Dimerization and Néel Order in Different Quantum Spin Chains Through a Shared Loop Representation

The ground-states of the spin-S antiferromagnetic chain [Formula: see text] with a projection-based interaction and the spin-1/2 XXZ-chain [Formula: see text] at anisotropy parameter [Formula: see text] 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 [Formula: see text] . 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 [Formula: see text] at all [Formula: see text] , and Néel order for [Formula: see text] at [Formula: see text] . 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 [Formula: see text] . Altogether, the quantum manifestation of the change between [Formula: see text] and [Formula: see text] is a transition from a gapless ground-state to a pair of gapped and extensively distinct ground-states.