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.