  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.
