Arboreal Topological and Fracton Phases

TL;DR

This paper explores topological and fracton phases on novel arboreal geometries constructed from tree graphs, revealing new fractonic behaviors, dualities, and classifications of topological orders in these unconventional lattice structures.

Contribution

It introduces a framework for studying topological and fracton orders on arboreal geometries, including dualities and classifications of phases on these novel structures.

Findings

Even simple ${ m Z}_2$ gauge theory on 2d arboreal arena is fractonic.

The X-cube model on 3d arboreal arena is fully fractonic.

Identified three classes of arboreal toric code orders and four classes of X-cube fracton orders.

Abstract

We describe topologically ordered and fracton ordered states on novel geometries which do not have an underlying manifold structure. Using tree graphs such as the $k$-coordinated Bethe lattice ${\cal B}(k)$ and a hypertree called the $(k,n)$-hyper-Bethe lattice ${\cal HB}(k,n)$ consisting of $k$-coordinated hyperlinks (defined by $n$ sites), we construct multidimensional arboreal arenas such as ${\cal B}(k_1) \square {\cal B}(k_2)$ by the notion of a graph Cartesian product $\square$. We study various quantum systems such as the ${\mathbb Z}_2$ gauge theory, generalized quantum Ising models (GQIM), the fractonic X-cube model, and related X-cube gauge theory defined on these arenas. Even the simplest ${\mathbb Z}_2$ gauge theory on a 2d arboreal arena is fractonic -- the monopole excitation is immobile. The X-cube model on a 3d arboreal arena is fully fractonic, all multipoles are rendered immobile. We obtain variational ground state phase diagrams of these gauge theories. Further, we find an intriguing class of dualities in arboreal arenas as illustrated by the ${\mathbb Z}_2$ gauge theory defined on ${\cal B}(k_1) \square {\cal B}(k_2)$ being dual to a GQIM defined on ${\cal HB}(2,k_1) \square {\cal HB}(2,k_2)$. Finally, we discuss different classes of topological and fracton orders on arboreal arenas. We find three distinct classes of arboreal toric code orders on 2d arboreal arenas, those that occur on ${\cal B}(2) \square {\cal B}(2)$, ${\cal B}(k) \square {\cal B}(2), k >2$, and ${\cal B}(k_1) \square {\cal B}(k_2)$, $k_1,k_2>2$. Likewise, four classes of X-cube fracton orders are found in 3d arboreal arenas -- those on ${\cal B}(2)\square{\cal B}(2)\square {\cal B}(2)$, ${\cal B}(k) \square {\cal B}(2)\square {\cal B}(2), k>2$, ${\cal B}(k_1) \square {\cal B}(k_2) \square {\cal B}(2), k_1,k_2 >2$, and ${\cal B}(k_1) \square {\cal B}(k_2) \square {\cal B}(k_3), k_1,k_2,k_3 >2$.