A minimal model of many body localization

TL;DR

This paper provides an analytical model of many-body localization (MBL) transition inspired by the SYK model, demonstrating key MBL features and validating them with numerical simulations, offering new theoretical insights into high-dimensional quantum localization.

Contribution

It introduces a fully analytical, microscopically defined model of MBL transition based on a SYK-like system, deriving and solving an effective theory from first principles.

Findings

Displays a transition between ergodic and localized phases.

Shows non-ergodic extended states with specific wave function statistics.

Matches analytical predictions with numerical results for systems up to 15 fermions.

Abstract

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.