Correlation function diagnostics for type-I fracton phases

Fracton phases are recent entrants to the roster of topological phases in three dimensions. They are characterized by subextensively divergent topological degeneracy and excitations that are constrained to move along lower-dimensional subspaces, including the eponymous fractons that are immobile in isolation. We develop correlation function diagnostics to characterize type-I fracton phases which build on their exhibiting partial deconfinement. These are inspired by similar diagnostics from standard gauge theories and utilize a generalized gauging procedure that links fracton phases to classical Ising models with subsystem symmetries. En route, we explicitly construct the space-time partition function for the plaquette Ising model which, under such gauging, maps into the $X$-cube fracton topological phase. We numerically verify our results for this model via Monte Carlo calculations.

Figure 1

The Euclidean time representation of the Wilson-loop and horseshoe generalizations for the PGT and its dual, which realize the $X$-cube topological phase. The blue circles represent $\tau$'s (which lie on vertices), red represent $\sigma$'s (which lie on the spatial plaquettes in the PGT but on spatial links in its dual), and the green lines represent the auxiliary spin $\lambda$'s (which lie on the links along the imaginary time direction). Nonequal time operators are shown projected to a 2 + 1D subspace with the time direction pointing “up” on the page. The three possible cut orientations are labeled by $a,b$, and $c$.