Topological defects in multicomponent Ginzburg-Landau theory

当某些固体材料(例如锡)冷却到很低的温度时，它们所包含的电子开始不再作为单独的独立粒子，而是作为一个集体、协作实体，一种电子对的气体。这使得它们可以在没有摩擦的情况下移动，因此电流可以通过材料而绝对没有能量损失。这种现象被称为超导性，具有巨大的技术潜力，但已经被部分开发(例如，现在大多数医用核磁共振扫描仪使用超导磁体)。进一步开发的一个主要障碍是超导电性通常发生的非常低的温度(约-270摄氏度)，这需要用液氦制冷。自2001年年中以来，人们设计出了在相对较高的温度下表现出超导电性的复合材料，并具有几种不同的相互渗透的协同电子“气体”。尽管传统低温超导体的基本机制已为人们所熟知，但这些新的多带材料的超导基础到目前为止还相对神秘。这个项目的目的是对一类被称为多组分Ginzburg-Landau模型的多带超导体模型进行彻底的数学研究。该模型的精确数学结构是由关于导致超导的电子配对机制的基本假设决定的。这些模型具有数学上有趣的解，称为“拓扑孤子”，即平滑的空间局域能量块，不能通过系统的任何连续变形来耗散。我们的想法是确定这些孤子的性质(最重要的性质是存在和稳定)如何取决于模型的数学结构。这些孤子在实际超导体中的存在、存在和特征可以用来推断这些材料超导电性背后的电子配对机制的信息。

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.