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The ground states of the spin-$ S $ antiferromagnetic chain $H_\textrm{AF}$with a projection-based interaction and the spin-$ 1/2$ XXZ-chain $H_\textrm{XXZ} $ at anisotropy parameter $\Delta=\cosh(\lambda) $ share acommon loop representation in terms of a two-dimensional functional integralwhich 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 directlyrelate the distinct forms of translation symmetry breaking which are manifestedin the ground states of these two models: dimerization for $H_\textrm{AF}$ atall $S> 1/2$, and N\'eel order for $ H_\textrm{XXZ} $ at $\lambda >0$. Theresults presented include: i) a translation to the above quantum spin systemsof the results which were recently proven by Duminil-Copin-Li-Manolescu for abroad class of two-dimensional random-cluster models, and ii) a short proof ofthe symmetry breaking in a manner similar to the recent structural proof byRay-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 transitionfrom a gapless ground state to a pair of gapped and extensively distinct groundstates.

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