Flux confinement-deconfinement transition of dimer-loop models on three-dimensional bipartite lattices

Motivated by recent work that mapped the low-temperature properties of a class of frustrated spin $S=1$ kagome antiferromagnets with competing exchange and single-ion anisotropies to the fully-packed limit (with each vertex touched by exactly one dimer or nontrivial loop) of a system of dimers and nontrivial (length $s > 2$) loops on the honeycomb lattice, we study this fully-packed dimer-loop model on the three-dimensional bipartite cubic and diamond lattices as a function of $w$, the relative fugacity of dimers. We find that the $w \rightarrow 0$ O($1$) loop-model limit is separated from the $w \rightarrow \infty$ dimer limit by a geometric phase transition at a nonzero finite critical fugacity $w_c$: The $w>w_c$ phase has short loops with an exponentially decaying loop-size distribution, while the $w<w_c$ phase is dominated by large loops whose loop-size distribution is governed by universal properties of the critical O($1$) loop soup. This transition separates two {\em distinct} Coulomb liquid phases of the system: Both phases admit a description in terms of a fluctuating divergence-free polarization field $P_{\mu}(\mathbf{r})$ on links of the lattice and are characterized by dipolar correlations at long distances. The transition at $w_c$ is a flux confinement-deconfinement transition. Equivalently, and independent of boundary conditions, half-integer test charges $q=\pm 1/2$ are confined for $w>w_c$, but become deconfined in the small-$w$ phase. Although both phases are unstable to a nonzero fugacity for the charge $\pm 1/2$ excitations, the destruction of the $w >w_c$ Coulomb liquid is characterized by an interesting slow crossover, since test charges with $q=\pm 1/2$ are confined in this phase.