Topological solitons

In classical physics, a soliton is a solution to a nonlinear partial differential equation that behaves like a delocalised (or wavelike) particle. Often, such equations also admit multisoliton solutions, which can be interpreted as a superposition of a certain number of solitons, and in some cases even as a combination of solitons and antisolitons. The existence of such superposition phenomena is a nontrivial feature of certain nonlinear equations. The set of rules describing nonlinear superposition is particular to a given equation, so there is no systematic way of studying (multi)solitons. This is in contrast with the principle of linear superposition, according to which the solutions to any linear equation form an affine vector space.

Originally, the word "soliton" was used to describe certain travelling bump solutions of the Korteweg-de Vries (KdV) equation, which models how water waves propagate in one dimension beyond the linear approximation. KdV solitons are very real and were vividly described by the engineer John Scott Russell, who was able to follow one on horseback for over a mile along the Union Canal near Edinburgh in 1834. The stability of such bump waves is due to the existence of a large number of conserved quantities in the KdV dynamical system; this feature is one aspect of the general concept of integrability. The structure of integrable systems can be very intricate, and it may manifest itself through striking properties of the solutions. For example, the speed of KdV solitons turns out to be determined by their height and width; in an overtaking process, there is a time shift effect at superposition, whereby the faster soliton advances and the slower soliton retards its march.

Topological solitons were so named in analogy with the solutions of the KdV equation: they are essentially also localised, superpose nonlinearly and retain individuality in their interactions. They owe their stability to topology rather than integrability --- either because they are maps between noncontractible spaces, or because they must obey nontrivial boundary conditions. Quite generally, the topology of a soliton is specified by one or more integers known as topological charges; mathematically, these could specify a homotopy class or a characteristic class. Topological charges are conserved under continuous dynamics, and this means that a party of solitons must retain the initial overall number as the individuals mingle and move about. The simplest examples of topological solitons are known as kinks, and they live on the real line just as KdV solitons do. Incidentally, equations for kinks such as the sine-Gordon equation are also integrable.

Many examples of topological solitons occur in field theory models in more than one spatial dimension, where the dynamics becomes more interesting. The equations of motion are then no longer integrable, so travelling solutions cannot be constructed explicitly. In theoretical physics, the most interesting examples of solitons occur in gauge field theories, where the equations of motion are invariant under gauge transformations forming an infinite group of internal symmetries; they generalise Maxwell's theory of electromagnetism and are the fundamental building blocks of the standard model of particle physics. A prototypical example of gauge theory solitons are magnetic monopoles, which exist in three-dimensional space. They can be thought of as sources of magnetic field in a theory that looks like Maxwell's from a distance, but has a smooth and much richer (nonabelian) structure at close range. The topological charge of a magnetic monopole can be interpreted as a net magnetic charge. Other examples of gauge theory solitons are vortices in two dimensions, which have been important in applications of field theory to superconductivity, the fractional quantum Hall effect and other systems in condensed matter physics. Skyrmions provide yet another example of topological solitons in a field theory with potentially interesting applications; they occur as classical solutions in a realistic model for the atomic nucleus, where the baryon number is realised as a topological charge.

Although not much hope can be put on finding exact solutions for solitons in more than one dimension, the models of interest are sometimes "self-dual", and this has allowed some progress on the dynamics of topological solitons to be made. Self-duality roughly means that the static field configurations of minimal energy satisfy first-order equations, which are usually more tractable than the time-independent equations of motion. One striking feature of such models is that there is a well-behaved space of all solutions to these first-order equations (modulo gauge transformations) with a given topological charge. This "moduli space" comes equipped with extra structures such as riemannian metrics (which allow one to measure angles and distances, and to define geodesics) or complex structures, so we can do geometry on them. The geometry of the moduli space encodes physical information about the system of solitons. For instance, geodesic arcs on the moduli space can be interpreted as approximating scattering processes in the true dynamics at low speed. As a concrete example, the following process illustrates slow scattering of topological solitons called lumps, propagating on an infinite cylinder:
(1) -> (2) -> (3)
Each picture is a snapshot of a soliton, displayed as a radial graph of its energy density, and corresponds to a different point on the same geodesic of the moduli space. The fact that the energy density at crossing has circular symmetry is just an example of how nonlinear superposition may give rise to surprising effects. For lumps, the first-order self-duality equations are just the well-known Cauchy-Riemann equations defining holomorphic maps. In gauge theories, the relevant first-order equations are often dimensional reductions of the self-duality equations defining instantons in Yang-Mills theory on four-manifolds, on which Donaldson's theory is based. The equations for instantons on euclidean space are integrable and, even more surprisingly, most integrable systems known can be obtained from them by dimensional reduction: the self-duality equation for magnetic monopoles and the sine-Gordon equation provide two examples. The geometry of the moduli spaces has been instrumental in approaching the problem of making sense of topological solitons at the quantum level, and in testing discrete symmetries known as "dualities" that one encounters in quantum field theories.

Further (technical) reading:

[1] M. Dunajski: Solitons, Instantons and Twistors, Oxford University Press, 2010
[2] N. Manton, P. Sutcliffe: Topological Solitons, Cambridge University Press, 2004

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