  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. The latter, namely the chiral Majorana modes are
protected by Chern numbers in even dimensions. Here we propose to observe novel
chiral Majorana modes by realizing Fulde-Ferrell-Larkin-Ovchinnikov state, i.e.
the pairing 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 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.
