Enforced Symmetry Breaking for Anomalous Valley Hall Effect in Two-Dimensional Hexagonal Lattices

The anomalous valley Hall effect (AVHE) is a pivotal phenomenon that allows for the exploitation of the valley degree of freedom in materials. A general strategy for its realization and manipulation is crucial for valleytronics. Here, by considering all possible symmetries, we propose general rules for the realization and manipulation of AVHE in two-dimensional hexagonal lattices. The realization of AVHE requires breaking the enforced symmetry that is associated with different valleys or reverses the sign of Berry curvature. Further manipulation of AVHE requires asymmetry operators connecting two states with opposite signs of Berry curvature. These rules for realizing and manipulating AVHE are extendable to generic points in momentum space. Combined with first-principles calculations, we realize the controllable AVHE in four representative systems, i.e., monolayer AgCrP_{2}Se_{6}, CrOBr, FeCl_{2}, and bilayer TcGeSe_{3}. Our work provides symmetry rules for designing valleytronic materials that could facilitate the experimental detection and realistic applications.