Winding numbers of nodal points in Fe-based superconductors

In this work, we analyze nodal points in multi-orbital Fe-based superconductors from the point of view of topology. We consider the case of $s^{+-}$ gap structure, with accidental nodes, and d-wave gap with nodes along symmetry directions. In both cases, the nodes are not monolithic and can be moved by varying an external parameter, e.g., a degree of inter-pocket pairing. Eventually, the nodes merge and annihilate and via Lifshitz-type transition. This happens in a multi-orbital model even if the gap symmetry is d-wave. We discuss Lifshitz transition in Fe-based superconductors from topological point of view. We show, both analytically and numerically, that in both s-wave and d-wave cases, the merging nodal points have winding numbers of opposite signs. This is consistent with a general reasoning that the total winding number is a conserved quantity in the Lifshitz transition.