Topological invariants for the Fermi surface of a time-reversal-invariant superconductor

A time-reversal-invariant (TRI) topological superconductor has a full pairing gap in the bulk and topologically protected gapless states on the surface or at the edge. In this paper, we show that in the weak pairing limit, the topological quantum number of a TRI superconductor can be completely determined by the Fermi-surface properties and is independent of the electronic structure away from the Fermi surface. In three dimensions, the integer topological quantum number in a TRI superconductor is determined by the sign of the pairing order parameter and the first Chern number of the Berry phase gauge field on the Fermi surfaces. In two dimensions and one dimension, the Z(2) topological quantum number of a TRI superconductor is determined simply by the sign of the pairing order parameter on the Fermi surfaces. Our results could directly aid the search for topological superconductors in real materials.