Quantization of fractional corner charge in Cn-symmetric higher-order topological crystalline insulators

In the presence of crystalline symmetries, certain topological insulators\npresent a filling anomaly: a mismatch between the number of electrons in an\nenergy band and the number of electrons required for charge neutrality. In this\npaper, we show that a filling anomaly can arise when corners are introduced in\n$C_n$-symmetric crystalline insulators with vanishing polarization, having as\nconsequence the existence of corner-localized charges quantized in multiples of\n$\\frac{e}{n}$. We characterize the existence of this charge systematically and\nbuild topological indices that relate the symmetry representations of the\noccupied energy bands of a crystal to the quanta of fractional charge robustly\nlocalized at its corners. When an additional chiral symmetry is present,\n$\\frac{e}{2}$ corner charges are accompanied by zero-energy corner-localized\nstates. We show the application of our indices in a number of atomic and\nfragile topological insulators and discuss the role of fractional charges bound\nto disclinations as bulk probes for these crystalline phases.\n