Charge-stripe order, antiferromagnetism, and spin dynamics in the cuprate-analog nickelate La$_4$Ni$_3$O$_8$

We report a muon spin rotation ($\mu$SR) study of the cuprate-analog nickelate La$_4$Ni$_3$O$_8$, which undergoes a transition at 105 K to a low-temperature phase with charge-stripe and antiferromagnetic (AFM) order on square planar NiO$_2$ layers. Zero-field $\mu$SR shows that this transition is abrupt and that the AFM configuration is commensurate. Comparison of observed muon precession frequencies with Ni dipolar field calculations yields Ni moments $\lesssim 0.5\mu_B$. Dynamic muon spin relaxation above 105 K suggests critical slowing of Ni spin fluctuations, but is inconsistent with corresponding $^{139}$La NMR results. Critical slowing and an abrupt transition are also observed in the planar cuprate AFM La$_2$CuO$_{4+\delta}$, where they are evidence for weakly interplanar-coupled two-dimensional AFM spin fluctuations, but our $\mu$SR data do not agree quantitatively with theoretical predictions for this scenario.