Planon-modular fracton orders

There are now many examples of gapped fracton models which are defined by the presence of restricted-mobility excitations above the quantum ground state. However, the theory of fracton orders remains in its early stages, and the complex landscape of examples is far from being mapped out. Here we introduce the class of planon-modular (𝑝-modular) fracton orders, a relatively simple yet still rich class of quantum orders that encompasses several well-known examples of type I fracton order. The defining property is that any nontrivial pointlike excitation can be detected by braiding with planons. From this definition, we uncover a significant amount of general structure, including the assignment of a natural number (dubbed the weight) to each excitation of a 𝑝-modular fracton order. We identify simple new phase invariants, some of which are based on weight, which can easily be used to compare and distinguish different fracton orders. We also study entanglement renormalization group (RG) flows of 𝑝-modular fracton orders, establishing a close connection with foliated RG. We illustrate our general results with an analysis of several exactly solvable fracton models that we show to realize 𝑝-modular fracton orders, including ℤ𝑛 versions of the X-cube, anisotropic, checkerboard, 4-planar X-cube, and four color cube (FCC) models. We show that each of these models is 𝑝-modular and compute its phase invariants. We also show that each example admits a foliated RG at the level of its nontrivial excitations, which is a new result for the 4-planar X-cube and FCC models. We show that the ℤ₂ FCC model is not a stack of other better-studied models but predict that the ℤ𝑛 FCC model with 𝑛 odd is a stack of ten 4-planar X-cubes, possibly plus decoupled layers of two-dimensional toric code. We also show that the ℤ𝑛 checkerboard model for 𝑛 odd is a stack of three anisotropic models.