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

TL;DR

This paper uses equivariant homotopy theory to classify fragile topological phases and topological order in 2D crystalline Chern insulators, revealing new mathematical invariants and implications for quantum computing.

Contribution

It introduces a novel application of equivariant 2-Cohomotopy to classify topological phases and order in crystalline insulators, a method not previously used in condensed matter physics.

Findings

Classified fragile topological phases using equivariant homotopy theory.

Computed adiabatic monodromy for non-abelian band topology groups.

Showed potential FQAH anyons are localized in momentum space.

Abstract

We apply methods of equivariant homotopy theory, which may not previously have found due attention in condensed matter physics, to classify first the fragile/unstable topological phases of 2D crystalline Chern insulator materials, and second the possible 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 space of these band topologies, we compute the latter in examples where this group is non-abelian, showing that any potential FQAH anyons must be localized in momentum space. We close with an outlook on the relevance for the search for topological quantum computing hardware. Mathematical details are spelled out in a supplement.