Defining and extracting crystalline-symmetry topological invariants in fractionalized topological orders

Develop general procedures to define and extract topological invariants that arise from crystalline symmetry in fractionalized topologically ordered phases with anyons, ensuring applicability beyond special cases and model systems to robustly capture symmetry fractionalization patterns and defect responses.

Background

Crystalline symmetry enriches topological phases by enabling symmetry fractionalization of anyons and quantized responses to lattice defects. While extensive theoretical frameworks exist for symmetry-enriched topological phases, determining experimentally and numerically accessible invariants that fully encode crystalline symmetry data remains challenging, especially in phases with intrinsic topological order.

The paper proposes extracting crystalline invariants from ground-state expectation values of partial rotations centered at high-symmetry points, develops a conformal field theory and G-crossed braided tensor category analysis, and validates predictions via Monte Carlo studies of projected parton wave functions for fractional Chern insulators. Despite these advances, the authors emphasize that, in general, defining and extracting crystalline-symmetry-induced topological invariants—particularly in fractionalized phases with anyons—remains an important open direction.

References

Despite significant progress over the last several decades, there are still important open questions about how to define and extract topological invariants that arise due to crystalline symmetry, particularly in fractionalized topologically ordered phases with anyons.

— Crystalline invariants of fractional Chern insulators  (2405.17431 - Kobayashi et al., 2024) in Introduction, paragraph 1 (page 1)

If several charge sectors contain equally admissible unitary candidates after the global check, the local classification is unresolved and requires explicit global-symmetry factorization or an independent ground-state projective-representation calculation.

— Krylov Edge Spectroscopy of Symmetry-Protected Topological Phases  (2609.10676 - Menzler et al., 9 Sep 2026) in Section 10.1, subsection “Final decision rule for the clock endpoint search”

This failure suggests an unreliable readout of the diagonal translation fractionalization phases $e{i\theta_\alpha}\approx e{-i2\pi/3}$ for $\alpha=1,2,3$ in~\cref{eq:translation lift ambiguity}, and may reflect an emergent nontrivial stabilizer of the optimized fusion tensor that is not captured by the minimal fitting scheme $\mathsf{GG}=\mathrm{GL}(N;\mathbb C)m\rtimes S_m$. A more systematic study involving projective symmetry group (PSG) analysis and parton band-structure calculations is therefore needed, and we leave it for future work.

— HyperDet Wavefunction: A Phase-Agnostic Ansatz for Strongly Correlated Systems  (2609.04146 - Hu et al., 3 Sep 2026) in Section 4.4, subsection “Translation Fractionalization”