$S = 1$ 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 $Δ=J + μ$ ($|μ| \ll J$) 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 $T_{c_1} \approx 0.98 μ$.

💡 Research Summary

The paper investigates the low‑temperature behavior of S = 1 pyrochlore magnets in which a dominant Ising‑like exchange J (favoring spin components along the local tetrahedral body diagonals) competes with a comparably large easy‑plane single‑ion anisotropy Δ = J + μ, with |μ| ≪ J. The sign of μ determines whether the easy‑plane term is slightly stronger (μ > 0) or weaker (μ < 0) than the Ising exchange. By fixing the ratio μ/T and introducing the fugacity w = exp(−μ/T), the authors focus on the T/J → 0 limit where only configurations that minimize the J term (i.e., those satisfying the “ice rule” Sᶻ_tet = 0 on each tetrahedron) contribute. These configurations can be mapped onto a divergence‑free polarization field P on the diamond lattice, whose global flux vector ϕ takes integer values.

Using a worm Monte‑Carlo algorithm that creates pairs of ±1 (or ±2) charge defects and measures the head‑to‑tail distance histograms h(1)(r) and h(2)(r), the authors identify three distinct thermodynamic phases as a function of w:

1. Short‑range correlated paramagnet (w ≈ 0) – all spins are in the Sᶻ = 0 state, the structure factor is featureless, and the flux ϕ is pinned to zero.

2. Flux‑deconfined Coulomb liquid (w > w_c1 ≈ 0.3609) – the polarization field remains divergence‑free, giving rise to pinch‑point singularities in the spin‑flip structure factor. The flux ϕ fluctuates freely over all integer values; the transition at w_c1 is in the 3D XY universality class, with a stiffness‑like quantity ⟨ϕ²⟩/L scaling as L at criticality.

3. Z₂‑confined Coulomb liquid (w > w_c2 ≈ 1.8904) – the same Coulomb physics persists, but the Z₂ part of the flux becomes confined: only even‑integer components of ϕ survive in the thermodynamic limit, while odd components are suppressed. This constitutes a Z₂ flux‑confinement transition with critical exponent ν ≈ 0.63.

For μ > 0, w can only increase up to 1, so only the first transition (at w_c1) is accessible. It occurs at a temperature T_c1 ≈ 0.98 μ and is continuous, XY‑like, separating the paramagnet from the flux‑deconfined Coulomb liquid. For μ < 0, w can become larger than 1, allowing the system to pass through both transitions. The second transition appears at T_c2 ≈ 1.57 |μ| and drives the system from the flux‑deconfined to the Z₂‑confined Coulomb phase.

Both Coulomb phases exhibit the characteristic pinch‑point singularities in the static structure factor S(q), but the intensity and background differ across the Z₂ confinement transition, providing an experimental signature. The specific heat shows two sharp peaks corresponding to the two transitions, confirming their thermodynamic nature.

The work demonstrates that a modest imbalance between Ising exchange and easy‑plane anisotropy in S = 1 pyrochlores can generate rich emergent gauge‑theoretic phenomena: a classical Coulomb spin liquid, a Z₂‑confinement transition, and an XY‑type ordering transition. These predictions are directly testable in candidate materials where μ can be tuned by chemical substitution, pressure, or strain, offering a new platform to explore topological phases and confinement physics in three‑dimensional frustrated magnets.