How to realize compact and non-compact localized states in disorder-free hypercube networks

We present a method for realizing various zero-energy localized states on disorder-free hypercube graphs. Previous works have already indicated that disorder is not essential for observing localization phenomena in noninteracting systems, with some prominent examples including the 1D Aubry-André model, characterized solely by incommensurate potentials, or 2D incommensurate Moiré lattices, which exhibit localization due to the flat band spectrum. Moreover, flat band systems with translational invariance can also possess so-called compact localized states, characterized by exactly zero amplitude outside a finite region of the lattice. Here, we demonstrate that both compact and non-compact (i.e., Anderson-like) localized states naturally emerge in disorder-free hypercubes, which can be systematically constructed using Cartan products. This construction ensures the robustness of these localized states against perturbations. Furthermore, we show that the hypercubes can be associated with the Fock space of interacting spin systems exhibiting localization. Viewing localization from the hypercube perspective, with its inherently simple eigenspace structure, offers a clearer and more intuitive understanding of the underlying Fock-space many-body localization phenomena. Our findings can be readily tested on existing experimental platforms, where hypercube graphs can be emulated, e.g., by photonic networks of coupled optical cavities or waveguides. The results can pave the way for the development of novel quantum information protocols and enable effective simulation of quantum many-body localization phenomena.