Skyrmion Knots in Frustrated Magnets

A magnetic Skyrmion is a stable two-dimensional nanoparticle describing a localized winding of the magnetization in certain magnetic materials. Skyrmions are the subject of intense experimental and theoretical investigation and have potential technological spintronic applications. Here we show that numerical computations of frustrated magnets predict that Skyrmions can be tied into knots to form new stable three-dimensional nanoparticles. These stable equilibria of twisted loops of Skyrmion strings have an integer-valued topological charge, known as the Hopf charge, that counts the number of particles. Rings are formed for low values of this charge, but for higher values it is energetically favorable to form links and then knots. This computational study provides a novel impetus for future experimental work on these nanoknots and an exploration of the potential technological applications of three-dimensional nanoparticles encoding knotted magnetization.

Figure 1

The stable charge one soliton. (a) An isosurface (blue) enclosing the region of the magnetic material where the magnetization points in the direction $\mathbf{m}=(0,0,-1)$, showing the location of the Skyrmion string. An isosurface (red) enclosing the region where $\mathbf{m}=[0,-(1/\sqrt{2}),-(1/\sqrt{2})]$ shows the twist of the Skyrmion string. This image clearly shows that the linking number, $Q$, is equal to 1. For scale, the outline of the displayed simulation box is a cube of side length 15 dimensionless units. (b) A cross section of the magnetization $\mathbf{m}$ in a plane containing the axis of rotational symmetry. The color represents the value of $m_3$ and the arrows are proportional to the two-component vector $(m_1,m_2)$. This shows that the right-hand side of the cross section contains a Skyrmion and the left-hand side contains an anti-Skyrmion. Note that the color schemes in (a) and (b) are not related because they represent different quantities in the two figures.

Figure 2

Stable higher charge solitons. Isosurfaces enclosing the region of the magnetic material where the magnetization points in the directions $\mathbf{m}=(0,0,-1)$ (blue) and $\mathbf{m}=[0,-(1/\sqrt{2}),-(1/\sqrt{2})]$ (red), for a selection of stable solitons: (a) the charge 3 circular ring, (b) the charge 6 buckled ring, (c) the charge 7 link with components of charge 3 and 2, (d) the charge 10 trefoil knot.

Figure 3

Soliton energies. The energy per unit charge $E/Q$, in units of the charge one energy, for stable equilibria that take the form of circular or buckled rings (black circles), links (red squares), and knots (blue triangles).