CAREER: Mathematical Aspects of Topological Phases

Developing innovative materials with revolutionary properties is at the forefront of modern research. This project focuses on topological phases of matter, a novel class of materials that conduct electricity along their edges in a way that remains unaffected by defects or noise. This makes them ideal to improve quantum computations - which are notoriously sensitive to noise.

This project aims to stimulate innovations in quantum technology by unlocking a deeper understanding of how electrons behave in these materials. The project investigates how stable currents emerge between topologically distinct phases; compute their profile and their speed; and derive effective equations for their propagation. The educational component of this project seeks to inspire students by emphasizing the vital role of mathematical skills in today's professional landscape.

The Principal Investigator (PI) plans to organize a monthly talk series, "Y Math?", which features industry leaders showcasing how mathematics drives innovation across diverse fields.

The first research track of this project focuses on topological insulators, materials whose Hamiltonians have a spectral gap and a topologically non-trivial Fermi projector. For straight interfaces, the bulk-edge correspondence predicts that the conductance along an interface between insulating phases of matter is equal to the Hall conductance jump across the interface.

The PI seeks to extend this principle to curved interfaces by extracting from the edge conductance an intersection number that accounts for the non-trivial geometry. The physical interpretation is that each connected component of the interface supports a number of edge modes equal to the Hall conductance difference. The PI also intends to compute high-frequency corrections to the bulk-edge correspondence in the continuum - where this principle is sometimes violated.

The second research track of the project focuses on Weyl semimetals, three-dimensional crystals with conical intersections in their band structure. Despite an extensive physics literature, there is no known example of Schrödinger operator with Weyl points. The PI plans to show that Weyl points nonetheless appear generically in the band spectrum of Schrödinger operators in three dimensions and produce explicit examples as parity-breaking perturbations of face-centered and body-centered cubic crystals.

By studying resolvent expansions for adiabatic modulations of these crystals, the PI plans to show that interfaces support topologically protected surface states: three-dimensional analogues of the robust edge states emerging between two-dimensional topological phases. These projects require tools from spectral theory, multiscale analysis and topology that will serve the mathematical community at large.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.