Fusion rules for Majorana Kramers pairs in time-reversal invariant topological superconductors

We theoretically investigate the fusion rules of Majorana Kramers pairs in time-reversal invariant topological superconductors. We find that the fusion of Majorana Kramers pairs is a process that Ising anyons fuse independently in the two distinct time-reversal sectors. Considering the full fusion including the initialization and the fusion, we explore the observation of a supersymmetry that emerges in time-reversal invariant topological superconductors, and design the schemes for the nontrivial fusion and the trivial fusion to show the non-Abelian statistics of Majorana Kramers pairs. We also show the possible influence of local adiabatic mixing on the fusion can generally be negligible and the differentiation between distinct fusion processes remains feasible even in the presence of such mixing. Our proposals are applied in ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave topological superconductors, and the theoretical framework can be extended to the fusion of multiple Majorana zero modes protected by unitary symmetry.