  Topological semimetals in ferromagnetic materials have attracted enormous
attention due to the potential applications in spintronics. Using the
first-principles density functional theory together with an effective lattice
model, here we present a new family of topological semimetals with a fully
spin-polarized nodal loop in alkaline-metal monochalcogenide $MX$ ($M$ = Li,
Na, K, Rb, Cs; $X$ = S, Se, Te) monolayers. The half-metallic ferromagnetism
can be established in $MX$ monolayers, in which one nodal loop formed by two
crossing bands with the same spin components is found at the Fermi energy. This
nodal loop half-metal survives even when considering the spin-orbit coupling
owing to the symmetry protection provided by the $\mathcal{M}_{z}$ mirror
plane. The quantum anomalous Hall state and Weyl-like semimetal in this system
can be also achieved by rotating the spin from the out-of-plane to the in-plane
direction. The $MX$ monolayers hosting rich topological phases thus offer an
excellent materials platform for realizing the advanced spintronics concepts.
