Topological phases in twisted Rashba superconductors

We study the topological properties of a twisted superconducting bilayer with spin-singlet pairings and Rashba spin-orbital coupling. By introducing the chirality basis, we obtain the effective odd-parity superconductors with the help of spin-orbital coupling. For the twisted bilayer with $d$-wave pairings, two non-Abelian topological phases with Chern numbers $C=\ensuremath{-}1$ and $C=\ensuremath{-}5$ are identified, and the analytical expressions for the boundary of non-Abelian phase are derived as well within the circular Fermi surface approximation. We perform numerical calculations at the twist angle of the moir\'e lattice which further verify the topological phase diagram from the effective odd-parity Hamiltonian. For the bilayer with $d$-wave and ${s}_{\ifmmode\pm\else\textpm\fi{}}$-wave pairings, we reveal the second-order topological superconductor with a Majorana zero mode at each corner by analyzing the relative configuration of the pairing nodes of superconductors and the Fermi surface of the normal state. We find that the regions of second-order topological phase are narrowed when the bilayer is twisted.