Crystalline Responses for Rotation-Invariant Higher-Order Topological Insulators

Two-dimensional higher-order topological insulators can display a number of exotic phenomena, such as half-integer charges localized at both corners and disclination defects. Here, we analyze these phenomena in the continuum limit, and present a topological field theory description of the mixed geometry-charge responses. Our theory provides a unified description of the corner and disclination charges in terms of a physical geometry (which encodes disclinations), and an effective geometry (which encodes corners). We extend this analysis to interacting systems, and predict the existence of fractional quadrupole insulators, which exhibit charge e/2(2k+1) bound to corners and disclinations.