Evolution of Magnetic and Orbital Properties in the Magnetically-Diluted \textit{A}-Site Spinel Cu$_{1-x}$Zn$_x$Rh$_2$O$_4$

In frustrated spinel antiferromagnets, dilution with non-magnetic ions can be a powerful strategy for probing unconventional spin states or uncovering interesting phenomena. Here, we present X-ray, neutron scattering and thermodynamic studies of the effects of magnetic dilution of the tetragonally-distorted A-site spinel antiferromagnet, CuRh$_2$O$_4$, with non-magnetic Zn$^{2+}$ ions. Our data confirm the helical spin order recently identified at low-temperatures in this material, and further demonstrate a systematic suppression of the associated N\'eel temperature with increasing site dilution towards a continuous transition with critical doping of $x_{spin} \sim 0.44$. Interestingly, this critical doping is demonstrably distinct from a second structural critical point at $x_{JT} \sim 0.6$, which is consistent with the suppression of orbital order on the A-site through a classical percolative mechanism. This anomalously low value for $x_{spin}$ is confirmed via multiple measurements, and is inconsistent with predictions of classical percolation theory, suggesting that the spin transition in this material is driven by an enhancement of pre-existing spin fluctuations with weak dilution.