On the cover: skyrmions

Sven Bjarke Gudnason and Calum Ross introduce you to the wonderful world of skyrmions


Let’s face it, physics is hard. Dirac may have opined that his eponymous equation “explains all of chemistry and most of physics”. However, even the most enthusiastic acolyte would admit that this is impractical at best. In particular, first principle calculations are only really possible for the simplest systems. If we want to understand anything more complicated than a harmonic oscillator or the electron energy levels of the hydrogen atom we need another approach.

This is where effective theories and simplified models come in. An effective field theory is essentially a model where you admit that you cannot describe a system in complete detail; instead, you try to describe what is effectively happening. A good example of this is how we study a fluid like water. We do not try and understand it at the level of the individual atoms; instead, we zoom out to a scale where we can describe it as a continuous substance.

An area where these effective models are particularly useful is when trying to understand the nuclei of atoms. You may remember from physics and chemistry lessons that atoms are like mini solar systems with a central structure, known as the nucleus, which contains most of the mass, and much lighter particles called electrons orbit around it. We can go one step further in and talk about the objects that make up the nucleus, the protons and neutrons. They are bound together by an incredibly strong force that acts over very short distances, and is creatively called the strong force. At this level the nucleus is a collection of protons and neutrons exchanging a triplet of other particles known as pions, $\pi$, which keep them stuck together. This is already starting to sound pretty complicated, but there is more. Protons, neutrons, and pions are not fundamental objects: they are formed from smaller particles known as quarks, bound together via the strong force. If you think that this is starting to sound ridiculous then you are not alone. We have started from a mini solar system and ended up with a churning sea of energy and mass.

A model that can sidestep some of these complications is the Skyrme model, introduced in the 1960s by Tony Skyrme describing nucleons, a catch-all term for protons and neutrons used in atomic physics. It has been extended to a model for the nuclei of atoms where we ignore the messy internal structure of protons and neutrons. Instead we zoom out to a scale where what we care about are pions, with nuclei appearing as static lumps of energy in the pion field.

Mathematically the Skyrme model is a field theory, a continuous model of a physical system, described by a non-linear partial differential equation. In other words, the equation describing how the pion fields evolve in time depends on how they change spatially and how the fields interact with themselves. In contrast to the integrable solitons met in Ricky’s solitons article, we can only solve the equations in the Skyrme model numerically. Another difference is that the Skyrme model involves a matrix field called $𝙐$ and is given by
\[𝙐(x)=\begin{pmatrix} \sigma(x)+\mathrm{i}\pi^{0}(x)&\mathrm{i}\pi^{1}(x)+\pi^{2}(x)\\ \mathrm{i}\pi^{1}(x)-\pi^{2}(x)&\sigma(x)-\mathrm{i}\pi^{0}(x) \end{pmatrix},\] subject to the constraint that the determinant of $𝙐(x)$ is $1$. The field $\sigma(x)$ is called the sigma field and is completely determined by physical constraints, so that we only care about finding the pion fields. We also need to include boundary conditions that specify how the pions behave asymptotically; to avoid describing a scenario with infinite energy, we set $\pi(\infty)=0$. Surprisingly, the skyrmion configurations found in this way have a conserved quantity: the baryon number, or the number of nucleons that the solution is equivalent to.

The nucleon

If we’re thinking about the classical model of atoms, the baryon number of a skyrmion is the same thing as an atomic mass number. However, this does not mean that skyrmions look like collections of well defined billiard-ball-like nucleons. On the contrary, there is a weird and wonderful world of skyrmion configurations so appetising that one of the authors referred to this as `a smörgåsbord of skyrmions’ in the title of a recent paper.

Skyrmions are examples of topological solitons: particle-like lumps of energy that are stabilised by the topology of the fields. In other words, the configurations described by $\pi^0, \pi^1$ and $\pi^2$ do not dissipate over time; all the energy has to stay in there somewhere. There is no continuous transformation of the field configurations to the one representing the vacuum: we say that they cannot be continuously deformed to it. This is where topology comes into play: it’s all about looking at structures, and asking whether they’re preserved by continuous transformations.

Now, where is the topology of the skyrmions coming from? It comes from the constraint that the determinant of $𝙐$ must be equal to $1$, which we can write as $\sigma^2+(\pi^0)^2+(\pi^1)^2+(\pi^2)^2=1$: the constraint tells us that the sum of the squared real numbers equals one. You might recall the formula for a circle of radius $r$: $x^2+y^2=r^2$. If we want to describe a (2-dimensional) sphere, we just extend it to $x^2+y^2+z^2=r^2$. We can keep going, building a formula with $n+1$ terms on the left to describe an $n$-sphere. So our constraint on the matrix field $𝙐$ is that the elements are coordinates for the 3-sphere of radius 1: skyrmion configurations start off with points $x,y,z$ in 3-dimensional Cartesian space, and map them to points on the 3-sphere.

We can even think of this as a map from one 3-sphere to another. To make it work, we use the clever trick of treating infinity as one point, which gives us a stereographic projection:

Stereographic projection

This lets us use a cool branch of mathematics known as homotopy theory, which tells us that maps from a circle to a circle are characterised by the number of times one circle is wrapped around the other circle. Think of wrapping a shoelace around your ankle lots of times before you tie it: to work out what’s happened to your leg, you just need to count the number of wraps.

