Theory of Odd-Parity Superconductivity: from Gap Function to Topological Invariants and Surface Andreev Bound States

Three-dimensional superconductors with a nodeless pairing gap can be classified by an integer topological invariant, and gapless Andreev bound states exist on the surface when the topological invariant is nontrivial. In this letter we give a general criterion for determining the topological invariant for centrosymmetric superconductors with \emph{odd-parity} pairing. We show that in a general system with spin-orbit coupling, the superconducting gap function can be expressed by a pseudospin d-vector, and the topological invariant can be determined from the total winding number of the pseudospin d-vector on the Fermi surfaces. We also discuss Andreev surface states, and we found gapless surfaces states when the topological invariant in the bulk is topologically nontrivial.