LAUSR

We study models with fractonlike order based on ${\mathbb{Z}}_{2}$ lattice gauge theories with subsystem symmetries in two and three spatial dimensions. The three-dimensional (3D) model reduces to the 3D toric code when subsystem symmetry is broken, giving an example of a subsystem symmetry enriched topological phase. Although not topologically protected, its ground-state degeneracy has as leading contribution a term which grows exponentially with the square of the linear size of the system. Also, there are completely mobile gauge charges living along with immobile fractons. Our method shows that fractonlike phases are also present in more usual lattice gauge theories. We calculate the entanglement entropy ${S}_{A}$ of these models in a subregion $A$ of the lattice and show that it is equal to the logarithm of the ground-state degeneracy of a particular restriction of the full model to $A$.