Gapped lineon and fracton models on graphs

Abstract

We introduce a ${\mathbb{Z}}_{N}$ stabilizer code that can be defined on any spatial lattice of the form $\mathrm{\ensuremath{\Gamma}}\ifmmode\times\else\texttimes\fi{}{C}_{{L}_{z}}$, where $\mathrm{\ensuremath{\Gamma}}$ 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 $\mathrm{\ensuremath{\Gamma}}$. In the special case when space is an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{z}$ 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.