Dimerization and N\'eel order in different quantum spin chains through a shared loop representation

The ground states of the spin-$S$ antiferromagnetic chain $H_{\text{AF}}$ with a projection-based interaction and the spin-$1\mathrm{/}2$ XXZ-chain $H_{\text{XXZ}}$ at anisotropy parameter $\mathrm{\Delta }=\cosh (\lambda )$ share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planar $Q$-state Potts model at $ \sqrt Q= 2S+1 =2\cosh(\lambda)$. The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the ground states of these two models: dimerization for $H_{\text{AF}}$ at all $S>1\mathrm{/}2$, and N'eel order for $H_{\text{XXZ}}$ at $\lambda >0$. The results presented include: i) a translation to the above quantum spin systems of the results which were recently proven by Duminil-Copin-Li-Manolescu for a broad class of two-dimensional random-cluster models, and ii) a short proof of the symmetry breaking in a manner similar to the recent structural proof by Ray-Spinka of the discontinuity of the phase transition for $Q>4$. Altogether, the quantum manifestation of the change between $Q=4$ and $Q>4$ is a transition from a gapless ground state to a pair of gapped and extensively distinct ground states.