Alexander Watson, from the University of Minnesota, will present a talk at an online Applied Mathematics Colloquium.

Title: "Existence and computation of exponentially localized Wannier functions for non-periodic materials"

Abstract: When modeling the electronic properties of periodic (crystalline) materials, the electronic Hamiltonian is diagonal in the basis of Bloch functions: plane wave-like functions which extend over the entire material. Taking the Fourier transform of the set of Bloch functions with respect to wave-number yields an alternative basis consisting of functions which decay rapidly, known as Wannier functions. The transformation from Bloch functions to Wannier functions is non-unique, and can be tricky to implement effectively, because it depends on the choice of Bloch function gauge, with more regular gauges yielding more rapidly decaying Wannier functions. Despite this difficulty, Wannier functions play a crucial role in modeling materials’ electronic properties. To give one important example, computing Wannier functions is the first step to computing accurate tight-binding models: discrete approximations to continuum models which are ubiquitous in condensed matter physics. Thouless observed in 1984 that the existence of Wannier functions which decay exponentially fast can be topologically obstructed. This fact is now central to our understanding of the quantum Hall effect and topological insulators. I will recap the rich theory of Wannier functions of periodic materials, and then present recent work joint with Kevin Stubbs and Jianfeng Lu (Duke) which extends this theory to non-periodic materials. Our work generalizes groundbreaking work of Kivelson and Nenciu-Nenciu on one-dimensional non-periodic materials to two dimensions and higher.

Bio: After earning my PhD in Applied Math at APAM with Michael Weinstein, I spent three years at Duke University as William E. Elliott Assistant Research Professor in the mathematics department, working primarily with Jianfeng Lu. I am currently Postdoctoral Associate at the University of Minnesota Twin Cities working with Mitch Luskin.

Host: Michael Weinstein

For the Zoom link, please send an email to [email protected].