  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 (SYK) 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 non-trivial wave function statistics indicating the
presence of `non-ergodic extended states'. We check our analytical description
of these phenomena by parameter free comparison to high performance numerics
for systems of up to $N=15$ fermions. In this way, our study becomes a testbed
for concepts of high-dimensional quantum localization, previously applied to
synthetic systems such as Cayley trees or random regular graphs. We believe
that this is the first many body system for which an effective theory is
derived and solved from first principles. The hope is that the novel analytical
concepts developed in this study may become a stepping stone for the
description of MBL in more complex systems.
