$S = 1$ pyrochlore magnets with competing anisotropies: A tale of two Coulomb phases, $Z_2$ flux confinement and $XY$-like transitions

Abstract: 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 $Δ=J + μ$ ($|μ| \ll J$) that favors the plane perpendicular to these local axes will exhibit interesting new phenomena due to the competition between $J$ and $Δ$. In the $T/J \rightarrow 0$ 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 $Z_2$ 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 {\em even} integers, while it takes on all integer values in the other Coulomb phase. Experimental realizations with $|μ| \ll J$ 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 $T_{c_2} \approx 1.57|μ|$, while realizations with positive $μ\ll J$ 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 $T_{c_1} \approx 0.98 μ$.