This talk illustrates the use of analytic microlocal methods in a problem motivated by condensed matter physics. It concerns the structures of eigenstates corresponding to magic angles. These are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letter by Tarnopolsky--Kruchkov--Vishwanath.
By adapting analytic hypoellipticity results of Kashiwara, Sjoestrand, Trepreau and Himonas, we show that the eigenfunction decay exponentially in geometrically determined regions as the angle of twisting decreases. This is joint work with Michael Hitrik.
The results will be illustrated by colourful numerics (coming from joint works with Simon Becker, Mark Embree and Jens Wittsten). They suggest many open problems.