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. In this paper, we analyze these focusing on the paradigmatic example quadrupole insulator with ${C}_{4}$ rotation symmetry, present field theory description mixed geometry-charge responses. Our provides unified corner in terms physical geometry (which encodes disclinations), an effective corners). We extend analysis to interacting systems, predict response fractional insulators, which exhibit charge $e/2(2k+1)$ bound disclinations.