Operator dynamics in chaotic long-range interaction sytems

We use out-of-time-order correlator (OTOC) to diagnose the propagation of chaos in one dimensional long-range power law interaction system. We consider a model called Brownian quantum circuit, which allows us to derive the master equation governing the operator dynamics and therefore transforms the evolution of OTOC to a classical stochastic dynamics problem. We find that the chaos propagation relies on the number of qubits N on each site and the power law exponent α. We use OTOC to define light cone and find that in the small N limit (N=1), there are three light cone regimes as we vary α: (1) a log light cone regime when 1<α<2, (2) a sublinear power law light cone regime when 2<α<4 and (3) a linear light cone regime when α>4. We further study the scaling behaviors of OTOC in the vicinity of light cone. Moreover, we also study the operator growth in the large N limit and find it to be remarkably different. Our result provides a unified physical picture for chaos dynamics in power law interaction system and can be generalized to higher spatial dimensions.