Entanglement and operator dynamics in unitary circuits with generic symmetries.*

1. Random unitary circuits are minimally structured models of chaotic dynamics which recover two key features of generic many-body Hamiltonian systems - unitarity and locality. Despite their apparent simplicity, they capture a number of universal aspects of dynamics in chaotic systems, such as the spreading of local operators measured by out-of-time-ordered correlators (OTOCs), as well as the ballistic growth of entanglement with time. The inclusion of additional global symmetries is expected to result in qualitative changes to the dynamics. For instance, the existence of a U(1) charge in random unitary circuits has been shown to lead to additional spatial structure in the OTOC and inhibit the redistribution of entanglement. Here we investigate the growth of entanglement and the spreading of operators in random unitary circuits possessing arbitrary non-abelian symmetries. By tapping into a number of powerful methods recently derived in the context of tensor networks, we construct a mapping to a classical spin model and assess the dominant saddle points which dictate the evolution of the OTOC and the Renyi entanglement entropies. We also discuss the possibility of detecting new measurement-induced critical phases found in circuits with non-abelian symmetries.