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Hidden magnetic order of Sr$_2$Ir$_{1-x}$Rh$_x$O$_4$, $x = 0.05$ and 0.1, hasbeen studied using muon spin relaxation spectroscopy. In zero applied field andweak longitudinal fields ($\mu_0H_L \lesssim 2$~mT), muon spin relaxation datacan be well described by exponentially-damped static Lorentzian Kubo-Toyabefunctions, indicating that static and dynamic local fields coexist at each muonsite. For $\mu_0H_L \gtrsim 2$~mT, the static rate is completely decoupled, andthe exponential decay is due to dynamic spin fluctuations. In both zero fieldand $\mu_0H_L = 1$--2~mT, the temperature dependencies of the exponential muonspin relaxation rate exhibit maxima at 215~K for $x = 0.05$ and 175~K for $x =0.1$, suggesting critical slowing down of electronic spin fluctuations. Thefield dependencies of the dynamic spin fluctuation rates can be well describedby the Redfield relation. The correlation time of this electronic spinfluctuation is in the range of~2--5~ns for Sr$_2$Ir$_{0.9}$Rh$_{0.1}$O$_4$, andshorter than 2~ns for Sr$_2$Ir$_{0.95}$Rh$_{0.05}$O$_4$. The rms fluctuatingfield is on the order of 1 mT, which is consistent with the polarized neutrondiffraction cross-section.

Unusual slow magnetic fluctuations and critical slowing down in Sr$_{2}$Ir$_{1-x}$Rh$_{x}$O$_{4}$