$Z_3$ 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 $\Delta = J+\mu/2$ (with $|\mu| \ll J$) favoring the perpendicular planes. For a precise analysis, we study the $T/J \rightarrow 0$ limit in which $w \equiv \exp(-\mu/T)$ 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 $\mu$ ({\em i.e.}, $-J \ll \mu < 0$) are therefore predicted to exhibit signatures of this topological transition when cooled below $T_c \approx 1.42|\mu|$.