Absence of fast scrambling in thermodynamically stable long-range interacting systems

In this study, we investigate out-of-time-order correlators (OTOCs) in systems with power-law decaying interactions such as R^-α, where R is the distance. In such systems, the fast scrambling of quantum information or the exponential growth of information propagation can potentially occur according to the decay rate α. In this regard, a crucial open challenge is to identify the optimal condition for α such that fast scrambling cannot occur. In this study, we disprove fast scrambling in generic long-range interacting systems with α>D (D: spatial dimension), where the total energy is extensive in terms of system size and the thermodynamic limit is well-defined. We rigorously demonstrate that the OTOC shows a polynomial growth over time as long as α>D and the necessary scrambling time over a distance R is larger than tgtrsim R^{frac{2α-2D}{2α-D+1}}.