Absence of Fast Scrambling in Thermodynamically Stable Long-Range Interacting Systems
Tomotaka Kuwahara, Keiji Saito
Abstract
In this study, we investigate out-of-time-order correlators (OTOCs) in systems with power-law decaying interactions such as ${R}^{\ensuremath{-}\ensuremath{\alpha}}$, 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 $\ensuremath{\alpha}$. In this regard, a crucial open challenge is to identify the optimal condition for $\ensuremath{\alpha}$ such that fast scrambling cannot occur. In this study, we disprove fast scrambling in generic long-range interacting systems with $\ensuremath{\alpha}>D$ ($D$: 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 $\ensuremath{\alpha}>D$ and the necessary scrambling time over a distance $R$ is larger than $t\ensuremath{\gtrsim}{R}^{[(2\ensuremath{\alpha}\ensuremath{-}2D)/(2\ensuremath{\alpha}\ensuremath{-}D+1)]}$.