Recently, exactly solvable 3D lattice models have been discovered for a new kind of phase, dubbed fracton topological order, in which the topological excitations are immobile or are bound to lines or surfaces. Unlike liquid topologically ordered phases (e.g. Z_2 gauge theory), which are only sensitive to topology (e.g. the ground state degeneracy only depends on the topology of spatial manifold), fracton orders are also sensitive to the geometry of the lattice. This geometry dependence allows for new physics which was forbidden in topologically invariant phases of matter. 

 

In this talk, I will review the X-cube model [1] of fracton order and how geometry dependence allows for braiding of point-like particles in this 3D phase. I'll summarize how the X-cube model can be described by a quantum field theory, which is analogous to a topological quantum field theory (TQFT). [2] We will see that the gauge invariance of the field theory results in the mobility restrictions of the topological excitations by imposing a new kind of geometric charge conservation. I will conclude by briefly discussing other remarkable geometry-dependent phenomenology of fracton order. For example, I will explain why even on a manifold with trivial topology, spatial curvature can induce a robust ground state degeneracy.

 

[1] Vijay, Haah, Fu 1603.04442 

[2] Slagle, Kim 1704.03870