Translation-invariant topological superconductors on a lattice

Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. In this paper we introduce four Z 2 topological indices ␨ k =0,1 at k = ͑0,0͒, ͑0,␲͒, ͑␲ ,0͒, and ͑␲ , ␲͒ 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 ␨ k is the number of electrons on k = ͑0,0͒, ͑0,␲͒, ͑␲ ,0͒, or ͑␲ , ␲͒ orbitals ͑mod 2͒ in the ground state. For three-dimensional superconducting states with only translation symmetry, topological indices give rise to 256 different types of topological superconductors.