Beyond Topological Order and Back Again: Foliated TQFT of Fractons

Abstract

Fracton order is a new kind of phase of matter which is similar to topological order, except its excitations have mobility constraints. The excitations are bound to various n-dimensional surfaces with exotic fusion rules that determine how excitations on intersecting surfaces can combine.

This talk will focus on one of the simplest fracton models: the X-cube model. I will explain how this fracton model can be thought of as starting from a 3D toric code and stacks of 2D toric code layers, and then condensing certain excitations on the 2D layers. This is the first TQFT picture of the X-cube model as a 3D order with non-invertible 2D "defects". When we zoom out so that the layers appear close together, I'll show that the model can be described by what I call a foliated field theory. I define a foliated field theory to be a field theory that couples to a foliation field, which describes the geometry of the foliating layers (in analogy to a metic which describes the geometry of spacetime). In this case, the foliated field theory takes the form of a 3+1D BF theory which is coupled to stacks of 2+1 BF theories, with stacking structure described by the foliation field.

This talk is based on arXiv:1812.01613 and another forthcoming work.