Collaborative Research: Spectral Theory and Edge States in Topological Insulators

About This Grant

Topological insulators are a class of materials that are insulators in their bulk and that nevertheless allow wave propagation along their edge. Such materials are at the forefront of the rapidly developing industry of low-energy consumption electronics technologies as they yield a fundamentally new type of electronic transport with ultra-low resistance. Mathematically, such transport is modeled by wave functions corresponding to the eigenvalues appearing, after introduction of a boundary, in a gap of the spectrum of the material without boundary. The principal goal of the project is to develop mathematical methods for measuring the number of such states based on readily computable information about the physical models of the corresponding material. The principal investigators investigate discrete boundary value problems for the tight-binding Hamiltonians describing electronic transport in topological insulators. One of the main themes is the derivation of explicit formulas for the number of eigenvalues in the gaps of the essential spectrum, which correspond to edge states localizing along the boundary. The formulas are expressed through mathematical tools stemming from symplectic geometry, such as the Maslov index and its discretization, the Duistermaat index, with the latter being particularly amenable to numerical computation. In addition to eigenvalues in the gaps, the new formulas give access to the integrated density of states (in the bands of the continuous spectrum) and shed new light on the bulk-boundary correspondence by connecting the gap indices with the band indices. The discrete boundary value problems are described using the theory of boundary triplets of non-densely defined symmetric operators. 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.