This paper presents the discovery of a Chern semimetal state in HgCr₂Se₄, a known ferromagnetic compound, which exhibits a quantized anomalous Hall effect (QAHE) without an external magnetic field. The Chern semimetal state is a 3D realization of Weyl fermions, characterized by topologically protected band crossings at the Fermi level. These crossings, called Weyl nodes, are separated in momentum space and are topologically unavoidable. The system is predicted to exhibit magnetic monopoles and fermi arcs on its surfaces. The study uses first-principles calculations to show that HgCr₂Se₄ has a Chern number of 2 in certain k_z regions, indicating a nontrivial topological state. The electronic structure of HgCr₂Se₄ is analyzed, revealing a band inversion around the Γ point, which leads to the formation of Weyl nodes at k_z = ±k_z^c. The Chern number is calculated for different k_z planes, showing that it is zero outside the region around k_z = ±k_z^c and 2 within that region. The Weyl nodes are topological defects in momentum space, associated with the Berry curvature and gauge flux. The presence of these nodes leads to the emergence of fermi arcs on the surfaces of HgCr₂Se₄, which are non-closed and can be measured by ARPES. The QAHE is observed in the quantum-well structure of HgCr₂Se₄, where the Hall conductance is quantized as σ_xy = C e²/ħ, with C being the Chern number. This effect is a unique consequence of the topological nature of the Chern semimetal state. The study also discusses the implications of the Chern semimetal state, including the existence of chiral edge states and the possibility of measuring the QAHE experimentally. The results are supported by first-principles calculations and comparisons with other materials, highlighting the unique properties of HgCr₂Se₄ as a topological material.This paper presents the discovery of a Chern semimetal state in HgCr₂Se₄, a known ferromagnetic compound, which exhibits a quantized anomalous Hall effect (QAHE) without an external magnetic field. The Chern semimetal state is a 3D realization of Weyl fermions, characterized by topologically protected band crossings at the Fermi level. These crossings, called Weyl nodes, are separated in momentum space and are topologically unavoidable. The system is predicted to exhibit magnetic monopoles and fermi arcs on its surfaces. The study uses first-principles calculations to show that HgCr₂Se₄ has a Chern number of 2 in certain k_z regions, indicating a nontrivial topological state. The electronic structure of HgCr₂Se₄ is analyzed, revealing a band inversion around the Γ point, which leads to the formation of Weyl nodes at k_z = ±k_z^c. The Chern number is calculated for different k_z planes, showing that it is zero outside the region around k_z = ±k_z^c and 2 within that region. The Weyl nodes are topological defects in momentum space, associated with the Berry curvature and gauge flux. The presence of these nodes leads to the emergence of fermi arcs on the surfaces of HgCr₂Se₄, which are non-closed and can be measured by ARPES. The QAHE is observed in the quantum-well structure of HgCr₂Se₄, where the Hall conductance is quantized as σ_xy = C e²/ħ, with C being the Chern number. This effect is a unique consequence of the topological nature of the Chern semimetal state. The study also discusses the implications of the Chern semimetal state, including the existence of chiral edge states and the possibility of measuring the QAHE experimentally. The results are supported by first-principles calculations and comparisons with other materials, highlighting the unique properties of HgCr₂Se₄ as a topological material.