Universal behavior beyond multifractality in quantum many-body systems

How many states of a configuration space contribute to a wave-function? Attempts to answer this ubiquitous question have a long history in physics and chemistry, and are keys to understand e.g. localization phenomena. Quantifying this aspect has often been overlooked for interacting many-body quantum systems, mainly due to the exponential growth of the configuration (Hilbert) space. Here, we introduce two Monte Carlo schemes to calculate Shannon-Renyi entropies for ground-states of large quantum many-body systems that are out of reach for any other exact method. Our simulations reveal that the very nature of quantum phases of matter and associated transitions is captured by universal subleading terms in these entropies, on top of a generic dominant multifractal behavior.