Entanglement patterns in many-body quantum systems constrained by spatial locality

A long-standing question in the field of statistical mechanics has been describing the universal structure of quantum state ensembles characterizing physical many-body quantum systems. The widely-accepted expectation for "quantum chaotic" systems is that their eigenspectra and eigenstates display universal properties described by random matrix theory (RMT). However, eigenstates of physical systems also encode structure beyond RMT, notably spatial locality and symmetries. In this talk, I discuss how locality is imprinted in the structure of eigenstates in local Hamiltonian systems, leading to deviations from commonly-used RMT ensembles. I will show how, by appropriately constraining RMT ensembles, one can accurately describe the entanglement patterns of eigenstates in quantum chaotic systems. Second, I define a metric that compares the microcanonical entanglement distributions of eigenstates and appropriately-constrained RMT ensembles. Remarkably, we find rare regions in Hamiltonian parameter space where deviations from RMT are minimal, thus suggesting that "maximally chaotic" Hamiltonians---those exactly described by constrained RMT ensembles---may only exist in fine-tuned pockets of parameter space.