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.