Topological and Fracton Order in Arboreal Arenas

In this work, we introduce and theoretically explore, systems constructed by qubit connectivities that do not tessellate space, i. e., non-manifold structures. We construct and study quantum models, including generalized transverse field models, gauge theories, and fractonic models, on structures – arboreal arenas – based on tree graphs. This reveals many unexpected results that could provide many new opportunities: 1) We show that even the simplest Ising gauge theory on an arboreal lattice is fractonic (excitations have restricted mobilities) with a large ground state degeneracy. 2) We show that the X-cube model (a standard example of a fracton model) is fully fractonic, i. e., neither the magnetic monopole nor any of its multipoles are mobile. 3) We study the phase/phase transitions of these models, paying careful attention to boundary conditions, and obtain their phase diagrams. 4) We uncover a new class of arboreal dualities, not only providing key insights into the physics of the phases and phase transition but also offering a comprehensive and generalized view of known dualities in lattice Ising systems. 5) We undertake the classification of topological orders on arboreal arenas and discover a completely unexpected and remarkable result. There are only three types of inequivalent arboreal toric code orders on two-dimensional arboreal arenas and four types of X-cube fracton orders on three-dimensional arboreal arenas.