Unconventional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-wave and finite-momentum superconductivity induced by altermagnetism through the formation of Bogoliubov Fermi surface

Altermagnets are an exotic class of magnetic materials in which the Fermi surface exhibits a momentum-dependent spin-splitting while maintaining net-zero magnetization. Previous studies showed that this distinctive spin splitting can induce chiral $p$-wave superconductors or Fulde-Ferrell (FF) superconducting states carrying finite momentum. However, the underlying mechanisms of such unconventional superconductivities remain elusive. Here, we propose that the formation of the Bogoliubov Fermi surface (BFS) through the exchange field can play a significant role in such phenomena. Through a systematic self-consistent mean-field analysis of the extended attractive Hubbard model combined with the $d$-wave spin splitting induced by the exchange field, as observed in ${\mathrm{RuO}}_{2}$, we demonstrate that the formation of the BFS suppresses conventional spin-singlet superconducting states with $s$-wave characteristics. In contrast, the chiral $p$-wave state maintains a fully gapped spectrum without the Fermi surface, thereby becoming the ground state in the strong-field regime. In the intermediate regime, we find that the FF state becomes the predominant state through the optimization of available channels for Cooper pairing. Moreover, we illustrate how the prevalence of the chiral $p$-wave and FF states over the $s$-wave state changes under the variation of the field strength or chemical potential. Our findings provide valuable insights into potential pathways for realizing sought-after topological $p$-wave superconductivity and finite-momentum pairing facilitated by altermagnetism.