Real Space Topology

Much of the theory of topologically nontrivial systems has come from the periodic structures studied in the condensed matter physics community, where tools like Bloch's theorem, closed Brillouin zones, and reciprocal space are standard. However, there are innumerable situations (many physical) where periodic boundary conditions do not hold, and the math breaks down, even approximately. Quasicrystals, amorphous and glass phases, grain boundaries, lattice defect sites, and interfacial regions are all examples.

Fortunately, there has been a great effort by a number of physicist and mathematicians (Alexi Kitaev, Raffaello Bianco, Raffaele Resta, and Terry Loring, to name a few) to expand the mathematics of topological insulators to real space, which frees up the possibility to study these non-periodic systems in detail. This project seeks to explore the limits of these and other methods, and to understand how symmetries and topology in real space can profoundly influence the behavior of condensed matter. From a pragmatic side, it also is another step towards creating practical devices out of these exotic materials by relaxing requirements of periodicity and various symmetries.

Associated Publications

1. Z. Xu, X. Kong, R. J. Davis, D. Bisharat,Y. Zhou, X. Yin, and D. F. Sievenpiper, “Topological valley transport under long-range deformations,” Phys. Rev. Research, vol. 2, no. 1, p. 013209, Feb. 2020, doi: 10.1103/PhysRevResearch.2.013209. (arxiv)