Topologically ordered quantum phases of matter are often characterized by their topological excitations (or anyons). Recently, exactly solvable lattice models (similar to toric code) have been discovered for a new kind of topological order in which the anyons exhibit remarkable mobility restrictions [1,2]. For example, these orders can have so-called fracton excitations, which are immobile when isolated from other fractons. For a given phase, if all of the topological excitations are immobile fractons, then the phase is a self-correcting quantum memory [2] which could e.g. be used as a quantum computer hard drive.

In this talk, I will briefly review topological order and fracton topological order. I will then present a quantum field theory description of the X-cube model of fracton topological order [3], which is analogous to a topological quantum field theory (TQFT). Previously, it was unknown if a field theory of fracton order even exists. I will describe how the gauge invariance of the field theory results in the previously mentioned mobility restrictions of the anyons by imposing a new kind of "charge conservation", and how the braiding statistics of the anyons are reproduced by the field theory. I will also show that even on a manifold with trivial topology, spatial curvature can induce a stable ground state degeneracy.

[1] Vijay, Haah, Fu 1505.02576
[2] Haah 1101.1962
[3] Slagle, Kim 1704.03870