Rainbow scar states in the fracton model

The eigenstate thermalization hypothesis describes the nonequilibrium dynamics of an isolated quantum many-body system, during which a pure state becomes locally indistinguishable from a thermal ensemble. The discovery of quantum many-body scars (QMBS) shows a weak violation of ergodicity, characterized by coherent oscillations of local observables after a quantum quench. These states consist of the tower of regular eigenstates which are equally spaced in the energy spectrum. Although subextensive entanglement scaling is a primary feature widely used to detect QMBS numerically as entropy outliers, rainbow scars exhibit volume-law scaling, which may challenge this property. In this work, we construct the rainbow scar state in the fracton model on a two-leg ladder. The fracton model is composed of four-body ring-exchange interactions, exhibiting global time-reversal symmetry <inline-formula><tex-math id="M2">\begin{document}$ \hat{{{\cal{T}}}}={{\cal{K}}} {\mathrm{i}} \hat{\sigma}^y $\end{document}</tex-math></inline-formula> and subsystem <inline-formula><tex-math id="M3">\begin{document}$ {\hat{U}(1)}=\displaystyle \prod\nolimits_{j \in \{\text {row/col}\}} {\mathrm{exp}}\Big({{\mathrm{i}} \dfrac{\theta}{2} \hat{\sigma}_j^z}\Big)$\end{document}</tex-math></inline-formula> symmetry. The subsystem symmetry constrains particle mobility, hindering the establishment of thermal equilibrium and leading to a series of anomalous dynamical processes. We construct the rainbow scar state with distributed four-body GHZ states whose entanglement entropy follows the volume law. By calculating the eigenstates of the fracton model with exact diagonalization, the rainbow scar state consists of a series of degenerate high-energy excited states that are not significant outliers among other eigenstates in the spectrum. By introducing the on-site interaction to break the time-reversal symmetry, the degeneracy of rainbow scar states is lifted into an equally spaced tower of states, ensuring the revival of the initial state. However, when subsystem <inline-formula><tex-math id="Z-20250513211643">\begin{document}$\hat U(1) $\end{document}</tex-math></inline-formula> symmetry is broken, the scar state is quickly thermalized, indicating that the weak thermalization may be protected by subsystem <inline-formula><tex-math id="Z-20250513211649">\begin{document}$\hat U(1) $\end{document}</tex-math></inline-formula> symmetry. Additionally, we propose a scheme for preparing the rainbow scar state by modulating the strength of the four-body interaction and <inline-formula><tex-math id="M4">\begin{document}$ \hat{\sigma}^z$\end{document}</tex-math></inline-formula> operations, analyzing the influence of noise on the strength of the four-body interaction. This work provides new insights into the weak thermalization processes in fracton model and aids in understanding the nature of ETH-violation in various nonequilibrium systems.