Crystal fields and magnetic structure of the Ising antiferromagnet Er$_3$Ga$_5$O$_{12}$

Rare earth garnets are an exciting playground for studying the exotic magnetic properties of the frustrated hyperkagome lattice. Here we present a comprehensive study of the single ion and collective magnetic properties of the garnet Er$_3$Ga$_5$O$_{12}$. Using inelastic neutron scattering, we find a crystal field ground state doublet for Er$^{3+}$ with strong Ising anisotropy along local [100] axes. Magnetic susceptibility and heat capacity measurements provide evidence for long-range magnetic ordering with $T_N$~$=$~0.8~K, and no evidence for residual entropy is found when cooling through the ordering transition. Neutron powder diffraction reveals that the ground state spin configuration corresponds to the six-sublattice, Ising antiferromagnetic state ($\Gamma_3$) common to many of the rare earth garnets. However, we also found that $\mu$SR appears to be insensitive to the ordering transition in this material, in which a low-temperature relaxation plateau was observed with no evidence of spontaneous muon precession. The combined muon and neutron results may be indicative of a dynamical ground state with a relatively long correlation time. Despite this potential complication, our work indicates that Er$_3$Ga$_5$O$_{12}$ is an excellent model system for studying the complex metamagnetism expected for a multi-axis antiferromagnet.