Fock space localization in the Sachdev-Ye-Kitaev model*

We study the physics of many body localization in the Majorana Sachdev-Ye-Kitaev (SYK) model perturbed by a one-body Hamiltonian. Specifically, we consider the statistics of many body wave functions and spectra as the strength of the one-body term is ramped up from an ergodic phase into a (Fock space) Anderson localized phase. Our results are obtained from an effective low energy theory, derived from the microscopic model by matrix integral techniques standard in the theory of disordered electronic systems. The analytical results produced by this formalism are compared to exact diagonalization for systems containing up to 30 Majorana fermions. The statistics of many body spectra and wave functions, and the indications of the localization transition are in quantitative agreement with numerics. We believe that this is the first many body system where a localization transition is observed in parameter free agreement with first principle analytical calculations.