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.
Highlights
Quantum wave functions subject to strong static randomness may show nonergodic localized behavior
We distinguish between two broad universality classes of quantum localization: Anderson localization [1] in lowdimensional single-particle systems and many-body localization (MBL) in random many-particle systems [2,3]
There is no fundamental distinction between these two. They both reflect the lack of ergodicity of wave functions on random lattices due to massive quantum interference
Summary
Quantum wave functions subject to strong static randomness may show nonergodic localized behavior. One of the most controversial topics concerns the presence or absence of a phase of nonergodic but extended (NEE) states intermediate between the regime of ergodic wave functions at weak and localized wave functions at strong disorder If existent, such a phase must be born out of the main principles distinguishing MBL from low-dimensional Anderson localization: the high coordination number of Fock space lattices, the strong correlation of their disorder potentials, and the sparsity of the hopping matrix elements in Fock space (see the section for a more detailed discussion). The analytical approach is based on matrix integral techniques imported from the theory of high-dimensional random lattices We apply these techniques subject to a number of assumptions which should generalize to other many-body systems of small spatial extension and/or a high degree of connectivity. Technical parts of our analysis are relegated to a number of Appendixes
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