  Nodal-chain fermions, as novel topological states of matter, have been hotly
discussed in non-magnetic materials. Here, by using first-principles
calculations and symmetry analysis, we propose the realization of fully
spin-polarized nodal chain in the half-metal state of LiV$_2$O$_4$ compound.
The material naturally shows a ferromagnetic ground state, and takes on a
half-metal band structure with only the bands from the spin-up channel present
near the Fermi level. The spin-up bands cross with each other, which form two
types of nodal loops. These nodal loops arise from band inversion and are under
the protection of the glide mirror symmetries. Remarkably, we find the nodal
loops conjunct with each other and form chain-like nodal structure.
Correspondingly, the w-shaped surface states are also fully spin-polarized. The
fully spin-polarized nodal chain identified here has not been proposed in
realistic materials before. An effective model is constructed to describe the
nature of nodal chain. The effects of the electron correlation, the lattice
strains, and the spin-orbit coupling are discussed. The fully spin-polarized
bulk nodal-chain and the associated nontrivial surface states for a half-metal
may open novel applications in spintronics.
