## Research

The focus of my research during my PhD has been on Skyrmions. These are topological solitons - solutions of a nonlinear (ie tricky) field theory which are stable and exist thanks to the topology of the system. To find a nice analogy: take off your belt and flip over one side. A knot will form and, if you clamp each side of the belt, it won't go away. You can move the knot around and even assign it a position and velocity if you'd like. Now replace the belt with all of space and the knot is a Skyrmion. This topological origin of the Skyrmion makes it an interesting mathematical object. But there's also a physical interest.

Witten showed that, in a certain limit, these knot-like Skyrmions should model nuclei: things like protons, neutons, alpha-particles and Uranium-238. The B-Skyrmion models a nucleus with B protons and neutrons. To make contact with experimental data you have to quantise the Skyrmions. This is tricky and the simplest method (called the zero mode approximation) leads to some problems. My PhD focuses on quantisation beyond the zero mode approximation.

Below are details on some individual projects. Click on an image to find out more!

## Visualising Skyrmions

The basic field which makes up a Skyrmion is the pion field, **π**(x) = (π_{1},π_{2},π_{3}). This is a field vector; if it points up at x is means there is a concentration of π_{3} at that point. So you can visualise a Skyrmion by plotting this vector at every point in space. This looks ok but is not very informative. We're interested in things like the symmetry of each Skyrmion so just plotting these vectors is not overly helpful. Instead we could plot contours of constant energy densit. You can think of these as follows: **most** of the energy of the Skyrmion is contained within these shapes. These plots are also good but the arrows did contain useful information so we use a hybrid method.

How we visualise Skyrmions. On the right, we plot the pion vector at every point in space then colour them in. In the middle we plot a contour of constant energy density. On the right we combine these two methods.

First, we assign each direction a colour: if the arrow points up we colour it white, if it points in the x-direction we colour it red and so on. We plot the contour of constant energy density (like in Figure 2) then at each point on the surface we ask: what colour is the vector here? And colour it in accordingly. We then the colourful and information-filled plots that are dotted around this website!

## Modelling Lithium-7

Figure 1: The dodecahedral Skyrmion

The ground state of Lithium-7 has spin 3/2. The dodecahedral symmetry does not allow for this spin due it's rigid symmetry so we must allow the Skyrmion to break apart to describe the state. The easiest way to break the Skyrmion apart is shown in Figure 2: you pull on a pair of opposite verticies, splitting the 7-Skyrmion into two clusters. Physically this breaks Lithium-7 into tritium and an alpha particle.

Figure 2: Breaking the 7-Skyrmion into two clusters.

**is**allowed at the dodecahedron. Hence it need not seperate so much and has less of this "seperation energy" and looks something like Figure 3 (right). However, as it has a higher spin, it has a lot more "spin energy". In fact, the difference in spin energy is larger than the difference in seperation energy so the spin 7/2 state has more energy in total than the spin 3/2 state, matching experimental data.

Figure 3: A rough classical picture of what each quantum state looks like. The spin 3/2 ground state is on the left while the spin 7/2 excited state is on the right.

So, already, the dodecahedral symmetry results in different interpretations of the quantum states. The spin 7/2 state is a comact dodecahedron while the spin 3/2 state is an elongated, cluster-like object. This has consequences for the radii of the states and the electromagnetic transtitions between them which have never been measured. These predictions from the dodecahedron model are in conflict with those of more conventional models so if we measure them we can see which model is correct. Unfortunately these quantities are hard to measure experimentally, but not impossible.

Find out more details in the paper I wrote.

## The spin-orbit force

Magic nuclei, such as Calcium-40, are especially stable, inert objects thanks to their specific number of protons and neutrons. A nucleus with one extra nucleon (such as Calcium-41) has a nice structural interpretation: a single nucleon orbits a large stationery core. To match experimental data, it is known that the nucleon experiences a spin-orbit force. This means that it is favorable for the single nucleon's spin and orbital angular momentum to be aligned.

Does this spin-orbit force have a natural interpretation in the Skyme model..? Of course!

If two Skyrmions are seperated they either attract each other or repel. This (roughly) depends on the colours of closest contact. If the colours match the Skyrmions attract. Now suppose we are trying to describe the type of nucleus discussed above. The large core is described by a large Skyrmion while the orbitting nucleon is a 1-Skyrmion. This situation is shown below (left) and the colouring (note: it uses different colours, sorry!) around the equators is shown more clearly on the right.

Two Skyrmions in the "attractive channel". The colours of closest contact (red) are matching. We model the Skyrmions as discs, like the ones displayed on the right.

The Skyrmions want to stay in the attractive channel (ie they want their colours of closest approach to match up) at all times. So, the smaller orbitting Skyrmion wants to **roll** around the larger one, instead of sliding. Hence its spin (rotation around its own axis) and orbital angular momentum (rotation around the larger Skyrmion) are in the same direction. This is exactly what the spin-orbit force does.

In this paper my supervisor (Nick Manton) and I take this classical idea and find out what happens when we transport it to the quantum world.

## Modelling Oxygen-16

Oxygen-16 is a fascinating nucleus. It's been studied for a long long time but there is stil a lot of debate about what it looks like. Most people agree that the ground state looks like four alpha-particles arranged in a tetrahedron. The Skyrme model version of this is shown in Figure 1 (left). There is much less agreement about the first excited state. Some people think it looks like a square arrangement of alpha-particles (like the right-most picture in Figure 1) while others think it's something called a *breather*. This means that the alpha-particles can move in and out of where they like to be, but they keep the tetrahedral symmetry so it looks like the tetrahedron is breathing. I find this imagine slightly unsettling.

Figure 1: Alpha-particle configurations in the Skyrme model. The alpha-particles are the component cubes. On the left we see them make a tetrahedral Skyrmion while on the right they form a square.

The Skyrme model has a key advantage versus other models - we can generate dynamics. So, we can fire alpha-particles at each other and see what happens. By doing this we found a path which naturally connects the tetrahedral and square configurations discussed above. A video of this path can be found here, and a picture is displayed in Figure 2.

Figure 2: A dynamical path connecting the tetrahedral and square configurations. We fire to 8-Skyrmions at each other who pass through the tetrahedron, the square, come out into a different tetrahedron then fly off again having picked up a 90 degree twist.

Using this path we can generate lots of configurations and use these in our quantisation procedure to figure out what the first excited state of Oxygen-16 looks like in the Skyrme model. We discover a novel interpretation: it looks like a quantum average of the tetrahedron, the square and lots of other configurations in between!

To figure this out we developed some new techniques and theory. In doing so we realised that we could create new quantum states - such as a spin 0 state with negative parity - which have never been seen in alpha-particle models before. We were also able to describe an energy gap between spin-parity 2^{+} and 2^{-} states which had not been done before. Find out more in this paper which I wrote with Chris King and our supervisor Nick Manton.

Any questions? Get in touch at cjh95@cam.ac.uk.