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.

Figures (6)

• Figure 1: (a) Schematic illustration of the Power of Two model with open boundary conditions. The interaction graph couples spins on sites separated by a power of two (see Eqns. \ref{['eq:Ham']} and \ref{['eq:Ham2']}). (b) The dynamics of the PWR2 model in the disorder-free limit ($h=0$). The left panel shows the time-evolution of the occupation, $\langle n_j \rangle = \langle S_j^{+} S_j^{-} \rangle$ for the single-magnon state $\vert \psi \rangle_{\rm SM}$ (Eq. \ref{['eq:SM-state']}). The right panel shows the time-evolution of the OTOC, $C_{1j}(t)$ (Eq. \ref{['eq:OTOC']}) in the zero-magnetization sector. These results demonstrate that this system exhibits fast scrambling and consequent rapid thermalization.
• Figure 2: The time-evolution of the 1-magnon initial state, $\vert \psi \rangle_{\rm SM}$ (Eq. \ref{['eq:SM-state']}) for N=256 for various values of the disorder strength, $h$. While, increasing disorder strength leads to greater localization, the spread of the magnon excitation does not follow a light-cone. Instead at strong disorder, the excitation number $\langle n_j \rangle$ exhibits a non-monotonic dependence on $\vert j - L/2 \vert$ due to the non-local nature of the model. The results have been obtained from disorder-averaging over 1000 realizations.
• Figure 3: (a) The OTOC, $C_{1j}$ for 3 representative values of disorder strength ranging from weak to strong disorder. In the weak disorder regime, the OTOC spreads super-ballistically. However, in the presence of stronger disorder, the growth slows down. Furthermore, $C_{1j}$ exhibits a non-monotonic dependence on $j$ due to the non-local nature of the interactions. (b) The time-evolution of the spatially averaged OTOC, $\overline{C}(t)$ (Eq. \ref{['eq:AvgOTOC']}). In the weak-disorder regime, $\overline{C}(t)$ saturates to $C_{\rm chaotic} \sim 0.125$ within a very short time. Increasing the disorder strength leads to a slower growth of $\overline{C}(t)$, and in the strong disorder regime $\overline{C}(t) \sim 0$. These results have been obtained for $N=12$ by averaging over 100 disorder realizations.
• Figure 4: (a) The time-evolution of the survival probability, $\mathcal{L}(t)$ (Eq. \ref{['eq:Fidelity']}) for different values of $h$. In the weak disorder regime, $\mathcal{L}(t)$ quickly decays to zero due to the fast information scrambling. Increasing disorder leads to a slower decay and a higher saturation value of $\mathcal{L}(t)$. (b) The growth of the half-chain entanglement entropy, $S(t)$ for various value of the disorder strength, $h$. In the weak disorder regime, $S(t)$ quickly grows and saturates to its maximum value. Increasing disorder suppresses the rate of growth and the saturation value of $S$. These results have been obtained for the domain wall initial state, $\vert \psi \rangle_{\rm DW}$; $N$ has been set to 12, and the disorder averaging has been performed for 100 disorder realizations.
• Figure 5: The adjacent gap ratio, $\langle r \rangle$ as a function of disorder strength $h$. The results have been averaged over 60000 realizations for $N=8$, 20000 realizations for $N=10$ and $12$, and 200 realizations for $N=14$ and $16$ respectively. These results have been obtained for the middle 1/3 spectra. Increasing $h$ leads to a transition of the spectral statistics from Wigner-Dyson to Poisson, thereby indicating localization. The $\langle r \rangle$ curve shifts towards the right with increasing $N$ indicating that the critical disorder strength for localization, $h_c \rightarrow \infty$ in the thermodynamic limit.