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 $w\to 0$ O(1) loop-model limit is separated from the $w\to \infty$ 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 distinct Coulomb liquid phases of the system: Both phases admit a description in terms of a fluctuating divergence-free polarization field $P_{\mu }\left(\mathbf{r}\right)$ on links of the lattice and are characterized by dipolar correlations at long distances. Away from the critical point, both the phases are described by an effective Gaussian action in terms of the coarse-grained analog of the microscopic polarization field $P_{\mu }\left(\mathbf{r}\right)$. The transition at $w_c$ is a flux confinement-deconfinement transition: For periodic samples, half-integer fluxes (along each periodic direction) of the polarization field proliferate in the $w<w_c$ phase, while being confined to integer values in the $w>w_c$ phase. 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.

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• Spin liquid
• Topological phase transition