  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 N\'eel 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.
