Skyrmions

Prerequisites: Solitons, Topology. Full path (14 pages)

Intuitive
Concrete
Abstract
History

Intuitive

Picture a large flat sheet covered with small arrows, one at every point, all pointing straight up. Now, in a small region, rotate the arrows so that the one in the middle points straight down and those around it tilt smoothly from down, through sideways, back to up as you move outward. The arrows on a circle around the centre wind once around while doing so. Far away nothing has changed.

This pattern is a skyrmion. It cannot be removed by any smooth rearrangement of the arrows. You could shrink it or move it, but to get rid of it you would have to tear the pattern somewhere, which costs a lot of energy. So the twist behaves like a particle: it has a position, it can move, and it is stable because of its shape rather than because of some conserved charge put in by hand. The number of times the arrows wrap around all possible directions is an integer, and it can only change by such a tearing.

Skyrmions show up in two very different places.

In magnets, the arrows are the electron spins. In certain magnetic materials the interactions between neighbouring spins favour a slight twist, and skyrmions of a few nanometres to micrometres form and can be pushed around by small electric currents. This is what the Nature article below is about.

In nuclear physics, Tony Skyrme proposed in the early 1960s that protons and neutrons are twisted lumps of this kind in the pion field, now in three dimensions instead of two. The integer that counts the wrapping is then identified with the number of baryons. The surprising part, quoted below, is that a lump made of a field of spin-zero particles can behave as a spin-1/2 fermion after quantisation.

Elusive spin textures discovered in Nature

Concrete

The thing about Skyrmions [I] that is surely hardest to understand is how a lump-like solution (soliton) of a classical scalar field theory can, and in some cases even must, be quantized as a fermion. How can you add integers together and get a half left over?

Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison

Abstract

Topological charge. A skyrmion is a soliton whose stability comes from a map between spheres. For the Skyrme model of baryons the field is $U(x) \in SU(2)$ with the boundary condition $U \to \mathbb{1}$ as $|\vec x| \to \infty$ . This compactifies space to $S^3$ , so a static field is a map $U: S^3 \to SU(2) \cong S^3$ . Finite energy configurations fall into classes labelled by $\pi_3(S^3) = \mathbb{Z}.$ With $L_\mu = U^\dagger \partial_\mu U$ (an $\mathfrak{su}(2)$ -valued current), the degree of the map is $B = \frac{1}{24\pi^2} \int d^3x\, \epsilon^{ijk} \operatorname{Tr}\left(L_i L_j L_k\right),$ up to an overall sign that depends on orientation conventions. It is the integral of the time component of the identically conserved current $B^\mu = \frac{1}{24\pi^2}\epsilon^{\mu\nu\rho\sigma}\operatorname{Tr}(L_\nu L_\rho L_\sigma)$ , and Skyrme identified it with baryon number.

Lagrangian and Derrick's theorem. With metric signature $(+,-,-,-)$ the Skyrme Lagrangian is $\mathcal{L} = \frac{f_\pi^2}{4}\operatorname{Tr}\left(\partial_\mu U \partial^\mu U^\dagger\right) + \frac{1}{32 e^2}\operatorname{Tr}\left([L_\mu, L_\nu][L^\mu, L^\nu]\right),$ with $f_\pi \approx 93\,\text{MeV}$ in this normalisation and $e$ a dimensionless constant. The first term is the nonlinear sigma model for pions. It alone admits no stable static solitons in three dimensions: under a rescaling $x \to \lambda x$ its static energy scales as $\lambda$ , so any configuration lowers its energy by shrinking (Derrick's theorem). The four-derivative Skyrme term scales as $\lambda^{-1}$ and balances it, fixing a size of order $1/(e f_\pi)$ . The static energy obeys the Faddeev-Bogomolny bound $E \geq 6\pi^2 \frac{f_\pi}{e}\,|B|,$ which is not saturated in flat space. The $B=1$ minimum is the hedgehog $U = \exp\left(i F(r)\, \hat x \cdot \vec\tau\right)$ with $F(0) = \pi$ , $F(\infty) = 0$ and $\vec\tau$ the Pauli matrices.

