Higher-dimensional magnetic Skyrmions

TL;DR

This work extends magnetic Skyrmions from two to three spatial dimensions by promoting the magnetization to a 4-component vector valued on $S^3$, enabling a topological charge from $\pi_3(S^3)=\mathbb{Z}$. The authors construct a higher-dimensional Dzyaloshinskii–Momaori interaction (DM) compatible with $SO(3)_{\rm diag}$ symmetry, obtaining two invariant DM structures (alpha and beta) but finding the beta part irrelevant for spherical solutions; the simplest model yields a stable Skyrmion and an unstable sphaleron, while including the Skyrme term yields a small metastable Skyrmion, an unstable sphaleron, and a large stable Skyrmion. They explore spherically symmetric solutions, analyze full and reduced equations of motion, and map out a rich spectrum including anti-Skyrmions, sphalerons, and phase-transition-like coalescences through Derrick-type arguments. A deformation that connects to Hopfions is discussed, showing a non-smooth topology-driven connection between sectors; finally, the potential to realize these objects in condensed matter systems via synthetic dimensions is outlined, highlighting future pathways for experimental exploration. The results significantly broaden the landscape of magnetic solitons and establish a framework for linking higher-dimensional solitons to well-studied Hopfions and Skyrmions.

Abstract

We propose a generalization of the theory of magnetic Skyrmions in chiral magnets in two dimensions to a higher-dimensional theory with magnetic Skyrmions in three dimensions and an $S^3$ target space, requiring a 4-dimensional magnetization vector. A physical realization of our theory could be made using a synthetic dimension, recently promoted and realized in condensed matter physics. In the simplest incarnation of the theory, we find a Skyrmion and a sphaleron - the latter being an unstable soliton. Including also the Skyrme term in the theory enriches the spectrum to a small metastable Skyrmion, an unstable sphaleron and a large stable Skyrmion.

Figures (14)

• Figure 1: Profile functions for 3D magnetic Skyrmions with higher-dimensional $\mathop{\mathrm{SO}}\nolimits(3)_{\rm diag}$-invariant DM term. Left panels (a,c,e) show the profile on a regular scale and the right panels (b,d,f) on a log scale. The $m_{\rm crit}$ solution is shown with a dashed line. The rows of panels correspond to the potential power parameter $p=1,\tfrac{3}{2},2$. No solutions exist for $m>m_{\rm crit}$ nor for $m=0$.
• Figure 2: Energy density and topological charge density for 3D magnetic Skyrmions with higher-dimensional $\mathop{\mathrm{SO}}\nolimits(3)_{\rm diag}$-invariant DM term. Left panels (a,c,e) show the total energy densities weighted with $r$ (for visualization purposes) as black lines and the DM contribution as red lines. Right panels (b,d,f) show the topological charge density, weighted with $r^2$. The $m_{\rm crit}$ solution is shown with a dashed line. The rows of panels correspond to the potential power parameter $p=1,\tfrac{3}{2},2$. No solutions exist for $m>m_{\rm crit}$ nor for $m=0$.
• Figure 3: Profile functions for magnetic sphalerons with higher-dimensional $\mathop{\mathrm{SO}}\nolimits(3)_{\rm diag}$-invariant DM term, for (a) $p=1$, (b) $p=\tfrac{3}{2}$ and (c) $p=2$. The $m=0$ solution is shown with a green line and the $m_{\rm crit}$ solution with a dashed line. No solutions exist for $m>m_{\rm crit}$.
• Figure 4: Energy density and topological charge density for magnetic sphalerons with higher-dimensional $\mathop{\mathrm{SO}}\nolimits(3)_{\rm diag}$-invariant DM term. Left panels (a,c,e) show the total energy densities weighted with $r^2$ as black lines and the DM contribution as red lines. Right panels (b,d,f) show the topological charge density, weighted with $r^2$. The $m=0$ solution is shown with a green line (light-green for the DM term) and the $m_{\rm crit}$ solution is shown with a dashed line. The rows of panels correspond to the potential power parameter $p=1,\tfrac{3}{2},2$.
• Figure 5: (a) Sizes and (b) energies of magnetic sphalerons (bottom series) versus 3D magnetic Skyrmions (top series) with higher-dimensional $\mathop{\mathrm{SO}}\nolimits(3)_{\rm diag}$-invariant DM term. (a) The size is measured simply by the radius where $\chi=\frac{\pi}{2}$. (b) $-E$ is shown for the magnetic Skyrmions, as they have negative energy (i.e. they are stable solitons). Black pluses correspond to $p=1$, red crosses to $p=\tfrac{3}{2}$ and blue plus-crosses to $p=2$. The critical masses, $m_{\rm crit}$, correspond to the largest sphalerons and the smallest Skyrmions.