Pinning Down Quantum Particles: Electrons, Excitons, and Anyons in Real Space

Date: Nov. 12, 2026
Speaker: Dr. Wladimir Benalcazar
Affiliation: Emory University

Abstract:

Where the electrons sit in a material determines a great deal, for example, how a crystal electrically polarizes, and whether it belongs to one of the topological phases discovered over the past two decades. Yet quantum mechanics describes electrons in crystals as waves spread over the whole sample, and asking where they are turns out to be subtle. In one dimension, the question has a complete answer. In two and three dimensions, it does not, because the position of an electron along different directions obeys an uncertainty principle of its own once the electron is confined to a set of energy bands. I will introduce the ideas needed to appreciate this problem (energy bands, Wannier functions, Berry phases, and topological insulators) assuming no prior background. I will then describe the Spatial Localizer, a tool we developed that finds where electrons are, and the most compact states they can form, by solving a single eigenvalue problem. Finally, I will show how the same idea locates more exotic particles: excitons, which carry an internal electric dipole, and anyons, the fractionally charged excitations of fractional quantum Hall liquids.