Gapped Lineon and Fracton Models on Graphs

We introduce a ${\mathbb{Z}}_N$ stabilizer code that can be defined on any spatial lattice of the form $\mathrm{\Gamma }\times C_{L_z}$, where $\mathrm{\Gamma }$ 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 ${\mathbb{Z}}_N$ 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 $\mathrm{\Gamma }$. In the special case when space is an $L\times L\times L_z$ 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 ${\mathbb{Z}}_N$ Laplacian model, which can be defined on any graph. It has fractons and a similarly strange GSD.