$Z_3$ confined and deconfined Coulomb liquids in $S_{\rm eff} = 3/2$ pyrochlore magnets

Abstract: We identify an interesting regime in the physics of pyrochlore magnets in which spin-orbit and crystal field effects lead to {\em two} low-lying magnetic doublets that can be modeled as an effective spin $S=3/2$ degree of freedom that sees a dominant easy-axis antiferromagnetic exchange $J>0$ favoring the local $[111]$ axes, which competes with a comparably strong single-ion anisotropy $Δ= J+μ/2$ (with $|μ| \ll J$) favoring the perpendicular planes. For a precise analysis, we study the $T/J \rightarrow 0$ limit in which $w \equiv \exp(-μ/T)$ is the control variable. In this limit, we find {\em two topologically distinct} zero-field Coulomb phases separated by a first-order $Z_3$ confinement transition at $w_c \approx 2.02$. Both Coulomb phases admit a description in terms of the fluctuations of a coarse-grained divergence-free polarization field. However, the flux of this polarization field is restricted to integer multiples of $3$, and only charges that are multiples of 3 are deconfined in one of these phases, while all integer fluxes are allowed and all integer charges are deconfined in the other phase. Experimental systems with small negative $μ$ ({\em i.e.}, $-J \ll μ< 0$) are therefore predicted to exhibit signatures of this topological transition when cooled below $T_c \approx 1.42|μ|$.

• The paper demonstrates that competing easy-axis exchange and easy-plane anisotropy produce two topologically distinct Coulomb liquids, separated by a first-order transition at w_c ≈ 2.02.
• Monte Carlo simulations show that the low-w phase deconfines unit charges and permits integer fluxes, while the high-w phase restricts flux to multiples of three and deconfines only charge-3 defects.
• The transition arises from percolation of ±1/2-spin clusters and could appear experimentally as a specific-heat anomaly near T_c ≈ 1.42|μ| despite nearly identical neutron-scattering pinch points.

Overview

Pandey, Kundu, and Damle study a class of pyrochlore magnets in which spin-orbit coupling and crystal-field effects leave two low-lying Kramers doublets, so that each magnetic ion is well modeled as an effective spin $S=3/2$ degree of freedom. The dominant interaction is an easy-axis antiferromagnetic Ising exchange $J>0$ along the local $[111]$ axes, which competes with a comparably strong easy-plane single-ion anisotropy $\Delta = J + \mu/2$ with $|\mu| \ll J$. The central result is that this competition produces two topologically distinct classical Coulomb phases separated by a first-order transition at $w_c \approx 2.02$, where $w = e^{-\mu/T}$ is the control parameter. Both phases exhibit the pinch-point structure factors characteristic of Coulomb spin liquids, but they differ in their emergent gauge structure: one admits all integer fluxes of the polarization field (with all integer charges deconfined), while the other restricts flux to multiples of 3 (with only charges that are multiples of 3 deconfined). The paper thus identifies a thermodynamic $Z_3$ confinement transition between two phases that are indistinguishable by conventional neutron-scattering signatures.

Effective model

The effective Hamiltonian is

$$H = \sum_{\langle r,r'\rangle}\left(J_\parallel S_r^z S_{r'}^z + J_{xy}(S_r^x S_{r'}^x + S_r^y S_{r'}^y)\right) + \Delta \sum_r (S_r^z)^2,$$

with $J_\parallel = J > 0$, $J>0$0, and the hierarchy $J>0$1. In the window $J>0$2, quantum fluctuations are negligible and only minimally frustrated configurations — those with zero total $J>0$3 on every tetrahedron — contribute appreciably to the partition function. Mapping spins onto a polarization field $J>0$4 on the bonds of the parent diamond lattice, the ice rule becomes the divergence-free condition $J>0$5, and each $J>0$6 spin carries a relative Boltzmann weight $J>0$7. The resulting low-energy theory is a 44-vertex model on the diamond lattice,

$J>0$8

where $J>0$9 counts the number of $[111]$0 spins. A naive coarse-graining argument would suggest that the microscopic allowed values of $[111]$1 are irrelevant, so that small-$[111]$2 and large-$[111]$3 physics should be continuously connected and essentially identical to pseudo-spin-1/2 spin ice. The paper's key claim is that this expectation fails: the vertex weights encode a topological distinction that survives coarse-graining.

Numerical evidence for two Coulomb phases

The phase diagram is mapped using a worm-algorithm Monte Carlo scheme on $[111]$4 periodic samples, with worm head-to-tail distance histograms serving as estimators of test-charge correlators $[111]$5. Three lines of evidence establish the two-phase structure:

Thermodynamic transition. The density $[111]$6 exhibits a size-independent jump at $[111]$7, and its histogram shows a clear two-peak structure at coexistence, establishing a first-order transition with sizeable latent heat.

Topological flux restriction. With periodic boundary conditions, the polarization flux $[111]$8 through any measurement surface is an integer three-vector invariant under local updates. The probability $[111]$9 of nonzero $\Delta = J + \mu/2$0 flux (some component not divisible by 3) vanishes at large $\Delta = J + \mu/2$1 for $\Delta = J + \mu/2$2 but remains finite for $\Delta = J + \mu/2$3. In both phases $\Delta = J + \mu/2$4 is Gaussian with diamond-lattice symmetry, as expected for a Coulomb liquid; the distinction is that only zero-$\Delta = J + \mu/2$5-flux sectors survive the thermodynamic limit at large $\Delta = J + \mu/2$6.

