Non-Abelian Majorana Modes Protected by an Emergent Second Chern Number

The search for topological superconductors and non-Abelian Majorana modes ranks among the most fascinating topics in condensed matter physics. There now exist several fundamental superconducting phases which host symmetry protected or chiral Majorana modes, of which the chiral modes are protected by Chern numbers in even dimensions. Here we propose to observe novel chiral Majorana modes by realizing a Fulde-Ferrell-Larkin-Ovchinnikov state, i.e., the pair density wave (PDW) phase in a Weyl semimetal which breaks time-reversal symmetry. Without symmetry protection, the 3D gapped PDW phase is topologically trivial. However, a vortex line generated in such a phase can host chiral Majorana modes, which are shown to be protected by an emergent second Chern number of a synthetic 4D space generalized from the PDW phase. We further show that these chiral modes in the vortex rings obey 3D non-Abelian loop-braiding statistics, which can be applied to topological quantum computation.