Gapped lineon and fracton models on graphs

We introduce a ZN stabilizer code that can be defined on any spatial lattice of the form Γ×CLz, where Γ is a general graph. We also present the low-energy limit of this stabilizer code as a Euclidean lattice action, which we refer to as the anisotropic ZN Laplacian model. It is gapped, robust (i.e., stable under small deformations), and has lineons. Its ground-state degeneracy (GSD) is expressed in terms of a "mod N-reduction"of the Jacobian group of the graph Γ. In the special case when space is an L×L×Lz cubic lattice, the logarithm of the GSD depends on L in an erratic way and grows no faster than O(L). We also discuss another gapped model, the ZN Laplacian model, which can be defined on any graph. It has fractons and a similarly strange GSD.