Abstract

By endowing the Hilbert space with a metric and a curvature, the modern theory of solids resorts to tools from differential geometry and topology to analyze the physical properties of electrons in a crystal. After introducing the concept of the quantum geometric tensor, I will explore the implications of the quantum geometry to flat bands, where the quasiparticles have zero group velocity. I will then address the possibility of using pumped light in flat Chern bands to create out-of-equilibrium excitons with finite vorticity in momentum space. Those excitons, called topological excitons, have their vorticity set by the difference between the Chern numbers in the conduction and valence bands. Topological excitons can be found optically through the non-linear Hall effect and can condense into a novel type of topological neutral superfluid with profile wavefunctions in momentum space that carry a finite vorticity.