The effects of strain in multi-orbital superconductors: the case of Sr$_2$RuO$_4$

Uniaxial strain experiments have become a powerful tool to unveil the character of unconventional phases of electronic matter. Here we propose a combination of the superconducting fitness analysis and density functional theory (DFT) calculations in order to dissect the effects of strain in complex multi-orbital quantum materials from a microscopic perspective. We apply this framework to the superconducting state of Sr$_2$RuO$_4$, and argue that the recently proposed orbitally anti-symmetric spin-triplet (OAST) order parameter candidate has unique signatures under strain which are in agreement with recent observations. In particular, we can account for the asymmetric splitting of the critical temperatures for compressive strain along the $\langle 100\rangle$ direction, and the reduction of the critical temperatures for compressive strain along the $\langle 001\rangle$ and $\langle 110\rangle$ directions with a single free parameter.