Spin Chern number in altermagnets

This paper explores the topological properties of altermagnets, a class of collinear magnetic materials. We employ equivariant K-theory of magnetic groups and Hamiltonian models to formulate a robust ${C}_{4}^{z}\mathbb{T}$ topological invariant to classify 2D and 3D altermagnetic systems. Our findings demonstrate that the spin Chern number serves as a robust topological index, corresponding to the half-quantized Chern number of the divided Brillouin zone. This indicator enables the prediction of a topologically protected 2D altermagnetic insulators and 3D Weyl altermagnetic semimetals, highlighting the relationship between altermagnetism and topological phases. Furthermore, our results provide a pathway to the exploration of topological applications in $d$-wave altermagnetic materials.