The paper discusses the prediction of a Chern semimetal state in the ferromagnetic compound HgCr$_2$Se$_4$ based on first-principles calculations. This state is characterized by topologically unavoidable band crossings at the Fermi level, leading to the presence of Weyl fermions, which are condensed-matter realizations of chiral fermions in (3+1)D. The authors predict that this Chern semimetal state can exhibit magnetic monopoles and fermi arcs, and that the quantized anomalous Hall effect (QAHE), a quantum Hall effect without an external magnetic field, can be achieved in the quantum-well structure of HgCr$_2$Se$_4$. The electronic structures of HgCr$_2$Se$_4$ are analyzed, showing a zero-gap half-metal behavior and band inversion around the $\Gamma$ point. The Chern number is evaluated for different $k_z$-fixed planes, revealing topologically non-trivial regions where Weyl nodes exist. The paper also discusses the experimental implications, including the possibility of observing fermi arcs and the quantized anomalous Hall effect in thin films of HgCr$_2$Se$_4$.The paper discusses the prediction of a Chern semimetal state in the ferromagnetic compound HgCr$_2$Se$_4$ based on first-principles calculations. This state is characterized by topologically unavoidable band crossings at the Fermi level, leading to the presence of Weyl fermions, which are condensed-matter realizations of chiral fermions in (3+1)D. The authors predict that this Chern semimetal state can exhibit magnetic monopoles and fermi arcs, and that the quantized anomalous Hall effect (QAHE), a quantum Hall effect without an external magnetic field, can be achieved in the quantum-well structure of HgCr$_2$Se$_4$. The electronic structures of HgCr$_2$Se$_4$ are analyzed, showing a zero-gap half-metal behavior and band inversion around the $\Gamma$ point. The Chern number is evaluated for different $k_z$-fixed planes, revealing topologically non-trivial regions where Weyl nodes exist. The paper also discusses the experimental implications, including the possibility of observing fermi arcs and the quantized anomalous Hall effect in thin films of HgCr$_2$Se$_4$.