Second Chern Number Protected Gapless Vortex Modes in Superconducting Weyl Semimetals

We have studied the vortex bound states of superconducting linear and quadratic Weyl semimetals with time reversal symmetry. For vortex with vorticity $n$ (vortex line along $z$ direction) and Weyl semimetals with Weyl point chirality $\chi$, there are $\mathrm{\mid }n\chi \mathrm{\mid }$ gapless vortex modes around the location of each Weyl point in superconducting Weyl semimetals. We show that the appearance of $\mathrm{\mid }n\chi \mathrm{\mid }$ gapless vortex modes originates from the imbalanced pumping of flux bound states between Chern insulators obtained at $k_z$ and $-k_z$ in Weyl semimetals. Such a physical picture is verified by numerical calculations on lattice models. Moreover, the gapless modes around each Weyl point can be characterized by the second Chern number defined in a synthetic four-dimensional space based on the effective BdG Hamiltonian. Finally, these gapless vortex modes are found to be robust against rotational symmetry-breaking terms.