Generalized Gauss Laws in Haah's Fracton Model

A fracton is a type of quasiparticle which has the defining property that it is immobile when isolated. These fractons exist in three dimensional topologically ordered states. Their immobility suggests it is possible to detect and count fractons by generalized Gauss laws. Existence of these laws is plausible because a fracton’s position must, by definition, be a conserved quantity even in the presence of perturbations (Haah, 2011). We study this hypothesis for Haah’s model on a cubic lattice. For this we derive families of surface and bulk operators, whose supports form fractal patterns. In one case we found the conserved flux emanating from a fracton excitation and flowing through these surfaces is contained in only half of the space. We also investigate the extensive topological ground state degeneracy that has been found in models with fracton excitations. To understand this, we derived an equation that describes the model’s logical operators, which should be in one-one correspondence with the ground states. This equation has a large number of solutions, elucidating the great ground state degeneracy of Haah’s code. However their number is greater than expected, suggesting a redundancy in the description of the ground state. This suggests an equivalence of logical operators with different geometric orientations, unlike in common examples of topological order, such as the toric code or FQH states.