Altermagnetic topological insulator on a 2D squre lattice

Altermagnetism emerges as the third fundamental collinear magnetism recently. Associating nontrivial band topology and quantum orders to this newly established magnetism is an important physical topic and may lead to interesting physics such as unusual quantum spin Hall effect with time-reversal-symmetry breaking, topological superconductivity and spin-valley locking. Here, we first propose a kp model for a two-dimensional altermagnetic lattice, instead of neglecting them, the spin-orbit couplings play the role of driving the system into a topological insulating region. We term this new topological phase as altermagnetic topological insulator, as opposed to the antiferromagnetic topological insulator such a MnBi2Te4. 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 degree of freedom of altermagnetic compounds through the light-induced anomalous Hall effects and longitudinal spin currents. Then, we generalize the tight-binding spinless Haldane model from the usual honeycomb lattice to the otherwise simpler square lattice. After restoring the doublet spin degree of freedom, the resulted spinful model resembles to the Kane-Mele model while is able to describe the unprecedented physics which intertwine C-paired spin-valley locking, Dirac physics, nontrivial band topology and altermagnetism. Our works shed lights on future studies and applications in spintronics and valleytronics for altermagnetic materials, the lattice theories we developed may also find its broad applications to optical lattice with ultra cold atoms, photonic lattice, etc. other than the usual electronic systems.