Mathematical Challenges in BCS Theory of Superconductivity

The superconducting phase transition is an important topic in condensed matter physics and material science, and progress in this field has been recognized by five Nobel prizes (1913, 1972, 1973, 1987 and 2003). Superconductivity is the phenomenon that certain materials completely lose their electrical resistance below a critical temperature. There are two mathematical descriptions of superconductivity, namely a microscopic one due to Bardeen, Cooper and Schrieffer (BCS) and a macroscopic one due to Ginzburg and Landau (GL). Our project will be centered around two questions. First, we attempt to understand whether there is a regime in which the microscopic BCS model can be rigorously derived from an underlying quantum many-body system. This is a long-standing problem and its solution is one of the long-term goals of our project. Second, we will study the relation between the microscopic BCS and the macroscopic GL model. The PIs were the first to put this relation on a firm mathematical footing, but several physically important questions still remain open. One of them concerns the inclusion of a self-generated magnetic field and thus ultimately the justification of the Meissner effect. Furthermore, the question of the effective boundary conditions on the macroscopic scale arises, which is a topic that has recently been taken up again in the physics literature.