Arboreal Topological and Fracton Phases

We investigate topologically ordered and fracton ordered states on arenas that do not have an underlying manifold structure. We focus on arenas built from tree graphs, such as the k-coordinated Bethe lattice B(k) and a hypertree called the (k,n)-hyper-Bethe lattice HB(k,n) consisting of k-coordinated hyperlinks (defined by n sites),  to construct ``multidimensional arboreal arenas'' using the notion of the generalized graph cartesian product □. We study various quantum systems such as the Z₂ gauge theory, generalized quantum Ising models (GQIM), the fractonic X-cube model, and related X-cube gauge theory defined on these arenas. Even the simple Z₂ gauge theory has a fractonic character on an arboreal arena -- the monopole excitation is fully immobile.  Similarly, the X-cube model on a 3-dimensional  arboreal arena is also fully fractonic as all multipoles are immobile.  Next, we find an intriguing class of dualities in arboreal arenas: e.g. the Z₂ gauge theory defined on B(k₁)□B(k₂) is dual to a GQIM defined on HB(2,k₁)□HB(2,k₂). Finally, we demonstrate, using entanglement renormalization, that there are only three classes of arboreal toric code topological orders on two-dimensional arboreal arenas, and four distinct arboreal X-cube fracton orders on 3d arboreal arenas.