Jan Philip Solovej - " Hartree-Fock-Bogolubov and BCS Theory"
In the lectures I will discuss the general mathematical formalism leading to the variational Hartree-Fock-Bogolubov (HFB) model. This requies introducing the notion of quasi-free states, quadratic Hamiltonians (Hamiltonians quadratic in creation and annihilation operators), and Bogolubov transformations. I will focus mainly on fermions, but I will also discuss briefly the case of bosons, which from an abstract level is more complicated. I will introduce the HFB model as the variational theory obtained by restricting to quasi-free (fermionic) states. I will then turn to the special case of translation invariant models and discuss the simplification that leads to the Bardeen-Cooper-Schrieffer (BCS) model used to explain superconductivity. I will discuss the gap equation characterizing the phase transition. Finally, I will give a brief sketch of the recent proof that BCS theory in an appropriate limit gives rise to the Ginzbug-Landau (GL) model of superconductivity. Most of what I will talk about was done in different collaborations with Bach, Frank, Hainzl, Hamza, Lieb, and Seiringer. For the latter part on BCS and GL there is an excellent recent review by Hainzl and Seiringer: http://arxiv.org/abs/1511.01995.
