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The superconducting gap of the Sr₂RuO₄ is still uncertain [1]. The most recent striking inconsistency is observed under uniaxial pressure. While high-sensitivity specific heat measurements detect only one transition temperature [2], muon spin relaxation experiments suggest two [3]. The possible superconducting gap functions are extremely intricate and complex due to the multiple orbitals involved and coupled by spin-orbit interaction [4]. We compute the electronic structure, free energy and specific heat obtained for different superconducting order parameters by using the Bogoliubov-de Gennes formalism. We focus on the non-trivial effects of spin-orbit coupling, relative phases between different spin-orbital sectors, and coexisting components.

[1] Mackenzie et al., npj Quantum Materials 2, 40 (2017).

[2] Li et al., Proc. of the Natl. Acad. Sci. 118 (10) e2020492118 (2021).

[3] Grinenko et al., Nature Physics 17, 748 (2021).

[4] Gingras et al., Phys. Rev. B 106, 064513 (2022).