Tighter thermalization bounds for perturbed quantum many-body scars

Abstract

Quantum many-body scars (QMBS) are exceptional eigenstates that defy thermalization, enabling long-lived coherent dynamics in strongly interacting systems. However, their stability under perturbations remains inadequately understood. In this work, we derive improved lower bounds on the thermalization time of QMBS under local perturbations with strength $λ$. Using both numerical simulations and analytical reasoning, we show that exact QMBS exhibit slow thermalization, with a timescale scaling as $τ\sim \mathcal{O}(λ^{-1/d})$ owing to the stabilizing restricted spectrum-generating algebra (RSGA), which is a significant improvement over previous bounds (e.g., $τ\sim \mathcal{O}(λ^{-1/(d+1)})$). Counterintuitively, approximate QMBS can thermalize even more slowly under generic perturbations, exhibiting $τ\sim \mathcal{O}(λ^{-2})$ scaling due to second-order perturbative effects in the absence of such protective structure. These distinct thermalization behaviors clarify how exact and approximate scars maintain coherence. Our work advances previous findings by establishing a tighter bound on the thermalization time, clarifying when scarred dynamics remain long-lived under weak but generic perturbations.

Figures (8)

• Figure 1: (a) Dynamics of the local observable $\langle \hat{O}^{+-}\rangle$ following a quench from the nematic Néel state in the spin-1 XY model with $L=12$. The dynamics are shown for the unperturbed case ($\lambda=0$) and a perturbed case ($\lambda=0.03$) under both PBC and OBC. (b) Extracted thermalization time $\tau$ as a function of perturbation strength $\lambda$ for system sizes $L=8, 12$ under different boundary conditions. Symbols represent numerical data obtained from Gaussian fits, while the dashed lines show the second-order polynomial fits of the decay rate $1/\tau$. In the weak-perturbation regime, the leading dependence is approximately linear, $1/\tau \propto \lambda$.
• Figure 2: Decay rate $1/\tau$ for the local observable $\langle \hat{O}^{z} \rangle$ in the deformed PXP model under PBC, plotted as a function of perturbation strength $\lambda$. The data for different system sizes are fitted to the leading-order term $1/\tau = \kappa_1 \lambda$ (dashed lines), consistent with the linear scaling characteristic of exact QMBS. Inset shows the scaling of the fitted slope $\kappa_1$ with inverse system size $1/L$. Error bars represent the uncertainty of the linear fits. The dashed line is a linear extrapolation to the thermodynamic limit ($1/L \to 0$), where the limiting value is indicated by the black star.
• Figure 3: Comparison of thermalization dynamics in the deformed PXP model for $L=24$. (a) Under PBC, the decay rate $1/\tau$ scales predominantly linearly with the perturbation strength $\lambda$. (b) Under OBC, the decay rate scales quadratically with the perturbation, as shown by the linear fit against $\lambda^2$. Insets show the dynamics at $\lambda=0$, confirming the transition from exact (PBC) to approximate (OBC) scarring.
• Figure 4: (a) Dynamics of the observable $\langle \hat{O}^{+-} \rangle$ in the spin-1 Kitaev model for $L=28$. The imperfect revival, even for the unperturbed case ($\lambda=0$), signifies approximate QMBS. (b) The decay rate $1/\tau$ plotted against $\lambda^2$ for different system sizes. The linear fit confirms the quadratic scaling $1/\tau =\kappa_2 \lambda^2$. Inset shows the fitted slope $\kappa_2$ extrapolates to a non-zero value in the thermodynamic limit.
• Figure 5: (a) Dynamics of $\langle \hat{O}^{x} \rangle$ in the exact DWC model for $L=20$, showing perfect revivals for the unperturbed case ($\lambda = 0$). (b) Decay rate $1/\tau$ versus perturbation strength $\lambda$ for various system sizes. The linear fits confirm the scaling $1/\tau = \kappa_1 \lambda$. Inset shows the fitted slope $\kappa_1$ as a function of inverse system size.