Many-body Localization and Poisson statistics in the Quantum Sun model

The Quantum Sun model is a many-body Hamiltonian model of interacting spins arranged on the half-line. Spins at distance $n$ from the origin are coupled to the rest of the system via a term of strength ${\alpha }^n$, with $\alpha \in (0,1)$. From theoretical and numerical considerations, it is believed that this model undergoes a localization-delocalization transition at the critical value $\alpha =\frac{1}{\sqrt{2}}$. We prove that, for $\alpha \ll \frac{1}{\sqrt{2}}$, the model is localized and that its spectral statistics is Poissonian. The main interest of this result is that the model is a genuine many-body model. In particular, the number of independent disorder variables grows only logarithmically with the Hilbert space dimension.