We apply modular flow—entanglement generated dynamics—to characterize quantum orders of ground state wave functions. In particular, we study the linear response of the entanglement entropy of a simply connected region with respect to modular flow. First, we apply it to $(1+1)\mathrm{D}$ conformal field theories and demonstrate its relationship to the chiral central charge—or, equivalently, the perturbative gravitational anomaly—which is shown to vanish. Next, we apply it to $(2+1)\mathrm{D}$ gapped ground states where it reduces to a recently proposed formula by Kim et al. that is conjectured to compute the edge chiral central charge. Modular flow provides an intuitive picture for this conjecture based on bulk-edge correspondence. We also provide numerics on free-fermion models that lend support to our picture.

• Received 2 August 2022
• Accepted 28 November 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.260403