Translation-invariant topological superconductors on a lattice

In this paper we introduce four Z(2) topological indices zeta(k)=0,1 at k=(0, 0), (0, pi), (pi, 0), and (pi, pi) characterizing 16 universal classes of two-dimensional superconducting states that have translation symmetry but may break any other symmetries. The 16 classes of superconducting states are distinguished by their even/odd numbers of fermions on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices. As a result, the 16 classes topological superconducting states exist even for interacting systems. For noninteracting systems, we find that zeta(k) is the number of electrons on k=(0, 0), (0, pi), (pi, 0), or (pi, pi) orbitals (mod 2) in the ground state. For three-dimensional superconducting states with only translation symmetry, topological indices give rise to 256 different types of topological superconductors.