Schreiber Crystalline Chern Order

An article that we are finalizing at CQTS:

---

• Hisham Sati$\;\;\;$and$\;\;\;$ Urs Schreiber:

  
  

  Fragile Topological Phases and Topological Order

  of 2D Crystalline Chern Insulators

  • main file:

    • pdf (v2, refs added)

    • arXiv:2512.24709

  • supplement draft: pdf (equivariant 2-cohomotopy of 2-tori)

  
  

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/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 moduli space of these band topologies, we compute the latter in various examples where this group is non-abelian, showing that any potential FQAH anyon states 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.

---

Following:

• Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems

Talk presentations:

• at QMATH16

• at Bilkent25

Related articles:

• Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems

• Orientations of Orbi-K-Theory measuring Topological Phases and Brane Charges

  in: Geometric Orbifold Cohomology

• Drinfeld Center as Quantum State Monodromy over Bloch Hamiltonians around Defects