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Motivated by properties-controlling potential of the strain, we investigatestrain dependence of structure, electronic and magnetic properties ofSr$_2$IrO$_4$ using complementary theoretical tools: {\it ab-initio}calculations, analytical approaches (rigid octahedra picture, Slater-Kosterintegrals), and extended $t-{\mathcal{J}}$ model. We find that strain affectsboth Ir-Ir distance and Ir-O-Ir angle, and the rigid octahedra picture is notrelevant. Second, we find fundamentally different behavior for compressive andtensile strain. One remarkable feature is the formation of two subsets of bond-and orbital- dependent carriers, a compass-like model, under compression. Thisoriginates from the strain-induced renormalization of the Ir-O-Ir superexchangeand O on-site energy. We also show that under compressive (tensile) strain,Fermi surface becomes highly dispersive (relatively flat). Already at a tensilestrain of $1.5\%$, we observe spectral weight redistribution, with thelow-energy band acquiring almost purely singlet character. These results can bedirectly compared with future experiments.

Evolution of electronic and magnetic properties of Sr$\mathbf{_2}$IrO$\mathbf{_4}$ under strain