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

The ground-states of the spin-S antiferromagnetic chain H-AF with a projection-based interaction and the spin-1/2 XXZ-chain H-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 root 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-AF at all S > 1/2, and N ' eel order for H-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.