Fragile Topological Phases and Topological Order of 2D Crystalline Chern Insulators

Abstract: We apply methods of equivariant homotopy theory, that may not previously have found due attention in condensed matter physics, to classify first the fragile topological phases of 2D crystalline Chern insulator materials, and second the potential topological order of their fractional cousins. We highlight that the phases are given by the equivariant 2-Cohomotopy of the Brillouin torus of crystal momenta (with respect to wallpaper point group actions) — which, despite the attention devoted to crystalline Chern insulators, seems not to have been considered before. Arguing then that any topological order must be reflected in the adiabatic monodromy of gapped quantum ground states over the covariantized moduli space of these band topologies, we compute the latter in various examples where this group is non-abelian, should that potential anyons must be localized in momentum space. We close with an outlook on the relevance for the search for topological quantum computing hardware.

algebraic topologycategory theory

Audience: researchers in the topic

Bilkent Topology Seminar

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Recordings of talks available at www.youtube.com/channel/UCLrmyGpqxyeVpTcA1b5HcMw/videos