skyrmion (panel (b)), the rotation $U_2^y$ corresponds to single point in SO(3) on the y-axis for any circular path of given radius. The rotation $U_2^z$ traverses a straight line path from the surface of SO(3), through the center of the ball, to the antipodal point as the skyrmion is encircled at any radius and $-\pi<\zeta<\pi$. This endows the $U_2$-skyrmion with a nontrivial spin-flux of $\pi$. Since antipodal points on the sphere of radius $\pi$ are identical rotations, the path $p_2$ cannot be continuously deformed to a single point. In other words, the paths $p_1 (p_1^{\prime})$ and $p_2$ are homotopically distinct.
• Figure 2: Behavior of the emergent magnetic field and vector potential for a conventional Neel skyrmion with spin rotation $\theta(r)= 2\tan^{-1}(r/\rho_0)$, and $U_1= \exp(-i\frac{\pi(\hat{n}\cdot\vec{\sigma})}{2})$, where $\vec{n}=(\sin\frac{\theta}{2}\cos\phi, \sin\frac{\theta}{2}\sin\phi,\cos\frac{\theta}{2})$. Panels (a-c) display the $\sigma_z$, $\sigma_x$, and $\sigma_y$ components of the magnetic field, respectively. Panels (d-f) show the corresponding components of the vector potential. The color bars represent the magnitude of the vector potential, while the arrows indicate its direction. The emergent vector potential and magnetic field are scaled in units of $\hbar/a$ and $\hbar/a^2$, respectively, where $a$ is the lattice constant.
• Figure 3: The emergent magnetic field and vector potential for a spin-flux skyrmion with $\theta(r)= 2\tan^{-1}(r/\rho_0)$, and unitary matrix field $U_2 = e^{-i\frac{\sigma_{z}\zeta}{2}} e^{-i\frac{\sigma_{y}\theta(r)}{2}}$. Panels (a) and (b) show the $\sigma_z$ and $\sigma_x$ components of the emergent magnetic field, respectively. Notably, the $\sigma_y$ component vanishes in this configuration. Unlike $U_1$, an emergent off-diagonal magnetic field of monopolar form appears in the $\sigma_x$ component. Panels (c-e) display the $\sigma_z$, $\sigma_x$, and $\sigma_y$ components of the vector potential. The color bars represent the magnitude of the vector potential, while the arrows indicate its direction. Panel (f) shows the singular radial dependence of the $\sigma_z$ component of the vector potential, denoted as $\mathcal{A}_{3}$. The emergent vector potential and magnetic field are scaled in units of $\hbar/a$ and $\hbar/a^2$, respectively, where $a$ is the lattice constant.
• Figure 4: Spatial profiles of components of the emergent magnetic field, for a lattice of skyrmions with core radius $\tilde{\rho_0}=5a$, within the large circular region of radius $\tilde{r}=20a$. Panel (a) shows the distribution of the $B_{zz}$ component in the $x-y$ plane for a lattice of either $U_1$ or $U_2$ skyrmions. For both skyrmion lattices, $B_{zz}$ remains nearly uniform throughout the interior but drops sharply near the boundary. Panel (b) presents the spatial variation of $B_{zx}$ for the $U_1$ skyrmion lattice. As expected, $B_{zx}$ is nearly zero across the central area but becomes nonzero near the boundary and displays a dipolar-like structure. In contrast, panel (c) illustrates the behavior of $B_{zx}$ for the $U_2$ skyrmion lattice. This has monopolar peaks at the center of each skyrmion, leading to a significant average value over the large circular area.
• Figure 5: Panels (a) and (b) show the behavior of the averaged magnetic field component $B_{zz}^{\text{av}}$ as a function of skyrmion density, $D_{\text{sk}}$, and skyrmion radius $\rho_0$, respectively. $B_{zz}^{\text{av}}$ remains the same for both $U_1-$skyrmion and spin-flux $U_2-$skyrmion lattices. Panels (c) and (d) display the behavior of the averaged off-diagonal field component $B_{zx}^{\text{av}}$ for $U_2-$skyrmion lattice, as a function of $D_{\text{sk}}$ and $\rho_0$, respectively. The amplitude of $B_{zz}^{\text{av}}$ is larger than that of $B_{zx}^{\text{av}}$, due to their distinct functional forms, as described in Eqs. \ref{['magcase1']} and \ref{['magcase2']}. The dependence of $B_{zz}^{\text{av}}$ and $B_{zx}^{\text{av}}$ on $\rho_0$ is qualitatively different as seen by comparing panels (b) and (d). For the $U_1-$skyrmion lattice, the average values of the off-diagonal field components are nearly zero. In all cases, the region containing skyrmions has a radius of $25a$, where $a$ is the lattice spacing between magnetic moments.