Mirror Chern Bands and Weyl Nodal Loops in Altermagnets

The electronic spectra of altermagnets are a fertile ground for non-trivial topology due to the unique interplay between time-reversal and crystalline symmetries. This is reflected in the unconventional Zeeman splitting between bands of opposite spins, which emerges in the absence of spin-orbit coupling (SOC) and displays nodes along high-symmetry directions. Here, we argue that even for a small SOC, the direction of the magnetic moments in the altermagnetic state has a profound impact on the electronic spectrum, enabling novel topological phenomena to appear. By investigating microscopic models for two-dimensional (2D) and three-dimensional (3D) altermagnets, motivated in part by rutile materials, we demonstrate the emergence of hitherto unexplored Dirac crossings between bands of same spin but opposite sublattices. The direction of the moments determines the fate of these crossings when the SOC is turned on. We focus on the case of out-of-plane moments, which forbid an anomalous Hall effect and thus ensure that no weak magnetization is triggered in the altermagnetic state. In 2D, the SOC gaps out the Dirac crossings, resulting in mirror Chern bands that enable the quantum spin Hall effect and undergo a topological transition to trivial bands upon increasing the magnitude of the magnetic moment. On the other hand, in 3D the crossings persist even in the presence of SOC, forming Weyl nodal loops protected by mirror symmetry. Finally, we discuss possible ways to control these effects in altermagnetic material candidates.