超越新生梦想：三维晶格模型中矩阵元胞自动机产生的经典分形自旋液体
Beyond the Freshman's Dream: Classical fractal spin liquids from matrix cellular automata in three-dimensional lattice models
AI总结：
本研究在两种三维晶格上构建经典分形自旋液体模型，发现其无法用标量元胞自动机描述，通过关联矩阵元胞自动机揭示新类别分形对称性，构建低能分形子激发并证实对数能垒与玻璃态动力学，为分形相关研究提供新途径。
摘要：
我们在两种共享顶点的三角形构成的现实三维（3D）晶格——三叶晶格（trillium）和超超笼目晶格（hyperhyperkagome, HHK）上，构建了承载经典分形自旋液体的模型。两种模型都包含与二维三角晶格上的Newman-Moore（NM）模型形式相同的三角格块三自旋伊辛相互作用。然而，与NM模型及其三维推广不同，它们的简并基态和低能激发无法用标量元胞自动机（CA）描述，因为对应的分形结构缺少一种常被称为“新生梦想”的简化代数性质。通过识别与矩阵元胞自动机（matrix CAs）的关联——该关联本质上利用了晶体结构——我们证明两种模型都展现出与NM型模型不同类别的分形对称性。我们设计了一种方法，通过翻转任意大的自相似自旋子集，显式构建由有限组不动缺陷即“分形子（fractons）”构成的低能激发，并解析计算了这些激发的分形维数。我们证明这些激发对应的能垒随系统尺寸对数增长，会导致“脆弱”的玻璃态动力学，我们通过经典蒙特卡洛模拟证实了其存在。我们还讨论了引入横向磁场带来的量子涨落时，自发分形对称破缺的相关结果，并提出了用于探测这类相变的多自旋关联函数诊断方法。我们的研究结果表明，矩阵元胞自动机可为识别晶格模型中的分形对称性和类分形子行为提供有效途径，对分形子拓扑序研究具有潜在意义。
英文摘要：
We construct models hosting classical fractal spin liquids on two realistic three-dimensional (3D) lattices of corner-sharing triangles: trillium and hyperhyperkagome (HHK). Both models involve the same form of three-spin Ising interactions on triangular plaquettes as the Newman-Moore (NM) model on the 2D triangular lattice. However, in contrast to the NM model and its 3D generalizations, their degenerate ground states and low-lying excitations cannot be described in terms of scalar cellular automata (CA), because the corresponding fractal structures lack a simplifying algebraic property, often termed the 'Freshman's dream'. By identifying a link to matrix CAs -- that makes essential use of the crystallographic structure -- we show that both models exhibit fractal symmetries of a distinct class to the NM-type models. We devise a procedure to explicitly construct low-energy excitations consisting of finite sets of immobile defects or "fractons", by flipping arbitrarily large self-similar subsets of spins, whose fractal dimensions we compute analytically. We show that these excitations are associated with energetic barriers which increase logarithmically with system size, leading to "fragile" glassy dynamics, whose existence we confirm via classical Monte Carlo simulations. We also discuss consequences for spontaneous fractal symmetry breaking when quantum fluctuations are introduced by a transverse magnetic field, and propose multi-spin correlation function diagnostics for such transitions. Our findings suggest that matrix CAs may provide a fruitful route to identifying fractal symmetries and fracton-like behaviour in lattice models, with possible implications for the study of fracton topological order.
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