S = 1 pyrochlore magnets with competing anisotropies: A tale of two Coulomb phases, Z 2 flux confinement, and XY-like transitions

We argue that the low-temperature physics of $S=1$ pyrochlore magnets with a predominantly Ising-like easy-axis exchange coupling J that favors the local tetrahedral body diagonals, and a comparably large easy-plane single-ion anisotropy $Δ=J + μ$ (|μ| J) that favors the plane perpendicular to these local axes will exhibit interesting new phenomena due to the competition between J and $Δ$. In the T/J → 0 limit, we find three low temperature phases as a function of $μ/T$: a short-range correlated paramagnetic phase, and two topologically-distinct Coulomb liquids separated by a Z₂ flux confinement transition. Both Coulomb liquids are described at long-wavelengths by a fluctuating divergence-free polarization field and have characteristic pinch-point singularities in their structure factor. In one Coulomb phase, the flux of this polarization field is confined to { even} integers, while it takes on all integer values in the other Coulomb phase. Experimental realizations with |μ| J and negative are predicted to exhibit signatures of a transition from a flux-deconfined Coulomb phase to the flux-confined Coulomb phase as they are cooled below Tc₂ ≈ 1.57|μ|, while realizations with positive μ J will show signatures of a transition from a flux-deconfined Coulomb liquid to a short-range correlated paramagnet via a continuous $XY$-like transition at Tc₁ ≈ 0.98 μ.