Physics Theory Seminar with Benoit Doucot on Topological Electrostatics

Two-dimensional electron gases under a strong magnetic field have tremendously expanded our understanding of many-body physics, with the discovery of integer and fractional quantum Hall effects, together with chiral edge states, fractional excitations, anyons. Another striking effect is the strong coupling between charge and spin/valley degrees of freedom, which takes place near integer filling of the magnetic Landau levels. More precisely, because of the large energy gap associated to cyclotron motion, any slow spatial variation of the spin background induces a variation of the electronic density proportional to the topological density of the spin background. Minimizing Coulomb energy leads to an exotic class of two-dimensional crystals, which exhibit a periodic non-collinear spin texture called a Skyrmion lattice. 

I will review the history of these concepts, with an emphasis on the notions of Berry phases and Berry curvature, which play a prominent role in all aspects of topological condensed matter physics. A main feature of these Skyrmion crystals is that their theoretical description involves effective theories with local and possibly non-Abelian gauge symmetries. It is therefore a theoretical challenge to identify the actual physical degrees of freedom in such systems. I will show how this may be achieved using some ideas from complex geometry. The main outcome of such analysis is the existence of two regimes depending on whether the topological charge per unit cell is smaller (unfrustrated case) or larger (frustrated case) than the number of internal states (spin/valley) accessible to electrons.