Identifying anyonic topological order in fractional quantum anomalous Hall systems

Abstract

Recently observed fractional quantum anomalous Hall (FQAH) materials are candidates for topological quantum hardware, but their required anyon states are elusive. We point out dependence on monodromy of the fragile band topology in 2-cohomotopy. An algebro-topological theorem of L. Larmore and E. Thomas, [Math. Scand. 47, 232 (1980)], identifies FQAH anyons over momentum space. Admissible braiding phases involve 2C-th roots of unity, with C the Chern number. This lays the foundation for understanding symmetry-protected topological order in FQAH systems, reducing the problem to computations in equivariant cohomotopy.