Signatures of integrability in the dynamics of Rydberg-blockaded chains

A recent experiment on a 51-atom Rydberg-blockaded chain observed anomalously long-lived temporal oscillations of local observables after quenching from an antiferromagnetic initial state. This coherence is surprising as the initial state should have thermalized rapidly to infinite temperature. In this Rapid Communication, we show that the experimental Hamiltonian exhibits nonthermal behavior across its entire many-body spectrum, with similar finite-size scaling properties as models proximate to integrable points. Moreover, we construct an explicit small local deformation of the Hamiltonian which enhances both the signatures of integrability and the coherent oscillations observed after the quench. Our results suggest that a parent proximate integrable point controls the early-to-intermediate time dynamics of the experimental system. The distinctive quench dynamics in the parent model could signal an unconventional class of integrable system.

Figure 1

Color plots of the mean level spacing ratio $\left[r\right]$ vs $h_{XZ}$ and $h_{YZ}$ for the model (2). (a) shows robust thermalization to the appropriate GOE/GUE ensemble in most of the parameter space. There is an integrable looking region in the vicinity of the origin $H_0$ near $h_{XZ}\approx -0.02$; this is clearer in the zoomed-in panels (c) and (d). (b) shows $\left[r\right]$ as a function of $h_{XZ}$ and $L$ for $h_{YZ}=0$, showing both a dip in $\left[r\right]$ towards the integrable value as a function of $h_{XZ}$, and a gradual drift of the dip value towards thermal with increasing $L$—indicating that the exact integrable point requires further deformations.

Figure 2

(a)–(c) Half-chain entanglement entropy (EE) density of eigenstates plotted against energy density for $L=22$ and $h_{YZ}=0$. (d)–(f) show the distribution of EEVs across the middle third of the eigenspectrum for the domain wall operator at the center of the chain, showing strong narrowing with increasing $L$ for $h_{XZ}\simeq 0.07$ (f) but almost no narrowing for $h_{XZ}\simeq -0.02$ (d). (g) shows the scaling of fluctuations of EEVs between neighboring eigenstates (3) as a function of Hilbert space dimension $\mathcal{D}$. This quantity narrows slower than the ETH prediction ${\mathcal{D}}^{-1/2}$ for $H_0$ and $H_{\mathrm{int}}$. (h) shows the scaling of the fluctuations $\mathrm{\Delta }$ with $\mathcal{D}$ as a function of $h_{XZ}$. The narrowing is slowest for $h_{XZ}\simeq -0.02$ but increases towards the ETH value of $-1/2$ on perturbing $h_{XZ}$ away from this value. The intermediate values of the slope (approximately $-1/4$ for $H_0$) are consistent with proximity to integrability [39].

Figure 3

(a) Time dynamics for a local domain wall operator starting from the $|{\mathbb{Z}}_2\rangle$ state for $L=22$ and three different quench Hamiltonians $H_{\mathrm{int}}$, $H_0$, and $H_{\mathrm{th}}$. The dynamics shows large initial oscillations in all cases, with the amplitude at $H_{\mathrm{int}}$ being much larger than that at $H_0$ or $H_{\mathrm{th}}$. (b) shows the difference between the late-time diagonal ensemble value and the canonical equilibrium prediction as a function of $L$ for the three different quench Hamiltonians. While this difference decreases slowly with $L$ for $H_0$, we see no significant flow for $H_{\mathrm{int}}$ and a fast decrease for $H_{\mathrm{th}}$.