Altermagnetic topological insulator with \(\mathcal{C}\)-paired spin-valley locking

Altermagnetism emerges as the third fundamental collinear magnetism recently. Associating nontrivial 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 generalize the spinless Haldane model from the usual honeycomb lattice to the otherwise simpler square lattice. We then restore the doublet spin degree of freedom, the resulted spinful model resembles the Kane-Mele model and is able to describe the unprecedented physics which intertwine $\mathcal{C}$ $\mathcal{C}$ -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 developed in this article may also find their broad applications to optical lattice with ultra cold atoms, photonic lattice, etc. other than the usual electronic systems.