Exploring a link between time crystals and many-body scars in long-range interacting systems

Time crystal is a non-equilibrium state of matter which spontaneously breaks time translation symmetry. While the existence of time crystals has been theoretically and experimentally established in periodically driven systems, resulting in a spontaneous breaking of a discrete $\mathbb{Z}_2$ symmetry, here we investigate the possibility of a \emph{continuous} time crystal, which has been proposed to occur in undriven, energy-conserving systems exhibiting prethermalization. Such systems are characterized by an exponentially long regime where thermalization is delayed, allowing the system to order and display long-lived oscillations of its order parameter, with the frequency set by the chemical potential. On the other hand, persistent oscillations have also recently been shown to arise due to a seemingly distinct mechanism of quantum many-body scarring: the emergence of a subspace of non-thermalizing eigenstates forming an su(2) algebra representation. In this paper we investigate a possible link between these two non-equilibrium phenomena in a realistic one-dimensional spin-1/2 model with long-range interactions. We identify a broad parameter regime with a weakly broken SU(2) symmetry, where the model hosts quantum many-body scars. On the other hand, our extensive numerical study did not find conclusive evidence of a time crystal phase, expected to arise for sufficiently long-range interactions. We relate the difficulty of observing a continuous time crystal to a lack of separation between the prethermalization and full thermalization time scales, found to be surprisingly insensitive to variation of the parameters of the model.