Professor Paul Sutcliffe

* *

Topological solitons

Topological solitons are stable, finite energy, particle-like solutions
of nonlinear wave equations. They arise in a variety of applications
in several areas including particle physics, cosmology and
condensed matter physics.

Below are some examples of 3D topological solitons.

Skyrmions are candidates for a solitonic description of nuclei, where the number of solitons is identified with baryon number. The Skyrme model solitons are only known through numerical computations. However, it is possible to use an approximation in which Skyrmions are constructed from rational maps between Riemann spheres, and this has proved useful in understanding the structure of Skyrmions. The figure displays baryon density isosurfaces for various soliton numbers plus models to help visualize the associated polyhedra. It can be seen that some Skyrmions are very symmetric, and this can be understood in terms of the existence of particularly symmetric rational maps.

Monopoles

Monopoles arise in Yang-Mills-Higgs gauge theories and are solitons that carry magnetic charge. The equations describing static BPS monopoles are integrable and this allows various sophisticated twistor methods to be applied. Monopole dynamics is not an integrable system but for slowly moving monopoles their dynamics can be approximated by geodesic motion on the moduli space of static solutions. The figure shows a particular scattering of three monopoles. Monopoles often resemble Skyrmions and although this is not yet completely understood there are some hints at a connection, as monopoles can also be described by rational maps.

Solitons stabilized by the Hopf invariant (which is a linking number between field lines) arise in the Skyrme-Faddeev model. The figure shows two field lines for each soliton with Hopf charge one to seven. It can be seen that the first few solitons consist of a single loop, which in some cases is twisted, but for higher Hopf charges links and knots appear. The charge seven configuration is a trefoil knot and many other links and knots appear with increasing Hopf charge.