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Description
The superconducting symmetry of Sr$_2$RuO$_4$ is one of the long standing problems of material science. Even though the normal state of the material can be well described in terms of density functional theory plus dynamical mean-field theory calculations, the character of the superconducting state remains elusive. In this paper we investigate the behavior of the superconducting order parameter under strain along the [1 0 0] direction, using density functional theory plus functional renormalization group simulations for a 2D model. We find a dominant $d_{x^2-y^2}$ superconductor in the $d_{xy}$-orbital with no closely related additional order parameter component. We can reproduce the experimentally observed change of $T_c$ under strain within this framework and propose a simple mechanism explaining the observed behavior. By comparing theory and experiment, we can construct further constrains for a possible degenerate partner of the $d_{x^2-y^2}$ superconducting state.