Many-body localization in a modified SYK model

We investigate the many-body spectrum and eigenfunctions of Sachdev-Ye-Kitaev model with additional random hopping terms. To preserve the symmetry class of the original model, the additional term we add is the square of standard two Majorana hopping which is integrable on its own. We examine the level statics of the many-body energy spectrum of this model numerically and find the crossover from Poisson statistics to Wigner-Dyson statistics when tuning the total energy of the system with finite number of Majorana. We study relations of the transition in level statistical transition to Anderson localization in the Fock space by examining the inverse partition ratios (IPR) and information entropy of the eigenstates with different total energy. Fitting IPR and information entropy with a scaling ansatz, one can access degree of the Fock space localization of the many-body eigenstates. We found that there exist three different regimes: ergodic delocalized, non-ergodic delocalized and localized, which closely resembles findings for the Anderson model on Bethe lattice and random regular graphs.