Topological defects in multicomponent Ginzburg-Landau theory

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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.