Quantum chaos dynamics in long-range power law interaction systems

We use an out-of-time-order commutator (OTOC) to diagnose the propagation of chaos in one-dimensional long-range power law interaction system. We map the evolution of OTOC to a classical stochastic dynamics problem and use a Brownian quantum circuit to exactly derive the master equation. We vary two parameters: The number of qubits N on each site (the on-site Hilbert space dimension) and the power law exponent α. Three light cone structures of OTOC appear at $N=1$: (1) logarithmic when 0.5<α0.8, (2) sublinear power law when 0.8α1.5, and (3) linear when α1.5. The OTOC scales as exp(λt)/x^2α and t^2α/ζ/x^2α, respectively, beyond the light cones in the first two cases. When α≥2, the OTOC has essentially the same diffusive broadening as systems with short-range interactions, suggesting a complete recovery of locality. In the large N limit, it is always a logarithmic light cone asymptotically, although a linear light cone can appear before the transition time for α1.5. This implies the locality is never fully recovered for finite α. Our result provides a unified physical picture for the chaos dynamics in a long-range power law interaction system.