Random cubic graph embedded in a hypercube: Entanglement spectrum and many-body localization

The schematic model of interacting spins is introduced, which combines the symmetry of hypercube with the simplicity of random regular graph with degree three, i.e. the random cubic graph. We study the localization transition in this model, which shares essential characteristics with the systems exhibiting many-body localization. Namely, we investigate the transition in terms of the entanglement entropy and entanglement spectrum. We also show that the most significant indicator of the localization transition is the failure of eigenstate thermalization hypothesis, when the distribution of matrix elements of local operators changes from Gasussian to bimodal. It also provides good estimate for the critical disorder strength.