Spin and statistics. Quantising the collective coordinates (position and an $SU(2)$ orientation) gives states of equal spin and isospin. Whether they are bosons or fermions is not fixed by the classical theory, because the configuration space of the $B=1$ sector has $\pi_1 = \mathbb{Z}_2$ and both choices of sign for the wavefunction under the noncontractible loop (a $2\pi$ rotation) are allowed. Witten showed that including the Wess-Zumino term required by QCD anomalies selects the fermionic choice when the number of colours $N_c$ is odd. This answers the question in the quote above.

Baby skyrmions and magnetic skyrmions. In two spatial dimensions the field is a unit vector $\vec n: \mathbb{R}^2 \to S^2$ with $\vec n \to \vec n_\infty$ at infinity, and the charge is the degree in $\pi_2(S^2) = \mathbb{Z}$ , $Q = \frac{1}{4\pi}\int d^2x\; \vec n \cdot \left(\partial_x \vec n \times \partial_y \vec n\right).$ In the pure $O(3)$ sigma model the static energy is scale invariant in two dimensions, so solutions have no preferred size. In chiral magnets the Dzyaloshinskii-Moriya interaction, which is linear in gradients, together with an external field or anisotropy, selects a finite size.

History

Tony Hilton Royle Skyrme was born in England in 1922 in the house of his maternal grandparents, c2~ His maternal great-grandfather Edward Robert knew and admired Kelvin, and was associated with the construc- tion of the Tidal Predictor under the direction of Kelvin and Tait. This machine was for predicting tides worldwide. The admiration led to his naming his son Herbert William Thomson Roberts. This machine was in the house where Tony was born. Tony has said in a speech(3~that he was greatly impressed by the ingenuity of its mechanism. Tony grew up in a world beset with increasing turbulence. In 1943, after Cambridge, he joined the British war effort in making the atomic bomb. It was only in 1946 that he began fundamental research. During 1946-61, he was associated with Cambridge, Birmingham, and Harwell and was engaged in wide-ranging investigations in nuclear physics. It was this work, especially the work on nuclear matter and the fluid drop model which eventually culminated in his beautiful proposition that nucleons are solitons made of pions. In the speech of Tony I mentioned above, he has described the reasons behind his extraordinary suggestions. He knew of Kelvin from the Tidal Predictor, and was vaguely aware of Kelvin's vortex atoms. Like Kelvin, he too desired a model of the nucleon which was visualizable and extended. He felt that fermions can emerge from self-interacting Bose fields just as bosons arise as bound states of fermions. Moved by these imprecise and intuitive desires, Tony began his work on nonlinear field theories, the sine- Gordon equation, and the chiral model, and was led to the proposition that nucleons are twisted topological lumps of pion fields. 16~ In his papers, Tony also had initiated ideas on bosonization, vertex opertors, and quantum theories on multiply connected spaces, all years and years ahead of his time, and all topics of central interest today. David Finkelstein had a grasp of topology and differential geometry which was exceptional for physicists in the 60's. Like Skyrme, he had under- stood that solitons can acquire spin 1/2 from the topology of the configura- tion space. He must have realized what little role relativistic quantum field theory played in the theory of solitons, and struck by the fact that all existing proofs of the spin-statistics theorem relied on relativistic quantum field theory. But solitons can acquire spin 1/2 in nonrelativistic models and can- not always be described by relativistic quantum fields. He and Rubinstein, I suppose, were led by such thoughts to seek and find an alternative proof of the spin-statistics theorem. Their proof was published in 1968.~71It is this proof which is important for chiral solitons. There are grounds to expect that it is the Finkelstein-Rubinstein approach which will be found signifi- cant in quantum gravity as well. An absolutely fundamental result, namely the spin-statistics theorem, is getting topologized. Even more striking, it is still not properly understood. TOPOLOGY IN PHYSICS - A PERSPECTIVE by A.P. Balachandran

Skyrmions

Intuitive

Concrete

Abstract

History

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