Classification of Topological Insulators

This project explores the mathematical structure underlying a class of materials known as topological insulators—substances that conduct electricity on the surface but not in the interior, with behavior that is both stable and robust due to deep topological principles. By investigating how mathematical concepts such as topology, operator theory, and quantum mechanics intersect in these systems, the research contributes to fundamental scientific understanding and has practical implications for quantum computing and nanotechnology. The project supports NSF’s mission by advancing science and engineering knowledge that could influence the design of novel electronic materials, foster interdisciplinary collaboration, and support the education of the next generation of researchers. A key societal benefit includes training a graduate student in a cutting-edge area at the intersection of physics and mathematics, thereby contributing to workforce development in STEM fields.

The investigator studies the classification of disordered quantum insulators by examining the topological structure of the space of quantum-mechanical Hamiltonians satisfying physical constraints such as locality, invertibility (the insulating condition), and symmetry. The primary mathematical challenge is to compute the path-components (π₀) of these spaces and understand how they relate to known topological invariants such as the non-commutative Chern number. Unlike previous approaches that assume translation invariance and rely on vector bundle methods, this project considers disordered and strongly disordered systems, where traditional K-theory methods are no longer adequate. The work is organized into a three-step program: first, the study of gapped disordered systems; second, systems with a mobility gap, replacing spectral constraints with dynamical ones; and third, systems involving interactions. The outcome will refine the mathematical framework for topological phases of matter, improve our understanding of the stability of physical systems, and provide tools with implications for both theoretical physics and future technologies.

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.

Award Number: 2510207
Principal Investigator: Jacob Shapiro
Funds Obligated: $149,000
State: NJ