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

TL;DR

This paper proves that in thermodynamically stable long-range interacting systems with decay exponent alpha greater than the spatial dimension D, fast scrambling does not occur, and information spreads polynomially over time.

Contribution

It rigorously shows that fast scrambling is absent in generic long-range systems with alpha > D, providing conditions for polynomial growth of OTOCs.

Findings

OTOCs grow polynomially for alpha > D

Scrambling time exceeds a certain polynomial bound

Fast scrambling is absent in thermodynamically stable long-range systems

Abstract

In this study, we investigate out-of-time-order correlators (OTOCs) in systems with power-law decaying interactions such as $R^{-\alpha}$, 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 $\alpha$. In this regard, a crucial open challenge is to identify the optimal condition for $\alpha$ such that fast scrambling cannot occur. In this study, we disprove fast scrambling in generic long-range interacting systems with $\alpha>D$ ($D$: 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 $\alpha>D$ and the necessary scrambling time over a distance $R$ is larger than $t\gtrsim R^{\frac{2\alpha-2D}{2\alpha-D+1}}$.