Berry Phase Enforced Spinor Pairing Order

We introduce a class of topological pairing orders characterized by a half-integer pair monopole charge, leading to Berry phase enforced half-integer partial wave symmetry. This exotic spinor order emerges from pairing between Fermi surfaces with Chern numbers differing by an odd integer. Using tight-binding models, we demonstrate spinor superconducting orders with monopole charges $\pm 1\mathrm{/}2$, featuring a single gap node and nontrivial surface states. Additionally, the superfluid velocity follows a fractionalized Mermin-Ho relation in spatially inhomogeneous pairing orders. The concept extends to spinor density waves and excitons.