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The pairing symmetry of Sr$_2$RuO$_4$ is a long-standing fundamental questionin the physics of superconducting materials with strong electroniccorrelations. We use the functional renormalization group to investigate thebehavior of superconductivity under uniaxial strain in a two-dimensionalrealistic model of Sr$_2$RuO$_4$ obtained with density functional theory andincorporating the effect of spin-orbit coupling. We find a dominant$d_{x^2-y^2}$ superconductor mostly hosted by the $d_{xy}$-orbital, with noother closely competing superconducting state. Within this framework wereproduce the experimentally observed enhancement of the critical temperatureunder strain and propose a simple mechanism driven by the density of states toexplain our findings. We also investigate the competition betweensuperconductivity and spin-density wave ordering as a function of interactionstrength. By comparing theory and experiment, we discuss constraints on apossible degenerate partner of the $d_{x^2-y^2}$ superconducting state.

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