Most non-trivial mathematical objects are hard to visualize, and topological solitons are no exception. Despite this, they seem to be one of physicists' long-time favourites, appearing in many fields from cosmology to condensed matter. But the mathematical elegance of quantum field theories isn't the only reason for their popularity: they describe real, measurable effects. Two-dimensional topological solitons have been observed in materials with magnetic or nematic order, and magnetic skyrmions are a promising route to creating spintronic devices. And now, in two recent papers, Paul Ackerman and Ivan Smalyukh have uncovered a host of three-dimensional solitons in liquid crystals (Nat. Mater. http://doi.org/bzh4; 2016 and Phys. Rev. X 7, 011006; 2017).

Credit: Macmillan Publishers Ltd

To understand these objects, let's start with two-dimensional magnetic skyrmions, or 'baby' skyrmions. These are easy to imagine as vortex-like spin configurations whose arrows twist around to point in all directions in three-dimensional space. This means that all possible magnetizations, if mapped as points on the unit sphere, would cover it entirely. This is the very source of topological protection for baby skyrmions. Now let's make things more complicated and move up a dimension where visualization fails. If one direction of magnetization used to correspond to a point on the unit sphere, for three-dimensional solitons we have closed loops, or 'preimages', instead of points. These preimages come in handy to describe these three-dimensional objects. For instance, a hopfion is a topological soliton whose preimages are closed loops on nested tori. Like in the case of the baby skyrmions, the hopfion preimages will completely fill the unit sphere.

We can think of the preimages as the projections of the higher-dimensional object onto a lower-dimensional plane. Putting them together allows us to visualize the three-dimensional solitons. Preimages are also the shadows left by the solitons in materials. Ackerman and Smalyukh directly probed the magnetization of a chiral ferromagnetic liquid crystal colloid to observe the preimages of hopfions (pictured). Then, with the help of computer simulations, they were able to reconstruct the three-dimensional solitons and study their stability and the role of chirality. Ackerman and Smalyukh got similar results in another experiment using a chiral nematic liquid crystal, where on top of hopfions they identified other exotic structures whose preimages are more complicated interlinked loops.

In terms of magnetic skyrmions' promise to spintronics, one can immediately imagine a practical use for three-dimensional solitons. But what is perhaps equally or even more exciting is the fact that such complex mathematical objects actually emerge in real materials, leaving their mark in the form of relatively long-lived micrometre-scale structures that can be imaged directly.