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 $R^{-\alpha }$, 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 $t\gtrsim R^{[(2\alpha -2D)/(2\alpha -D+1)]}$.

Figure 1

The OTOC (1) roughly determines the spreading of local operator $W_i$ by time evolution. We aim to approximate $W_i(t)$ in a local region $i[r]$, which has a maximum distance of $r$ from site $i$ [Eq. (4)]: If operator $W_i(t)$ is well approximated by using $W_{i[r]}^{(t)}$ as long as $t\lesssim \mathcal{O}(r^{\zeta })$ ($\zeta <1$), the OTOC exhibits polynomial growth, as in Eq. (3), because of Eq. (7).

Figure 2

We decompose time $t$ and length $r$ to $m_t$ pieces, namely, $\mathrm{\Delta }t≔t/m_t$ and $\mathrm{\Delta }r≔r/m_t$. We start from time evolution $W_i(\mathrm{\Delta }t)$ and approximate it by $W_{X_1}^{(1)}$, which is supported on an extended region $X_1$ as in Eq. (9). Then, we iteratively approximate $W_{X_m}^{(m)}(\mathrm{\Delta }t)$ by $W_{X_{m+1}}^{(m+1)}$, which finally yields the approximation (12). The main advantage of this method is that we need to estimate the local approximation of the time-evolved operators only for a short time.