Hybrid d/p-wave altermagnetism in Ca${}_3$Ru${}_2$O${}_7$ and strain-controlled spin splitting

The interplay of strong electronic correlations, sizable octahedral distortions, and pronounced spin-orbit coupling (SOC) makes perovskite oxides promising candidates for realizing altermagnetic phases. We study altermagnetic phases in Ca${}_3$Ru${}_2$O${}_7$, a non-centrosymmetric layered perovskite whose ground state is a Kramers-degenerate antiferromagnet. We show that an alternative Néel-type spin arrangement hosts a P-2 d-wave altermagnetic state with orbital selectivity similar to Ca${}_2$RuO${}_4$. Including SOC generates a symmetry-allowed p-wave component and yields a hybrid d/p-wave altermagnetic order. We further demonstrate that biaxial strain tunes both magnetic stability and band splitting: compressive strain beyond 2 % favors the altermagnetic phase over the antiferromagnetic ground state, while tensile strain increases altermagnetic splittings by up to 9 %. To quantify these trends, we define an altermagnetic figure of merit and trace its strain dependence to changes in electronic localization and octahedral geometry in this polar metal.