Mathematics of topological insulators
Columbia University, New York City
Daniel Freed, Gian Michele Graf, Rafe Mazzeo, and Michael I Weinstein
This workshop, sponsored by AIM, the Simons Foundation, and the NSF, will consider the role of topology in characterizing materials and in the prediction of their physical properties, particularly for two-dimensional material such as graphene. The focus will be on important mathematical questions at the interface of the analysis and topology in the context of the governing fundamental partial differential equations and other models. Two such questions are the bulk edge correspondence and the existence and robustness of edge states in aperiodic settings.
A revolution in Condensed Matter Physics and Materials Science has been brought about by recognition of the role of topology in characterizing materials and in the prediction of their physical properties. A second revolution is the emergence of the two-dimensional material: a one-atom-thick monolayer that extends in-plane to the macro-scale. At the crossroads of both is graphene, the best-known and most widely studied example. Graphene is a two-dimensional honeycomb arrangement of carbon atoms and is the most conductive known material, both electrically and thermally. Seen through the Schroedinger equation, honeycomb symmetry endows graphene with spectral properties (Floquet-Bloch band structure) which seed robust and physically important phenomena such as quasiparticle-wavepackets that propagate according to the laws of relativistic Fermions and, when time-reversal symmetry is broken, protected edge states. These properties can be understood (the bulk-edge correspondence) in terms of topological invariants (e.g. Chern number) computable from quantities obtained from the spectral theory of the bulk unperturbed material (Berry curvature of Floquet-Bloch bands). In general, the symmetries of the unperturbed bulk structure and the manner in which symmetry are broken give rise to different topological phases with associated wave properties, which are stable against local (even strong) perturbations.
Another approach to classifying topological phases is via a field theory description which applies at large distances and times. In this context homotopy theoretic methods have been brought to bear, and quite general classifications have been developed and successfully applied in condensed matter physics. Despite the already substantial successes obtained by applying the methods of field theory and of quantum mechanics on lattices to this set of problems, a central goal of this workshop is to build bridges between these two approaches.
Further, the recognition that such phenomena can be realized in other physical wave systems, such as photonics, acoustics and mechanics, has led to very significant activity and excitement in the fundamental and applied physics communities catalyzed by the problems of: predicting, explaining, controlling and applying such physical properties of graphene and related materials.
This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for funding to participate in this workshop has passed.
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Additional support for this workshop is provided by the Columbia University Department of Mathematics and Department of Applied Physics and Applied Mathematics.