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The iso-electronic $d^{4}$ compounds of the $4d$ series show rich phasediagrams due to competing spin, charge and orbital degrees of freedom inpresence of strong correlations and structural distortions. One suchiso-electronic series, $Ca_{2-x}Sr_{x}RuO_{4}$, is studied within the GGA (andspin-orbit coupled GGA) plus DMFT formalism using the hybridization expansionof continuous time Quantum Monte Carlo solver. While the local dynamicalcorrelations make $Sr_{2}RuO_{4}$ a Hund's metal, they drive $Ca_{2}RuO_{4}$ toa Mott insulating ground state. We study the dynamic and static single-particleand local irreducible vertex-corrected two-particle responses at threedifferent points ($x = 2.0, 0.5, 0.0$) to understand the anomalous cross-overfrom Hund's metal ($x = 2.0 $) to a Mott insulator ($x = 0 $) and find that astructural distortion is likely to be responsible for the cross-over. Further,dynamical correlations reveal that the band-width ($W$) of the Hund's metal islarger than its effective local Hubbard $U$, and a finite Hund's coupling$J_{H}$ helps it remain in a bad metallic and nearly spin-frozen state over alarge temperature range. $Ca_{2}RuO_{4}$, on the other hand, is intrinsicallydriven to the proximity of a Mott transition due to narrowing of band width($U/W > 1.5$), though its finite temperature excitations indicate badmetallicity. We show that there is a critical end point of second-orderstructural transition at $x = 0.5$, where spin fluctuations become criticallysingular and follow the exact scaling of conformally invariant boundary fieldtheory. The critical end point of quasi-$3D$ nature is associated with aneffective dimensional cross-over between the $x = 2.0$ and $x=0.0$ quasi-$2D$structures. Finally we draw a modified magnetic phase diagram of the material,showing a fan-like region starting from the quantum critical end point at $x =0.5.$

The Iso-electronic Series $Ca_{2-x}Sr_{x}RuO_{4}$: Structural Distortion, Effective Dimensionality, Spin Fluctuations and Quantum Criticality