Topological defects in multicomponent Ginzburg-Landau theory

When certain solid materials (for example, tin) are cooled down to very low temperatures, the electrons they contain start to behave not as individual, independent particles, but as a collective, collaborative entity, a kind of gas of electron pairs. This allows them to move without friction so that electrical currents can pass through the material with absolutely no energy loss. This phenomenon, called superconductivity, has immense technological potential, already partially exploited (most medical MRI scanners use superconducting magnets nowadays, for example). A major barrier to further exploitation is the very low temperatures at which superconductivity typically occurs (around -270 degrees C), which require refrigeration with liquid helium. Since mid 2001 complex materials have been engineered which exhibit superconductivity at relatively high temperatures and have several different inter-pervading collaborative electron "gases". Whereas the underlying mechanism for conventional low temperature superconductivity is well understood, the basis of superconductivity in these newer multiband materials is, so far, relatively mysterious.

The aim of this project is to make a thorough mathematical study of a class of models of multiband superconductors called multicomponent Ginzburg-Landau models. The precise mathematical structure of the model is determined by underlying assumptions about the electron pairing mechanisms which lead to superconductivity. These models possess mathematically interesting solutions called "topological solitons", smooth spatially localized lumps of energy which cannot be dissipated by any continuous deformation of the system. The idea is to determine how the properties (the most important property being existence and stability) of these solitons depend on the mathematical structure of the model. The absence, presence and characteristics of these solitons in real superconductors can then be used to infer information about the electron pairing mechanisms underlying superconductivity in these materials.

Superconductivity has immense technological potential, particularly in the fields of medical imaging, transport, and scientific instrumentation. A major barrier to exploiting this potential is the very low temperature at which superconductivity typically occurs -- hence the surge of excitement in the mid 1980's when high T_c superconductors were first developed. Currently the most promising materials are iron-based superconductors, which have multiband structure. In order to exploit these materials well, it is important that the (so far mysterious) mechanisms underlying their behaviour are properly understood. The proposed research will contribute to this goal by clarifying the mathematical predictions of the mathematical systems commonly employed to model several important classes of high T_c superconductors (including MgB_2, Sr_2RuO_4 and various iron pnictides), particularly the phenomenology of the various topological solitons supported by such models. This will give indirect evidence that can be used to support, or rule out, the various electron pairing mechanisms currently considered feasible for these materials.

It should be emphasized that the proposal concerns basic research at the interface of mathematics and theoretical condensed matter physics. While the potential non-academic beneficiaries are numerous and diverse (manufacturers of superconducting devices and, more broadly, any scientific or technological discipline relying on such devices), the timescale involved is likely to be long.