$Z_3$ 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 $T_c \approx 1.42|μ|$.

Figures (5)

• Figure 1: Left: Bravais lattice translations $\vec{a}_{1,2,3}$ of the pyrochlore lattice whose sites lie at the centers of the bonds of a diamond lattice. Right: The mapping from spin $S^z$ variables to the lattice polarization field $\vec{P}$ on bonds of the diamond lattice.
• Figure 2: (a) $\rho_{3/2} = \epsilon/\mu$, the energy density in units of $\mu$, has a clear jump at $w\approx 2.02$, corresponding to a first-order transition with a latent heat. (b) Its histogram shows a clear two-peak structure in the vicinity of this transition. (c) The spin-flip structure factor in both phases has pinch-point singularities characteristic of a Coulomb liquid.
• Figure 3: (a) The probability $P_{Z_3}$ of having a nonzero $Z_3$ flux, i.e., of the flux vector $\vec{\phi}$ having a component not divisible by $3$, tends to zero (remains nonzero) at large $L$ for $w>w_c$ ($w<w_c$). (b) The histogram $h^{(1)}(\vec{r})$ of head-to-tail distances of unit-charge worms goes to a nonzero constant at large $r$ for $w<w_c$, but decays rapidly to zero for $w>w_c$, while the corresponding $h^{(3)}(\vec{r})$ for charge $3$ worms goes to a nonzero constant at large $r$ for $w>w_c$. (c) The probability distribution $P(\vec{\phi})$ of $\vec{\phi}$ is a Gaussian for both $w>w_c$ and $w<w_c$, but only $\vec{\phi}$ with all components divisible by $3$ survive at large $L$ and follow the Gaussian for $w>w_c$.
• Figure 4: (a) The probability $P_{\rm cross}^{(1/2)}$ of having a connected cluster of spins with $S^z=\pm 1/2$ shows clear evidence of a percolation transition: it tends to zero at large $L$ for $w>w_c$, while being nonzero in this limit for $w<w_c$. The corresponding probability for connected clusters of spins with $S^z=\pm 3/2$ is featureless and such clusters play no role in this transition. (b) The ratio of the masses of the second largest and largest connected clusters of spins with $S^z = \pm 1/2$ also shows a clear indication of this percolation transition, tending to zero at large $L$ for $w<w_c$ while remaining nonzero in this limit for $w>w_c$. Again, the corresponding quantity for $S^z = \pm 3/2$ clusters shows no signs of this transition. (c) The mass of the largest connected cluster of spins with $S^z = \pm 1/2$ scales with $L^3$ for $w<w_c$, but remains $O(1)$ in the large $L$ limit for $w>w_c$, consistent with the other indications of a percolation transition. The corresponding quantity for spins with $S^z = \pm 3/2$ scales with $L^3$ on both sides of $w_c$.
• Figure 5: (a) For the sizes accessible to our numerics, the distribution of masses of clusters of spins with $S^z = \pm 1/2$ approximately follows a power-law form at criticality. (b) Indeed, this distribution approximately obeys the finite-size scaling form discussed in the Appendix. (c) The histogram $h^{(2)}(\vec{r})$ of head-to-tail displacements of charge-$2$ worms decays rapidly to zero at large $r$ for $w>w_c$. While this does not establish confinement of the corresponding test charge correlator $C^{(2)}(\vec{r})$, it is consistent with this.