AI 中文总结
该研究证明一类特殊U(1)对称自旋1链及周期Motzkin链不存在特定范围的局域守恒量，明确其连续对称性破缺源于无挫败结构而非守恒序参量。
AI 中文摘要
我们证明了一类具有U(1)对称性、近邻相互作用且部分四极耦合消失的自旋1链中不存在非平凡的局域守恒量，这类链并未被以往的研究覆盖。将Shiraishi的技术应用于这些系统，我们证明，对于周期边界条件下N个格点的这类链中的每一个模型，不存在满足3≤k≤N/2的k-局域守恒量。特别地，对于一维零温下表现出自发U(1)对称性破缺的无挫败自旋1链，我们证明，所有支撑范围不超过系统尺寸一半的局域守恒量，都是单位矩阵、总磁化强度S^z和哈密顿量本身的线性组合。这严格证实，与海森堡铁磁体不同，该模型不存在与哈密顿量对易的局域序参量，因此其连续对称性破缺由无挫败结构而非守恒序参量实现。我们还证明，周期Motzkin链（一类与原始Motzkin链密切相关的无挫败自旋1链，已有研究报道其在零温下存在自发U(1)对称性破缺）中不存在满足3≤k≤N/2的k-局域守恒量。
英文摘要
We prove the absence of nontrivial local conserved quantities in a class of $U(1)$-symmetric spin-$1$ chains with nearest-neighbor interactions in which some of the quadrupolar couplings vanish, a class that is not covered by previous studies. Applying the technique of Shiraishi to these systems, we show that, for every model in this class on a periodic chain of $N$ sites, there is no $k$-local conserved quantity for any $3\le k\le N/2$. In particular, for a frustration-free spin-$1$ chain that exhibits spontaneous $U(1)$ symmetry breaking at zero temperature in one spatial dimension, we prove that every local conserved quantity with support up to half of the system size is a linear combination of the identity, the total magnetization $S^z$, and the Hamiltonian itself. This rigorously establishes that, unlike the Heisenberg ferromagnet, the model admits no local order parameter commuting with the Hamiltonian, so that its continuous symmetry breaking is enabled by the frustration-free structure rather than by a conserved order parameter. We also prove the absence of $k$-local conserved quantities for $3\le k\le N/2$ in the periodic Motzkin chain, a frustration-free spin-$1$ chain closely related to the original Motzkin chain, for which spontaneous $U(1)$ symmetry breaking at zero temperature has also been reported.
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