Strain-Induced Quantum Anomalous Valley Hall Phase in Monolayer FeCl2: an Explicit Chern Number Criterion and Crystal-Field Mechanism

Two-dimensional ferrovalley semiconductors offer a compelling platform for topologically nontrivial valleytronics, yet an explicit topological criterion governing the strain-driven quantum anomalous valley Hall effect (QAVHE) and its microscopic origin has remained largely unexplored. Here, combining first-principles calculations, a effective model, and crystal-field analysis for the FeCl2 monolayer, we resolve the complete five-stage phase sequence driven by biaxial tensile strain. Central to our analysis is an explicit Chern number criterion derived analytically from the two-band Hamiltonian: a nonzero Chern number requires the renormalized gaps of the two valleys to carry opposite signs. Crystal-field analysis further reveals why this condition is microscopically inevitable: in-plane strain couples selectively to the E-type orbitals via σ-bond weakening while leaving the A1-type orbital nearly unperturbed—a symmetry-selective modulation that guarantees orbital inversion and opposite-sign valley gaps over a finite strain window. These results establish a complete causal chain from strain to topology for QAVHE engineering in 2D magnetic materials.