A central tenant in the classification of phases is that boundary conditions cannot affect the bulk properties of a system. In this work, we show striking, yet puzzling, evidence of a clear violation of this assumption. We use the prototypical example of an XYZ chain with no external field in a ring geometry with an odd number of sites and both ferromagnetic and antiferromagnetic interactions. In such a setting, even at finite sizes, we are able to calculate directly the spontaneous magnetizations that are traditionally used as order parameters to characterize the system's phases. When ferromagnetic interactions dominate, we recover magnetizations that in the thermodynamic limit lose any knowledge about the boundary conditions and are in complete agreement with standard expectations. On the contrary, when the system is governed by antiferromagnetic interactions, the magnetizations decay algebraically to zero with the system size and are not staggered, despite the AFM coupling. We term this behavior ferromagnetic mesoscopic magnetization. Hence, in the antiferromagnetic regime, our results show an unexpected dependence of a local, one--spin expectation values on the boundary conditions, which is in contrast with predictions from the general theory.

中文翻译：

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奇怪的挫败感：边界条件如何破坏局部秩序

相分类的一个核心特征是边界条件不能影响系统的整体特性。在这项工作中，我们展示了明显违反这一假设的惊人但令人费解的证据。我们使用 XYZ 链的原型示例，该链在具有奇数位点以及铁磁和反铁磁相互作用的环形几何结构中没有外场。在这样的设置中，即使在有限的尺寸下，我们也能够直接计算自发磁化强度，这些自发磁化强度传统上用作表征系统相位的顺序参数。当铁磁相互作用占主导地位时，我们恢复了在热力学极限内失去关于边界条件的任何知识并且与标准预期完全一致的磁化强度。相反，当系统受反铁磁相互作用控制时，尽管存在 AFM 耦合，但磁化强度随系统尺寸代数衰减为零，并且不会交错。我们将这种行为称为铁磁介观磁化。因此，在反铁磁体系中，我们的结果显示了局部单自旋期望值对边界条件的意外依赖性，这与一般理论的预测相反。