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

We present a three-dimensional cubic lattice spin model, anisotropic in the (z)over cap direction, that exhibits fractonlike order. This order can be thought of as the result of interplay between two-dimensional Z(2) topological order and spontaneous symmetry breaking along the (z)over cap 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 L-x x L-y x L-z three-torus, it has a 2(2Lz) topological degeneracy and an additional symmetry-breaking nontopological degeneracy equal to 2(LxLy-2). The fractons can be combined into composite excitations that move either in a straight line along the (z)over cap 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)over cap or (y)over cap directions and their absence on the surfaces normal to (z)over cap. This result can be explained using the properties of the two kinds of composite two-fracton mobile excitations.