Minimal model of many-body localization

We present a fully analytical description of a many-body localization (MBL) transition in a microscopically defined model. Its Hamiltonian is the sum of one- and two-body operators, where both contributions obey a maximum-entropy principle and have no symmetries except Hermiticity (not even particle number conservation). These two criteria paraphrase that our system is a variant of the Sachdev-Ye-Kitaev model. We will demonstrate how this simple zero-dimensional system displays numerous features seen in more complex realizations of MBL. Specifically, it shows a transition between an ergodic and a localized phase, and nontrivial wave-function statistics indicating the presence of nonergodic extended states. We check our analytical description of these phenomena by a parameter-free comparison to high performance numerics for systems of up to N = 15 fermions. In this way, our study becomes a test bed for concepts of high-dimensional quantum localization, previously applied to synthetic systems such as Cayley trees or random regular graphs. The minimal model describes a many-body system for which an effective theory is derived and solved from first principles. The hope is that the analytical concepts developed in this study may become a stepping stone for the description of MBL in more complex systems.