Fracton models, a collection of exotic gapped lattice Hamiltonians recently discovered in three spatial dimensions, contain some `topological' features: they support fractional bulk excitations (dubbed fractons), and a ground state degeneracy that is robust to local perturbations. However, because previous fracton models have only been defined and analyzed on a cubic lattice with periodic boundary conditions, it is unclear to what extent a notion of topology is applicable. In this paper, we demonstrate that the $X$-cube model, a prototypical type-I fracton model, can be defined on general three-dimensional manifolds. Our construction revolves around the notion of a singular compact total foliation of the spatial manifold, which constructs a lattice from intersecting stacks of parallel surfaces called leaves. We find that the ground state degeneracy depends on the topology of the leaves and the pattern of leaf intersections. We further show that such a dependence can be understood from a renormalization group transformation for the X-cube model, wherein the system size can be changed by adding or removing 2D layers of topological states. Our results lead to an improved definition of fracton phase and bring to the fore the topological nature of fracton orders.

中文翻译：

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通用三维流形上的Fracton模型

Fracton模型是一组最近在三个空间维度上发现的奇异带隙哈密顿量的集合，它们具有一些“拓扑”特征：它们支持分数整体激发（配音的分数），以及对局部扰动具有鲁棒性的基态简并性。但是，由于以前的fracton模型仅在具有周期性边界条件的立方晶格上定义和分析，因此尚不清楚拓扑概念在多大程度上适用。在本文中，我们证明了可以在常规三维流形上定义$ X $ -cube模型（一种典型的I型fracton模型）。我们的构造围绕着空间流形的奇异紧凑全叶的概念，该叶面是由相交的平行表面（称为叶）的相交堆栈构造而成的。我们发现基态的简并性取决于叶子的拓扑和叶子相交的模式。我们进一步表明，可以从X多维数据集模型的重归一化组转换中了解这种依赖性，其中，可以通过添加或删除2D拓扑状态层来更改系统大小。我们的结果导致对分数相的定义有了改进，并且使分数阶的拓扑性质突显出来。