Topological Markers for C_{n}-symmetric Topological Crystalline Insulators with and without Translation Symmetry

Bulk and surface topological properties of non-interacting topological phases can be diagnosed via symmetry-eigenvalue analysis of Bloch states at high symmetry points in the Brillouin zone, in the presence of crystalline symmetry. This symmetry-based diagnosis has allowed for a simplification of the calculation of topological invariants. However, when open boundaries are present, only the point group part of the symmetry group remains, and it is unclear how to utilize crystalline symmetries to diagnose bulk topology. In this work, we introduce basis-independent topological crystalline markers to characterize bulk topology in C_{n}-symmetric (n=2,3,4,6) topological crystalline insulators with and without translation symmetry. These markers are expressed solely in terms of the trace of the product of the crystalline symmetry operator and the ground state projector. First, we provide a general method for calculating topological markers in periodic systems with an arbitrary number of unit cells in each direction. This is because it is not possible to obtain the symmetry eigenvalues of Bloch states at all high-symmetry points in the Brillouin zone even if periodic boundaries are present; the quantization of the crystalline momentum along each direction spanning the Brillouin zone depends on the unit cell number along each direction spanned by the primitive lattice vector on the real-space lattice, thereby determining the presence of high-symmetry points. Second, we construct explicit mappings from the markers to the Chern number, bulk polarization, and sector charge for two-dimensional $C_{n}$-symmetric insulators in symmetry classes A, AI, AII, and D. Finally, we demonstrate how to numerically calculate the marker in finite-size systems with open boundaries, and how to diagnose the bulk topology from the marker.