Euler-obstructed Cooper pairing: Nodal superconductivity and hinge Majorana zero modes

Since the proposal of monopole Cooper pairing in [Phys. Rev. Lett. 120,\n067003 (2018)], considerable research efforts have been dedicated to the study\nof Cooper pairing order parameters constrained (or obstructed) by the\nnontrivial normal-state band topology at Fermi surfaces in 3D systems. In the\ncurrent work, we generalize the topologically obstructed pairings between Chern\nstates (like the monopole Cooper pairing) by proposing Euler obstructed Cooper\npairing in 3D systems. The Euler obstructed Cooper pairing widely exists\nbetween two Fermi surfaces with nontrivial band topology characterized by\nnonzero Euler numbers; such Fermi surfaces can exist in 3D PT-protected\nspinless-Dirac/nodal-line semimetals with negligible spin-orbit coupling, where\nPT is the space-time inversion symmetry. An Euler obstructed pairing channel\nmust have pairing nodes on the pairing-relevant Fermi surfaces, and the total\nwinding number of the pairing nodes is determined by the sum or difference of\nthe Euler numbers on the Fermi surfaces. In particular, we find that when the\nnormal state is time-reversal invariant and the pairing is weak, a\nsufficiently-dominant Euler obstructed pairing channel with zero total momentum\nleads to nodal superconductivity. If the Fermi surface splitting is small, the\nresultant nodal superconductor hosts hinge Majorana zero modes. The possible\ndominance of the Euler obstructed pairing channel near the superconducting\ntransition and the robustness of the hinge Majorana zero modes against disorder\nare explicitly demonstrated using effective or tight-binding models. Our work\npresents the first class of higher-order nodal superconductivity originating\nfrom the topologically obstructed Cooper pairing.\n