标题： 长程纠缠来自自发非局域对称性破缺 显示英文标题

标题： Long-range entanglement from spontaneous non-onsite symmetry breaking

摘要： 我们探索了由$\mathbb{Z}_2$非在线对称性自发对称性破缺（SSB）产生的物质状态。 在一维空间中，我们构建了一个无挫折的格点模型，表现出非在线对称性的SSB，该模型具有两个具有不同对称性保护拓扑（SPT）序的基态共存。 我们解析证明了两重基态简并和有限能隙的存在。 固定对称性子空间会产生一个具有非可逆电荷算符之间长程关联的长程纠缠基态。 我们还提出了一种常深度测量反馈协议，在热力学极限下以恒定的成功概率制备这种状态，这可能具有独立的兴趣。 在对称变形下，SSB持续到临界点，超过该点后出现由共形场理论表征的无能隙相。 在二维空间中，1形式非在线对称性的SSB导致了一个长程纠缠态（SPT汤）——任何闭合环路中1d SPT的凝聚。 在环面上，有四个这样的局部不可区分的状态，它们表现出局部算符之间的代数关联，我们通过映射到临界$O(2)$环模型推导出了这些结果。 这提供了一个引人入胜的“拓扑量子临界性”示例。 我们的工作揭示了非在线对称性SSB的奇特特性，这些特性可能超出了拓扑全息（SymTFT）的框架。 显示英文摘要

摘要： We explore the states of matter arising from the spontaneous symmetry breaking (SSB) of $\mathbb{Z}_2$ non-onsite symmetries. In one spatial dimension, we construct a frustration-free lattice model exhibiting SSB of a non-onsite symmetry, which features the coexistence of two ground states with distinct symmetry-protected topological (SPT) orders. We analytically prove the two-fold ground-state degeneracy and the existence of a finite energy gap. Fixing the symmetry sector yields a long-range entangled ground state that features long-range correlations among non-invertible charged operators. We also present a constant-depth measurement-feedback protocol to prepare such a state with a constant success probability in the thermodynamic limit, which may be of independent interest. Under a symmetric deformation, the SSB persists up to a critical point, beyond which a gapless phase characterized by a conformal field theory emerges. In two spatial dimensions, the SSB of 1-form non-onsite symmetries leads to a long-range entangled state (SPT soup) - a condensate of 1d SPT along any closed loops. On a torus, there are four such locally indistinguishable states that exhibit algebraic correlations between local operators, which we derived via a mapping to the critical $O(2)$ loop model. This provides an intriguing example of `topological quantum criticality'. Our work reveals the exotic features of SSB of non-onsite symmetries, which may lie beyond the framework of topological holography (SymTFT).