The ground-states of the spin-S antiferromagnetic chain \(H_{{\text {AF}}}\) with a projection-based interaction and the spin-1/2 XXZ-chain \( H_{{\text {XXZ}}}\) 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 \(H_{{\text {AF}}}\) at all \(S> 1/2\), and Néel order for \( H_{{\text {XXZ}}} \) 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.

中文翻译：

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通过共享循环表示，不同量子自旋链中的二聚化和Néel阶。

自旋S反铁磁链\（H _ {{\ text {AF}}} \\）的基态具有基于投影的相互作用，并且自旋1/2 XXZ链 \（H _ {{\ text {XXZ }}} \）在各向异性参数\（\德尔塔= \ COSH（\拉姆达）\）份额的二维函数积分换算的公共环路表示，它类似于经典的平面Q在-state Potts模型\（ \ sqrt {Q} = 2S + 1 = 2 \ cosh（\ lambda）\）。在多方面的关系在这里用来直接关系其表现在这两款机型的地面国平移对称性破的不同的形式：二聚化\（H _ {{\文本{AF}}} \）的所有\（S > 1/2 \）和奈尔为了\（H _ {{\文本{XXZ}}} \）在\（\拉姆达> 0 \） 。提出的结果包括：（i）将上述结果转化为上述量子自旋系统，最近，Duminil-Copin-Li-Manolescu已针对广泛的二维随机集群模型证明了这一点，并且（ii）简短关于对称性破坏的证明，类似于最近由Ray–Spinka提出的关于\（Q> 4 \）的相变不连续性的结构证明。总而言之，\（Q = 4 \）和\（Q> 4 \）之间的变化的量子表现是从无间隙基态到一对有间隙的且广泛不同的基态的过渡。