Basis-Independent Topological Crystalline Markers for Two-Dimensional C_n-symmetric Obstructed Atomic Insulators and Topological Crystalline Superconductors

Topological crystalline insulators and superconductors are topological phases which are protected by spatial symmetries such as reflections and rotations. These phases have been classified using quantities such as momentum-space rotation invariants and symmetry indicators. Previous works have shown that correspondences exist between two quantities: (i) strong topological invariants such as the Chern number, electromagnetic responses such as the bulk polarization, boundary signatures such as the corner charge and (ii) momentum-space rotation invariants at high-symmetry momenta in the Brillouin zone. In this work, we re-express these previously established relationships for 2D C_n-symmetric (n=2,3,4,6) obstructed atomic insulators and topological crystalline superconductors in terms of topological crystalline markers constructed from projected symmetry operators, which do not require a specific choice of basis such as a momentum-space basis. These topological crystalline markers are similar in form to quantities such as the local Chern marker, which is expressed in terms of the commutator of projected position operators. Our results show that quantities such as the bulk polarization, corner charge, and Chern number, depend on multiple topological crystalline markers based on projected symmetry operators over different symmetry centers. Our results provide a new interpretation of existing topological crystalline invariants that has applications to systems where translational symmetry is broken, but the crystalline symmetry is preserved.