Fractional disclination charge in two-dimensional C_n−symmetric topological crystalline insulators

Robust fractional charge localized at disclination defects has been recently found as a topological response in C₆ symmetric 2D topological crystalline insulators (TCIs). We thoroughly investigate the fractional charge on disclinations in C_n symmetric TCIs, with or without time-reversal symmetry, and including spinless and spin-1/2 cases. We find the disclination charge is fractionalized in units of e/n for C_n symmetric TCIs; and for spin-1/2 TCIs, with additional time-reversal symmetry, the disclination charge is fractionalized in units of 2e/n. Utilizing band representation theory, we construct topological indices of the fractional disclination charge for all 2D TCIs that admit a (generalized) Wannier representation. Moreover, we use an algebraic technique to generalize the indices for TCIs with non-zero Chern numbers, where a (generalized) Wannier representation is not applicable. Our results provide a quantitative and intuitive understanding of fractional charge at defects for 2D C_n-symmetric TCIs.