We report the emergence of Majorana flat bands (MFBs) in the vortex line of superconducting (SC) time-reversal symmetry-breaking Weyl semimetals. By considering a Weyl semimetal as a stack of Chern insulators with varying Chern numbers along one ($z$) direction, we decompose the vortex bound states of SC Weyl semimetals into those of $k_z$-resolved SC Chern insulators. Through analytical and numerical calculations of the topological phase diagram of the SC Chern insulators, we explain the appearance of MFBs and determine the exact boundaries of them. Notably, the tuning of chemical potential or pairing strength results in the MFBs along the entire $k_z$ axis. We find that the MFBs are absent in SC type-$\mathrm{II}$ Weyl semimetals due to the inversion symmetry-breaking term. To characterize the MFBs, we propose a $k_z$-resolved $Z_2$ Chern-Simons invariant as the topological indicator. Finally, we take an attractive Hubbard interaction into consideration, and the aforementioned SC Weyl semimetal with BCS pairing can be realized under appropriate parameters.

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• Chern insulators
• Flat bands
• Majorana bound states
• Topological superconductors
• Vortices in superconductors
• Weyl semimetal
• Bogoliubov-de Gennes equations
• Mean field theory