We study models with fractonlike order based on ${\mathbb{Z}}_2$ lattice gauge theories with subsystem symmetries in two and three spatial dimensions. The three-dimensional (3D) model reduces to the 3D toric code when subsystem symmetry is broken, giving an example of a subsystem symmetry enriched topological phase. Although not topologically protected, its ground-state degeneracy has as leading contribution a term which grows exponentially with the square of the linear size of the system. Also, there are completely mobile gauge charges living along with immobile fractons. Our method shows that fractonlike phases are also present in more usual lattice gauge theories. We calculate the entanglement entropy $S_A$ of these models in a subregion $A$ of the lattice and show that it is equal to the logarithm of the ground-state degeneracy of a particular restriction of the full model to $A$.

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来自子系统对称性的类分数维相

我们研究基于 ${\mathbb{ž}}_2$在两个和三个空间维度上具有子系统对称性的晶格规范理论。当子系统对称性破坏时，三维（3D）模型简化为3D复曲面代码，给出了子系统对称性丰富的拓扑阶段的示例。尽管不受拓扑保护，但其基态简并性作为一个主要贡献，该术语与系统线性大小的平方成指数增长。此外，还有完全可移动的标准装药和不可移动的氟利昂。我们的方法表明，在更常见的晶格规理论中也存在类分形相。我们计算纠缠熵${\mathrm{小号}}_{\mathrm{一种}}$ 一个分区中的这些模型 $\mathrm{一种}$ 证明它等于对整个模型的特定限制的基态简并性的对数 $\mathrm{一种}$。