Extracting the universal and non-universal data in integer and fractional Chern insulators

Edge states of chiral topologically ordered phases are commonly described by chiral Luttinger liquids, an effective theory that is exact only in the conformal limit; in crystalline systems, deviations from simple power-law scaling of correlators generally emerge. Motivated by recent bulk observations of fractional Chern insulators in two-dimensional materials and by synthetic realizations in ultracold atoms, we revisit this framework on lattices and quantify departures from the fractional quantum Hall case arising from lattice geometry and finite system size. Using a combination of analytical arguments and numerics, including time-dependent density-matrix renormalization group (tDMRG), we cleanly separate universal and non-universal edge information: from short-time dynamics and correlation functions, we extract the anomalous boundary exponent, which tracks the bulk filling factor, and independently determine the non-universal edge velocity and associated energy scales. Applied across integer and fractional Chern bands with realistic Berry-curvature inhomogeneity, our procedure provides stable estimators that connect edge responses to bulk topology beyond the flat-band limit. We outline experimental probes in excitonic FCIs and in ultracold atom systems, including time-resolved edge spectroscopy, which directly access the predicted exponents and velocities.