Competition between d-wave superconductivity and magnetism in uniaxially strained Sr₂RuO₄

The pairing symmetry of Sr₂RuO₄ is a long-standing fundamental question in the physics of superconducting materials with strong electronic correlations. We use the functional renormalization group to investigate the behavior of superconductivity under uniaxial strain in a two-dimensional realistic model of Sr₂RuO₄ obtained with density functional theory and incorporating the effect of spin-orbit coupling. We find a dominant \({d}_{{{{{\rm{x}}}}}^{2}-{{{{\rm{y}}}}}^{2}}\) $d_{{\mathrm{x}}^2-{\mathrm{y}}^2}$ superconductor mostly hosted by the d_xy-orbital, with no other closely competing superconducting state. Within this framework, we reproduce the experimentally observed enhancement of the critical temperature under strain and propose a simple mechanism driven by the density of states to explain our findings. We also investigate the competition between superconductivity and spin-density wave ordering as a function of interaction strength. By comparing theory and experiment, we discuss constraints on a possible degenerate partner of the \({d}_{{{{{\rm{x}}}}}^{2}-{{{{\rm{y}}}}}^{2}}\) $d_{{\mathrm{x}}^2-{\mathrm{y}}^2}$ superconducting state.