Nodal topology in $d$-wave superconducting monolayer FeSe

A nodeless $d$-wave state is likely in superconducting monolayer FeSe on SrTiO$_3$. The lack of nodes is surprising but has been shown to be a natural consequence of the observed small interband spin-orbit coupling. Here we examine the evolution from a nodeless state to the nodal state as this spin-orbit coupling is increased from a topological perspective. We show that this evolution depends strongly on the orbital content of the superconducting degrees of freedom. In particular, there are two $d$-wave solutions, which we call orbitally trivial and orbitally nontrivial. In both cases, the nodes carry a $\pm 2$ topological winding number that originates from a chiral symmetry. However, the momentum space distribution of the positive and negative charges is different for the two cases, resulting in a different evolution of these nodes as they annihilate to form a nodeless superconductor. We further show that the orbitally trivial and orbitally nontrivial nodal states exhibit different Andreev flat band spectra at the edge.