Coherence Effects of Caroli-de Gennes-Matricon Modes in Nodal Topological Superconductors

Coherence effects by the impurity scattering of Caroli--de Gennes--Matricon (CdGM) modes in a vortex for nodal topological superconductors have been studied. The coherence effects reflect a topological number defined on a particular momentum space avoiding the superconducting gap nodes. First, we analytically derived the eigenvalue and eigenfunction of the CdGM modes, including the zero-energy modes, in a nodal topological superconducting state without impurities, where we focused on a possible superconducting state of UPt$_3$ as an example. Then, we studied impurity effects on the CdGM modes by introducing the impurity self-energy, which are dominated by the coherence factor depending on the eigenfunction of the CdGM modes. For the zero-energy CdGM modes, the coherence factor vanishes in a certain momentum range, which is guaranteed by topological invariance characterized by the one-dimensional winding number.