Non-equilibrium dynamics of the disordered Power of Two model

Abstract

Motivated by recent experimental realizations of programmable spin models with long-range interactions, we investigate the non-equilibrium dynamics of the Power-of-Two (PWR2) model. This model consists of sparse long-range couplings between spin-$1/2$ objects separated by $d = 2^n$. In the absence of disorder, the system exhibits rapid scrambling and fast thermalization. We explore the impact of disorder in this system by analyzing the time evolution of the survival probability, half-chain entanglement entropy, and out-of-time-ordered correlators (OTOCs). We find that sufficiently strong disorder suppresses information spreading and induces localization. Remarkably, in the strong-disorder regime, the OTOCs display a non-monotonic spatial profile arising from the intrinsic nonlocality of the interactions, signaling qualitatively distinct scrambling dynamics compared to conventional long-range interacting systems. To characterize the localization transition, we extract the critical disorder strength $h_c$ from the spectral statistics and the eigenstate entanglement. We observe that $h_c$ increases with system size. Furthermore, at a fixed disorder strength, the eigenstate-averaged entanglement entropy increases with system size, while the inverse participation ratio decreases, indicating enhanced delocalization at larger sizes. These results collectively suggest that the PWR2 model remains ergodic in the thermodynamic limit for any finite disorder strength.