Altermagnetic topological insulator and the selection rules

Altermangetism is newly identified as the third fundamental class of collinear magnetism with zero net magnetization while hosting spin-split bands which break the time-reversal symmetry and Kramers degeneracy. Here, we propose a $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ model for a two-dimensional altermagnetic lattice, in which instead of neglecting them, the spin-orbit couplings play the role of driving the system into a topological insulating region. We term this topological phase as an altermagnetic topological insulator, as opposed to an antiferromagnetic topological insulator such as ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$. A spin Chern number is attributed as the topological invariant. We also derive the selection rules for optical conductivity measurements based on our minimal model, with which the elliptically polarized lights will serve as powerful probes to detect altermagnetism and excite individual spin degrees of freedom of altermagnetic compounds through light-induced anomalous Hall effects and longitudinal spin currents.