Unconventional superconductivity in altermagnets with spin-orbit coupling

We investigate some possible symmetries of the superconducting state that emerges in three-dimensional altermagnets in the presence of spin-orbit coupling. We demonstrate within a weak-coupling approach that these altermagnets, which naturally possess an order modulated by a vector form factor $\boldsymbol{g}_{\mathbf{k}}$, favor spin-triplet superconductivity described by gap functions given by $\boldsymbol{d}(\mathbf{k}) = \boldsymbol{u}(\mathbf{k}) \times \boldsymbol{g}_{\mathbf{k}}$, where $\boldsymbol{u}(\mathbf{k}) = - \boldsymbol{u}(-\mathbf{k})$. Consequently, this singles out $f$-wave spin-triplet superconductivity as the most favorable pairing state to appear in the vicinity of $d$-wave altermagnetism. Furthermore, we obtain that the combination of spin-singlet superconducting states with altermagnetism gives rise to Bogoliubov-Fermi surfaces, which are protected by a $\mathbb{Z}_2$ topological invariant. Using a Ginzburg-Landau analysis, we show that, for a class of spin-orbit coupled altermagnetic models, a superconducting phase is expected to appear at low temperatures as an intertwined $d + if$ state, thus breaking time-reversal symmetry spontaneously.