Dimerization and Néel order in different quantum spin chains through a shared loop representation

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 $Δ=\cosh(λ)$ 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 $\sqrt Q= 2S+1 =2\cosh(λ)$. 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 $λ>0$. 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.

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