Map between circles. When the first circle is mapped $N$ times to the second circle, we can visualise it as $N$ circles

This works in higher dimensions too: to understand how our maps $\pi^0, \pi^1, \pi^2$ work, we just need to look at the surface they describe, and work out how many times it coves the 3-sphere. This is the baryon number we mentioned earlier, and it’s a conserved quantity (so we should be able to predict the total). However, what we can’t predict is whether they are clustered or widely separated, just their total number throughout 3-dimensional space.

How do we understand what skyrmions look like? Well, we go back to the Skyrme model and look at the expression for the energy of these pion field configurations. We know that they correspond to physical atoms when the energy is minimised, so we minimise the energy and see what resulting configuration looks like.

We can think of the space of field configurations with a given baryon number as being a mountainous landscape. High energy configurations sit at the peaks of mountains and are very unstable, able to roll down the side of the hill to reach a valley where the low energy configurations live. Understanding this energy landscape is a challenging problem, particularly since we want to find the lowest valleys where our classical picture of an atomic nucleus lives.

Over hills and through the vales

Starting with a single skyrmion, its solution can be constructed by minimising the energy of a map that resembles a hedgehog. We draw the map of the surface described by the pions: usually by letting $\pi^0\propto z$ and $\pi^\pm=\pi^{1}\pm \mathrm{i}\pi^{2}\propto x\pm \mathrm{i}y$ (but any other permutation or rotation would do equally well). Why do we call it a hedgehog? If we plot all the vectors we get a sphere covered with arrows all of which are pointing outwards, like a hedgehog rolled up into a ball.

Skyrmions are best visualised with a colour that tells us the directions in which the pions are pointing. That is positive $\pi^0$ is white, negative $\pi^0$ is black, and $\pi^\pm$ is usually mapped to a standard choice of the colour wheel, going around from red to green to blue and back to red. The entire sphere of colours is also known as Runge’s colour sphere. This is what leads to the jazzy colours in our pictures of the nucleon.

What do we need to know about skyrmion interactions? Same colours attract. Opposite colours repel. It’s more or less that simple. If we place two skyrmions with the same colours close to each other, they will attract in a finite time and form a bound state of some sort. The first few shapes that will be formed are these: two skyrmions will attract and form a torus, three skyrmions will attract and form a tetrahedron, and finally four skyrmions in the attractive channel will form a cube:

Skyrmion solutions with Baryon number 2 through 4

How do we find a multi-skyrmion? This topic has been fascinating researchers for several decades and several very nice mathematical concepts have been invoked to cook up good approximations for skyrmions with a given baryon number. If we are happy to turn to a numerical algorithm and let the computer do the work, there is a nice simple recipe, that was in fact used in finding over 400 skyrmion solutions in the aforementioned smörgåsbord of skyrmions.

It goes like this: start with a single skyrmion $𝙐$. Rotate it with three random angles. Place it at a random position.

Add in another skyrmion, which you’ve rotated with random angles and placed at a random position (but not too far from the previous one). This ensures that we find a multi-skyrmion and not just several clustered skyrmions. Repeat $B-1$ times, until we reach baryon number $B$.

How do we combine the maps of these skyrmions? A simple way, which is not unique, is to use the fact that the determinant of the product of matrices is the product of determinants of the individual matrices. This guarantees that $𝙐=𝙐_1𝙐_2\cdots 𝙐_B$ has determinant $1$, if each $𝙐$ has determinant $1$. This gives us what is known as an initial condition.

An intuitive numerical method is to simulate the motion of a ball in a potential. If we think of the initial condition as the ball sitting at any hillside of the energy landscape, letting it go corresponds to evolving it in time. The ball accelerates toward the valley of the landscape.

Being interested in finding the valley, we need to measure the potential energy, ie the height of the position on the hill as we go. Once the height starts to increase, we have either crossed the minimum or a mountain pass—a saddle point—and the ball is now climbing up another hillside.

We alter the dynamical situation, by removing all the kinetic energy as soon as we discover that the ball is climbing another hillside. Then we start over and let the ball go from the new position. This numerical algorithm does not guarantee that we find the global minimum, that is the lowest valley of the landscape, but only that we have found a local minimum: the bottom of the nearest valley.

For skyrmions with baryon numbers 1–4, 6, and 7, there is only one known skyrmion solution. So whatever we start with for these baryon numbers, we will end up in the known lowest valley of the energy landscape. It becomes more complicated for larger baryon numbers. In fact, there are 16 skyrmions with $B=11$, including the one in the header picture, and more than 140 skyrmions with $B=16$ in the smörgåsbord.

$B=16$ skyrmion

Atoms and beyond

We know that atoms exist in the realm of quantum mechanics, so there is another chapter to this story. We have also not distinguished between protons and neutrons. However, it is well known that they are not exactly the same. The way to tell them apart comes from quantising the Skyrme model. However, that is a story for another day. Research into skyrmions and their role in understanding atoms is very much an active field. If you want to know more, then a great place to start is the Solitons at Work network. This is an online community of soliton researchers, giving talks about their work, organising conferences and workshops, and sharing pictures of skyrmions. As members of the committee we may be biased, but it is a great place to see more about the wonderful world of solitons.

Calum is a lecturer in physics at Edge Hill University where he studies topological solitons and enjoys talking about maths and physics to anyone who will listen. He can often be found on the side of a mountain or out on a bike, talking to himself about solitons.

Bjarke is a professor of mathematical physics at Henan University and an expert in all things skyrmions. He is often found near a blackboard drinking coffee and thinking about solitons, gravity and maths. His Smörgåsbord of Skyrmions paper was the inspiration for this article.

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