Flux fractionalization transition in anisotropic $S=1$ antiferromagnets and dimer-loop models

We demonstrate that the low temperature ($T$) properties of a class of anisotropic spin $S=1$ kagome (planar pyrochlore) antiferromagnets on a field-induced $\frac{1}{3}$-magnetization ($\frac{1}{2}$-magnetization) plateau are described by a model of fully-packed dimers and loops on the honeycomb (square) lattice, with a temperature-dependent relative fugacity $w(T)$ for the dimers. The fully-packed O(1) loop model ($w=0$) and the fully-packed dimer model ($w=\infty$) limits of this dimer-loop model are found to be separated by a phase transition at a finite and nonzero critical fugacity $w_c$, with interesting consequences for the spin correlations of the frustrated magnet. The $w>w_c$ phase has short loops and spin correlations dominated by power-law columnar order (with subdominant dipolar correlations), while the $w<w_c$ phase has dominant dipolar spin correlations and long loops governed by a power-law distribution of loop sizes. Away from $w_c$, both phases are described by a long-wavelength Gaussian effective action for a scalar height field that represents the coarse-grained electrostatic potential of fluctuating dipoles. The destruction of power-law columnar spin order below $w_c$ is driven by an unusual {\em flux fractionalization} mechanism, topological in character but quite distinct from the usual Kosterlitz-Thouless mechanism for such transitions: Fractional electric fluxes which are bound into integer values for $w>w_c$, proliferate in the $w<w_c$ phase and destroy power-law columnar order.