In this paper, we explore the interplay between symmetry and fracton order, motivated by the analogous close relationship for topologically ordered systems. Specifically, we consider models with 3D planar subsystem symmetry, and show that these can realize subsystem symmetry protected topological phases with gapless boundary modes. Gauging the planar subsystem symmetry leads to a fracton order in which particles restricted to move along lines exhibit a new type of statistical interaction that is specific to the lattice geometry. We show that both the gapless boundary modes of the ungauged theory, and the statistical interactions after gauging, are naturally captured by a higher-rank version of Chern-Simons theory. We also show that gauging only part of the subsystem symmetry can lead to symmetry-enriched fracton orders, with quasiparticles carrying fractional symmetry charge.

中文翻译：

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对称分形物质：扭曲和丰富

在本文中，我们探讨了对称性和分形序之间的相互作用，其动机是拓扑有序系统的类似密切关系。具体来说，我们考虑具有 3D 平面子系统对称性的模型，并表明这些模型可以实现具有无间隙边界模式的子系统对称保护拓扑相。测量平面子系统的对称性会导致分形阶，其中被限制为沿线移动的粒子表现出一种新型的统计相互作用，这种相互作用是特定于晶格几何的。我们表明，非测量理论的无间隙边界模式和测量后的统计相互作用，都自然地被陈 - 西蒙斯理论的更高等级版本所捕获。我们还表明，仅测量子系统对称性的一部分可以导致对称性丰富的分形阶数，