$Z_4$ Topological Superconductivity in UCoGe

Abstract: Topological nonsymmorphic crystalline superconductivity (TNCS) is an intriguing phase of matter, offering a platform to study the interplay between topology, superconductivity, and nonsymmorphic crystalline symmetries. Interestingly, some of TNCS are classified into $Z_4$ topological phases, which have unique surface states referred to as a M\"obius strip or an hourglass, and have not been achieved in symmorphic superconductors. However, material realization of $Z_4$ TNCS has never been known, to the best of our knowledge. Here we propose that the paramagnetic superconducting phase of UCoGe under pressure is a promising candidate of $Z_4$-nontrivial TNCS enriched by glide symmetry. We evaluate $Z_4$ invariants of UCoGe by deriving the formulas relating $Z_4$ invariants to the topology of Fermi surfaces. Applying the formulas and previous ab-initio calculations, we clarify that three odd-parity representations, out of four, are $Z_4$-nontrivial TNCS, while the other is also $Z_2$-nontrivial TNCS. We also discuss possible $Z_4$ TNCS in CrAs and related materials.