Charge deconfinement. The unit-charge worm histogram $\Delta = J + \mu/2$7 saturates to a nonzero constant at large separation for $\Delta = J + \mu/2$8 but decays rapidly for $\Delta = J + \mu/2$9, while the charge-3 histogram behaves oppositely. Hence unit charges are deconfined only in the small-$|\mu| \ll J$0 phase, whereas charge-3 defects are deconfined in both. This is direct evidence that the transition is a $|\mu| \ll J$1 confinement transition. Via the standard current-loop duality to a fictitious 3D XY model, the large-$|\mu| \ll J$2 phase corresponds to long-range order of $|\mu| \ll J$3 but short-range order of $|\mu| \ll J$4 — a $|\mu| \ll J$5 "nematic" — while the small-$|\mu| \ll J$6 phase has $|\mu| \ll J$7 itself long-range ordered, consistent with the dual $|\mu| \ll J$8 gauge description.

Notably, the spin-flip structure factor shows dipolar pinch points in both phases, differing only in overall intensity. The implication for experiment is sharp: neutron scattering cannot distinguish the two phases, even though they are topologically distinct.

Microscopic mechanism: percolation

The appendix identifies the geometric driver of the first-order transition: connected clusters of $|\mu| \ll J$9 spins undergo a weakly first-order percolation transition near $w_c \approx 2.02$0. Multiple diagnostics support this — the torus-winding probability $w_c \approx 2.02$1, the ratio of second-largest to largest cluster masses, and the scaling of the largest cluster mass ($w_c \approx 2.02$2 for $w_c \approx 2.02$3, $w_c \approx 2.02$4 for $w_c \approx 2.02$5) — while clusters of $w_c \approx 2.02$6 spins percolate on both sides and play no role. At criticality, the cluster-mass histogram approximately follows a finite-size scaling form $w_c \approx 2.02$7 with $w_c \approx 2.02$8 and $w_c \approx 2.02$9, mimicking second-order behavior at accessible sizes despite the underlying weak first-order character. The charge-2 worm histogram is confined for $w = e^{-\mu/T}$0, consistent with — though not conclusive proof of — confinement of $w = e^{-\mu/T}$1 in the $w = e^{-\mu/T}$2-confined phase.

Physical implications

For materials with small negative $w = e^{-\mu/T}$3 ($w = e^{-\mu/T}$4), cooling below $w = e^{-\mu/T}$5 drives the system from the $w = e^{-\mu/T}$6-deconfined to the $w = e^{-\mu/T}$7-confined Coulomb liquid, and the first-order transition should be visible in specific heat measurements. Because the distinction between the phases is topological, it is expected to survive the quantum fluctuations induced by small transverse exchanges $w = e^{-\mu/T}$8, implying two distinct quantum Coulomb ground states whose transition maps to the confinement transition of a 3+1-dimensional quantum $w = e^{-\mu/T}$9 gauge theory. Related work on anisotropic $Z_3$0 magnets has identified analogous $Z_3$1 flux-confinement transitions, suggesting a broader family of such phenomena arising from competing anisotropies; parallel work also reports sign-dependent slow quench dynamics that could serve as an experimental discriminator between the two equilibrium phases found here.

Limitations and open questions

Several caveats bear directly on these results. The analysis is confined to the strict limit $Z_3$2 with $Z_3$3; the stability of the two phases against realistic transverse couplings is argued but not demonstrated computationally. The identification of materials realizing the required hierarchy — two low-lying doublets with $Z_3$4 and $Z_3$5 — remains open, and the authors call for microscopic tight-binding studies of spin-orbit-coupled candidates. The confinement of charge-2 test charges in the large-$Z_3$6 phase is inferred rather than established, since short-ranged $Z_3$7 does not rigorously imply confinement of $Z_3$8. Finally, the percolation transition underlying the thermodynamic transition displays finite-size scaling resembling a continuous critical point, leaving the precise universality character of the weakly first-order transition incompletely resolved at accessible system sizes.

Conclusion

This paper demonstrates that effective $Z_3$9 pyrochlore magnets with competing easy-axis exchange and easy-plane single-ion anisotropy host two topologically distinct classical Coulomb liquids — one with unrestricted integer fluxes and deconfined unit charges, the other with flux restricted to multiples of 3 — separated by a first-order $$H = \sum_{\langle r,r'\rangle}\left(J_\parallel S_r^z S_{r'}^z + J_{xy}(S_r^x S_{r'}^x + S_r^y S_{r'}^y)\right) + \Delta \sum_r (S_r^z)^2,$$0 confinement transition at $$H = \sum_{\langle r,r'\rangle}\left(J_\parallel S_r^z S_{r'}^z + J_{xy}(S_r^x S_{r'}^x + S_r^y S_{r'}^y)\right) + \Delta \sum_r (S_r^z)^2,$$1, driven microscopically by a percolation transition of $$H = \sum_{\langle r,r'\rangle}\left(J_\parallel S_r^z S_{r'}^z + J_{xy}(S_r^x S_{r'}^x + S_r^y S_{r'}^y)\right) + \Delta \sum_r (S_r^z)^2,$$2 spin clusters. Since the two phases share qualitatively identical pinch-point structure factors, the result predicts a thermodynamic signature (specific heat anomaly near $$H = \sum_{\langle r,r'\rangle}\left(J_\parallel S_r^z S_{r'}^z + J_{xy}(S_r^x S_{r'}^x + S_r^y S_{r'}^y)\right) + \Delta \sum_r (S_r^z)^2,$$3) without a concomitant change in the equilibrium neutron-scattering pattern, posing a concrete experimental challenge for candidate rare-earth pyrochlores.