Arboreal Obstructed Atomic Insulating and Metallic Phases of Fermions

We explore phases of free fermions on arenas that do not tessellate a manifold. Specializing to arboreal arenas described by tree graphs which possess a notion of translation symmetry, we study possible fermionic phases in the BDI symmetry class on the $p$-coordinated Bethe lattice. We find that there are $p$ distinct obstructed atomic insulating phases that are characterized by distinct edge states, pattern of entanglement, and a winding characteristic that we define here. These distinct insulting phases are always separated by a metallic region in the parameter space rather than isolated quantum critical points. The metallic region itself comprises several distinct metallic phases that are distinguished by the winding characteristic and correlation functions. The correlation functions of distinct subsystems display non-analytic behavior at distinct points in the metallic region, signaling a cascade of subsystem transitions. An intriguing feature of these arboreal metals is the presence of truncated subsystems with zero energy boundary modes despite being gapless. This work suggests new opportunities for synthetic quantum systems to realize these novel phases.