Manifestation and classification of anyonic topological order in FQAH systems

Determine how anyonic topological order manifests in crystalline two-dimensional fractional Chern insulators exhibiting the fractional quantum anomalous Hall (FQAH) effect, and classify the resulting anyonic topological quantum states so as to guide experimental detection.

Background

The paper discusses recent experimental observations of fractional Chern insulators (FCIs) exhibiting the fractional quantum anomalous Hall (FQAH) effect and notes ongoing efforts to understand their anyonic topological order, particularly in the presence of crystalline symmetries. Despite these developments, the authors emphasize that there has been no theoretical prediction for how anyonic topological order would manifest in FQAH systems, leaving a gap in guidance for experimental detection.

To address this, the authors argue that topological order should be reflected in adiabatic monodromy of gapped ground states over the space of band topologies, and they develop a framework based on equivariant homotopy theory and covariantization by diffeomorphisms. The stated open problem motivates their subsequent formalism and computations of fundamental groups of the relevant covariantized mapping spaces.

References

However, there has been no theoretical prediction (in fact no consideration, it appears) for how anyonic topological order could actually manifest in FQAH systems, and hence no guide to experiment for how to go about detecting it. In order to address this open problem, we now step back and revisit the general question: What is the manifestation of topological quantum order, generally?

— Fragile Topological Phases and Topological Order of 2D Crystalline Chern Insulators (2512.24709 - Sati et al., 31 Dec 2025) in Section 2.2 (Topological Order and Monodromy), General Topological Order