Fermi-surface diagnosis for topological superconductivity with s-wave-like pairing symmetries

Theoretical prediction of topological superconductivity is key to their discovery. Recently, it is proven that in 199 out of 230 space groups, topological superconductivity coexists with an s-wave-like pairing symmetry, raising the hope of finding more candidates for this exotic phase. However, a comprehensive and efficient method for diagnosing topological superconductivity in realistic materials remains elusive. Here, we derive Fermi-surface formulas for gapped and gapless topological phases of time-reversal symmetric superconductors with s-wave-like pairing symmetries in all layer and space groups, applicable to thin-film and bulk materials. Our diagnosis uses only the sign of the pairing and the Fermi velocity at several Fermi points, and yields complete (partial) diagnosis for gapped and gapless superconducting phases in 159 (71) space groups. This provides a fundamental basis for the first-principles prediction of new topological superconductors. The authors derive formulas for topological invariants in time-reversal symmetric spinful superconductors invariant under all symmetry operations for a given space group. The formulas require only the sign of the intra-band pairing and sign of the Fermi velocity at several points on the Fermi surface.