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 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_\textrm{AF} at all S>1/2, and Néel order for H_\textrm{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.
