Exact spectral statistics in strongly localized circuits

Since the seminal work of Anderson, localization has been recognized as a standard mechanism allowing quantum many-body systems to escape ergodicity. This idea has acquired even more prominence in the last decade as it has been argued that localization - dubbed many-body localization (MBL) in this context - can sometimes survive local interactions in the presence of sufficiently strong disorder. A conventional signature of localization is in the statistical properties of the spectrum - spectral statistics - which differ qualitatively from those in the ergodic phase. Although features of the spectral statistics are routinely used as numerical diagnostics for localization, their derivation from first principles in the presence of nontrivial interactions has been lacking. Here we provide the example of a simple class of quantum many-body systems - which we dub strongly localized quantum circuits - that are interacting and localized and where the spectral statistics can be characterized exactly. Furthermore, we show that these systems exhibit a cascade of three different regimes for spectral correlations depending on the energy scale: at small, intermediate, and large scales they behave as disconnected patches of three decreasing sizes. We argue that these features appear in generic MBL systems, with the difference that only at the smallest scale do they become Poissonian.