Gapped Lineon and Fracton Models on Graphs

TL;DR

This paper introduces a new class of $ Z_N$ stabilizer codes on arbitrary graphs, revealing novel gapped phases with lineons and fractons, and analyzing their ground state degeneracy and stability.

Contribution

It presents a general framework for $ Z_N$ stabilizer codes on graph-based lattices and characterizes their low-energy limits as anisotropic Laplacian models, including GSD analysis.

Findings

The models are gapped and stable under small deformations.

Ground state degeneracy depends on the graph's Jacobian group mod N.

GSD exhibits irregular growth with system size, especially on cubic lattices.

Abstract

We introduce a $\mathbb{Z}_N$ stabilizer code that can be defined on any spatial lattice of the form $\Gamma\times C_{L_z}$, where $\Gamma$ 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 $\Gamma$. 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.