Topological crystalline superconductors in two dimensions

We propose a general framework to classify the topological edge modes in two dimensional topological crystalline superconductors with time-reversal symmetry, which are protected by different crystalline symmetries. The realization of topological crystalline phases is investigated in the superconductors and a criterion for distinguishing topological phases in crystalline superconductor both with mirror and mirror-inversion symmetry is founded. In these topological phases, Majorana edge modes can be identified along the mirror invariant line, while topological edge modes in the AIII class exist along the mirror-inversion invariant line. Furthermore, for the time-reversal invariant superconductor when only the mirror-inversion symmetry is preserving, topological edge modes in the AIII class with nearly flat band emerge in the mirror-inversion parity subspace.