Arboreal topological and fracton phases

We describe topologically ordered and fracton-ordered quantum systems on geometries which do not have an underlying manifold structure. Using 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 (links defined by n sites), we construct multidimensional arboreal arenas such as B(k1)???B(k2) using a generalized notion of a graph Cartesian product ???. We study various quantum systems such as the Z2 gauge theory, generalized quantum Ising models (GQIMs), the fractonic X-cube model, and related X-cube gauge theory defined on these arboreal arenas, finding several fascinating results. Even the simplest Z2 gauge theory on a two-dimensional arboreal arena is found to be fractonic???an isolated monopole excitation is rendered fully immobile on an arboreal arena. The X-cube model on a generic three-dimensional arboreal arena is found to be fully fractonic in the magnetic sector, i.e., all multipoles of magnetic excitations are rendered immobile on the arboreal arena. We obtain variational ground state phase diagrams of the gauge theories (both Z2 and X-cube gauge theories) which are shown to have deconfined and confined phases. These phases are usually separated by a first-order transition, while continuous transitions are obtained in some cases. Further, we find an intriguing class of dualities in arboreal arenas, as illustrated by the Z2 gauge theory defined on B(k1)???B(k2) being dual to a GQIM defined on HB(2, k1)???HB(2, k2). Finally, we discuss different classes of topological and fracton orders that appear on arboreal arenas. We find three distinct classes of arboreal toric code orders on two-dimensional arboreal arenas, those that occur on B(2)???B(2), B(k)???B(2), k > 2, and B(k1)???B(k2), k1, k2 > 2. Likewise, four classes of X-cube fracton orders are found on three-dimensional arboreal arenas which correspond to those on B(2)???B(2)???B(2), B(k)???B(2)???B(2), k > 2, B(k1)???B(k2)???B(2), k1, k2 > 2, and B(k1)???B(k2)???B(k3), k1, k2, k3 > 2.