Topological crystalline superconductors with linearly and projectively represented $C_{n}$ symmetry

We study superconductors with $n$-fold rotational invariance both in the presence and in the absence of spin-orbit interactions. More specifically, we classify the non-interacting Hamiltonians by defining a series of $Z$-numbers for the Bogoliubov-de Gennes (BdG) symmetry classes of the Altland-Zimbauer classification of random matrices in $1$D, $2$D, and $3$D in the presence of discrete rotational invariance. Our analysis emphasizes the important role played by the angular momentum of the Cooper pairs in the system: for pairings of nonzero angular momentum, the rotation symmetry may be represented projectively, and a projective representation of rotation symmetry may have anomalous properties, including the anti-commutation with the time-reversal symmetry. In 1D and 3D, we show how an $n$-fold axis enhances the topological classification and give additional topological numbers; in 2D, we establish a relation between the Chern number (in class D and CI) and the eigenvalues of rotation symmetry at high-symmetry points. For each nontrivial class in 3D, we write down a minimal effective theory for the surface Majorana states.