Systematic Construction of Scarred Many-Body Dynamics in 1D Lattice Models

We introduce a family of nonintegrable 1D lattice models that feature robust periodic revivals under a global quench from certain initial product states, thus generalizing the phenomenon of many-body scarring recently observed in Rydberg atom quantum simulators. Our construction is based on a systematic embedding of the single-site unitary dynamics into a kinetically constrained many-body system. We numerically demonstrate that this construction yields new families of models with robust wave-function revivals, and it includes kinetically constrained quantum clock models as a special case. We show that scarring dynamics in these models can be decomposed into a period of nearly free clock precession and an interacting bottleneck, shedding light on their anomalously slow thermalization when quenched from special initial states.

Figure 1

(a) A schematic of scarred clock models. Green clock can precess because both of its neighbors are in the unlocking state $|0⟩$ (white), unlike the frozen red clock. (b) Dynamics of fidelity, $|⟨\varphi |e^{-itH}|1010\dots ⟩{|}^2$, for $N_c=4$-color clock model in Eq. (5). Different curves correspond to several choices of $|\varphi ⟩$ indicated in the legend. (c) Overlap of all eigenstates of $N_c=4$-color clock model with the Néel state $|0101\dots ⟩$. Each dot corresponds to a single eigenstate $|E⟩$ with energy $E$ shown on the $x$ axis. Color scale indicates the density of data points. Scarred states are marked by red circles. (d) Entanglement entropy $S$ of all eigenstates of $N_c=4$-color clock model, plotted as a function of their energy $E$. Red circles indicate the matching scarred states from (c), while a few additional scar states, associated with the a “defected ${\mathbb{Z}}_4$” state, $|20002030103000⟩$, are marked by blue circles. Plots (b), (c) are for system size $N=16$, while (d) is for $N=14$. In all cases, we resolve translation and inversion symmetry, and plot both $[k=0,P=+]$ and $[k=\pi ,P=-]$ sectors.

Figure 2

The phase diagram of scarred models with $N_c=4$ and projector $P^0$. Shown in (a),(b) are two slices of the phase diagram obtained by varying the matrix elements of $C$, defined in the text. Color scale represents the maximum of the first fidelity revival for quenches from any of the states $|0101\dots ⟩$, $|0202\dots ⟩$, $|0303\dots ⟩$. Results are for system size $N=10$. Labels on the diagrams refer to special limiting cases defined in the text. Scarred models can be accurately predicted based on the commensurability of the eigenvalue spectrum of $C$, as denoted by lines and explained in the text.