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We present a muon spin relaxation study on the Ising pyrochloreNd$_2$Zr$_2$O$_7$ which develops an "all-in-all-out" magnetic order below0.4~K. At 20~mK far below the ordering transition temperature, the zero-fieldmuon spin relaxation spectra show no static features and can be well describedby a dynamical Gaussian-broadened Gaussian Kubo-Toyabe function indicatingstrong fluctuations of the ordered state. The spectra of the paramagnetic state(below 4.2~K) reveal anomalously slow paramagnetic spin dynamics and show onlysmall difference with the spectra of the ordered state. We find that thefluctuation rate decreases with decreasing temperature and becomes nearlytemperature independent below the transition temperature indicating persistentslow spin dynamics in the ground state. The field distribution width shows asmall but sudden increase at the transition temperature and then becomes almostconstant. The spectra in applied longitudinal fields are well fitted by theconventional dynamical Gaussian Kubo-Toyabe function, which further supportsthe dynamical nature of the ground state. The fluctuation rate shows a peak asa function of external field which is associated with a field-induced spin-fliptransition. The strong dynamics in the ordered state are attributed to thetransverse coupling of the Ising spins introduced by the multipoleinteractions.

Spin dynamics of the ordered dipolar-octupolar pseudospin-$^1/_2$ pyrochlore Nd$_2$Zr$_2$O$_7$ probed by muon spin relaxation