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Quantum Field Theory of Fracton Topological Order and Topological Degeneracy from Geometry

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