Glued tree lattices with only compact localized states

By constructing a tight--binding single particle system that has vanishing momentum dispersion in one or more bands, and subsequently including more particles and interactions, it is possible to study physics in strongly interacting regimes. Inspired by the glued trees that first arose in one of the few known examples of quantum supremacy, we define and analyze many infinite families of tight binding single particle Bose--Hubbard models that have only flat bands, and only compact localized states despite being connected and having any number of translation symmetries. The central mechanism that enables our constructions is Arahronov--Bohm caging. We produce models in any number of Euclidian or Hyperbolic spatial dimensions, but we focus on low dimensional models.