Superconductivity in the presence of strong correlations and spin-orbit interactions

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Much of what we understand about simple metals and semiconductors is based on a single-particle picture. Once we determine what one electron does, then any macroscopic current or response is determined by this single-particle behaviour, with slight modifications to account for the statistics of the many electrons. In recent decades, however, researchers have discovered more and more materials that support conduction --- that is, they show metallic behaviour, and turn out to have extremely interesting properties. By way of example, a class of the high temperature cuprate materials, epitomized by La2CuO4, are not conducting, but have fascinating magnetic behaviour. When doped, they are conductors, albeit poor ones, but eventually, at sufficiently low temperatures, they superconduct. The "low temperatures" at which this is achieved turn out to be much higher than in any previously discovered superconductor. Superconductivity is an example of a highly correlated state that cannot be understood in terms of single particles. Moreover, in these materials, even the normal state cannot be understood in terms of single-particle behaviour. There are now many families of materials where this type of scenario is played out; there are sufficiently different properties from family to family that family-specific explanations have been offered for the "glue" that gives rise to the correlations in each family. In this proposal we want to explore possible more universal origins of some of the peculiar behaviour in these materials. Our focus will be spin-orbit coupling, which is normally thought of as a single particle property, one in which the intrinsic angular momentum of an electron (i.e. its "spin") affects the motion of the electron and vice-versa. The usual theory of superconductivity treats spin and orbital motion separately, although many modifications have been made in the last 15 years because certain superconductors that consist of structures that lack inversion symmetry make this separation impossible. All superconductors, however, have some degree of spin-orbit coupling, and in this proposal we will revisit the theoretical formulation of superconductivity with spin-orbit coupling, particularly near the surface of the material, which is precisely where many experiments probe for superconducting properties. At the same time we will explore the nature of the superconducting state that arises not because of some "glue", but because of a more generic mechanism driven by quantum matter's tendency to always expand, i.e. through kinetic energy lowering. A deeper understanding of the driving mechanism will aid material scientists in their quest to develop materials that superconduct at higher temperatures, and ultimately make superconducting applications cheaper and more accessible.