Simple anisotropic three-dimensional quantum spin liquid with fractonlike topological order

We present a three-dimensional cubic lattice spin model, anisotropic in the \^{}z direction, that exhibits fractonlike order. This order can be thought of as the result of interplay between two-dimensional Z₂ topological order and spontaneous symmetry breaking along the \^{}z direction. Fracton order is a novel type of topological order characterized by the presence of immobile pointlike excitations, named fractons, residing at the corners of an operator with two-dimensional support. As other recent fracton models, ours exhibits a subextensive ground-state degeneracy: On an LₗLₘLₙ three-torus, it has a 2^2{Lₙ} topological degeneracy and an additional symmetry-breaking nontopological degeneracy equal to 2^{LₗLₘ-2}. The fractons can be combined into composite excitations that move either in a straight line along the \^{}z direction or freely in the xy plane at a given height z. While our model draws inspiration from the toric code, we demonstrate that it cannot be adiabatically connected to a layered toric code construction. Additionally, we investigate the effects of imposing open boundary conditions on our system. We find zero energy modes on the surfaces perpendicular to either the \^{}x or \^{}y directions and their absence on the surfaces normal to \^{}z. This result can be explained using the properties of the two kinds of composite two-fracton mobile excitations.