Fate of Quantum Many-Body Scars in the Presence of Disorder

Experiments performed on strongly interacting Rydberg atoms have revealed surprising persistent oscillations of local observables. These oscillations have been attributed to a special set of nonergodic states, referred to as quantum many-body scars. Although these states have enriched our understanding of thermalization in quantum systems, their interplay with disorder in realistic quantum simulators has remained unclear. We address this question by studying numerically and analytically the magnetization and spatiotemporal correlators of a scar model of disordered and interacting qubits that can be realized in present-day simulators. While the oscillation amplitudes of these observables decay with time as the disorder strength is increased, their oscillation frequency remains remarkably constant. This stability stems from resonances of the disordered spectrum in the overlap with the clean scar eigenstates that are approximately centered at the same scar energies of the clean system. We also find that multiple additional sets of scar resonances become accessible due to the presence of disorder and further enhance the oscillation amplitudes. Our results show the robustness of nonergodic dynamics in scar systems, and opens the door to understanding their behavior in near-term quantum devices and potentially using them as a resource in quantum-sensing protocols to calibrate quantum hardware.

Popular Summary

A scientific and technological revolution is underway to build quantum machines capable of studying the world around us. Quantum simulators constitute a class of such machines, which can be programmed to emulate complex physical systems. Remarkable discoveries have already been achieved using these devices. In a recent development, scientists emulated the behavior of a special magnet that was expected to rapidly lose its magnetization by reaching thermal equilibrium. Instead, it was found that the magnetization keeps oscillating between two opposite values at a constant rate. This remarkable behavior was found to be due to quantum scarring, a puzzling property of quantum systems that keeps them from thermalizing.

Although quantum simulators have been used to make discoveries such as quantum scarred magnets, imperfections in their components can impact the physics they attempt to emulate. In this work, we theoretically model the inner workings of simulators to analyze how imperfections influence scarred magnets. We discover a remarkable level of robustness: the magnetization continues to oscillate at the same rate as if there were no imperfections although it becomes gradually demagnetized. Our results shed new light on this emergent collective behavior: while it was previously assumed that oscillations are associated with a discrete set of equidistant eigenstates, we show that they persist even when these eigenstates are no longer present. Furthermore, disorder reveals the existence of new families of scarred eigenstates that are not visible in clean systems. These properties of realistic quantum simulators can be used to build improved calibrated devices.

Figure 1

(a) Schematic representation of the simulator modeled by Eq. (1). The excited and ground states of each qubit are separated by an energy $\mathrm{\Omega }$. Only when two neighboring qubits are in their excited states will the qubits interact, with energy $J_R$. When $J_R\gg \mathrm{\Omega }$, a kinematic constraint develops that leads to the formation of quantum many-body scars. (b) Diagram of the dynamical regimes obtained in this work as a function of the nearest-neighbor interaction $J_R$ and disorder strengths $W$: a regime that exhibits scar dynamics due to scar resonances, a regime that is ergodic without scar resonances, a constrained MBL phase, and a MBL without kinematic constraints.

Figure 2

(a) Overlap with the ${\mathbb{Z}}_2$ state, and (b) entanglement entropy for each eigenstate with $L=14$. Top figures correspond to the clean system, where clear signatures of scar states appear at the integer values of $E/{\omega }_{\mathrm{scar}}$. The bottom figures correspond to disorder strength $W=0.25{\omega }_{\mathrm{scar}}$ for a particular disorder realization. The scar signatures have become difficult to identify after disorder is introduced in the system.

Figure 3

(a) Mean level spacing ratios $[⟨r_n⟩]$ defined by Eq. (7) and (b) decay exponent of $\mathrm{\Delta }\hat{\mathcal{O}}$ defined by Eq. (8) for the domain-wall operator $\hat{\mathcal{O}}={\sigma }_{L/2}^z{\sigma }_{L/2+1}^z$, as a function of disorder strength. The error bars in both figures are of the order or less than the circle markers. The disorder strength $W_c$ at which nonergodic oscillations are lost is denoted by the purple line, whereas the disorder strength $W_{\mathrm{Th-L}}$ at which the constrained MBL phase sets in is denoted by the yellow line. “Th-L” refers to the thermal-to-localized transition.

Figure 4

Density plot of the matrix elements $|⟨{\varphi }_{l^{\prime }}^{(0)}|{\tilde{\sigma }}_{\pi }^y|{\varphi }_l^{(0)}⟩|$ with respect to the basis of energy eigenstates ordered by (a) energy and (b) towers, respectively, for $L=8$. The corresponding figure for $|⟨{\varphi }_{l^{\prime }}^{(0)}|{\tilde{\sigma }}_{\pi }^z|{\varphi }_l^{(0)}⟩|$ is similar.

Figure 5

Left: time evolution of $[{\mathcal{M}}_z(r,t)]$ for disorder strengths $W/{\omega }_{\mathrm{scar}}=0.1,0.19,0.63,1.58,7.41$. In these figures, $L=16$, but we show only the first six sites for visual clarity. Right: corresponding disorder-averaged Fourier component $[|{\overline{\mathcal{M}}}_z(k,\omega )|]$ at the same disorder strengths. For clarity, we subtract the component $[|{\overline{\mathcal{M}}}_z(0,0)|]$, which represents a constant homogeneous shift in the on-site magnetization. For weak disorder, oscillations occur at $(k,\omega )=(\pi ,{\omega }_{\mathrm{scar}})$. As the disorder strength is increased, the distribution broadens, and eventually concentrates at $(k,\omega )=(\pi ,0)$.

Figure 6

Fourier component $[|{\overline{\mathcal{M}}}_z(\pi ,\omega )|]$ as a function of $\omega$ and $W$ for the PXP model (a) and the free paramagnet (b). A pronounced broadening occurs in the paramagnetic case, whereas in the PXP model the dynamics is consistently centered and localized at the scar frequency.

Figure 7

(a) Behavior of $|{\overline{\mathcal{M}}}_z(\pi ,\omega )|/L$ as the system size is increased with $W=0.1{\omega }_{\mathrm{scar}}$, showing that there are no appreciable changes. (b) Behavior as a function of disorder strength (for three system sizes) of the magnetization Fourier components $[|{\overline{\mathcal{M}}}_z(\pi ,{\omega }_{\mathrm{scar}})|]$ and $[|{\overline{\mathcal{M}}}_z(\pi ,0)|]$ (values indicated on the left axis); and the normalized frequency distribution ${\mathrm{\Lambda }}_{\omega }^{(q=6)}$ (values indicated on the right axis). There is a clear and stable transition between a regime with scar dynamics and a dynamically trivial regime. In this figure, we choose $q=6$ for convenience as it accentuates the position of the peak, but other values of $q$ also work.

Figure 8

(a) Direct decay of scar states. The disorder induces transitions from the scar state into spectrally nearby generic states, with the rate controlled by matrix elements of the form $|⟨{\varphi }_{l^{\prime }\notin {\mathrm{\Lambda }}_s}^{(0)}|\hat{W}|{\varphi }_{l\in {\mathrm{\Lambda }}_s}^{(0)}⟩{|}^2$. Additionally, there is an energy shift of the form $⟨{\varphi }_{l\in {\mathrm{\Lambda }}_s}^{(0)}|\hat{W}|{\varphi }_{l\in {\mathrm{\Lambda }}_s}^{(0)}⟩$. (b) Decay via $J\ne 1$ tower states. The initial scar state couples (blue arrow) to a state in a separate tower (purple line), which then decays (blue arrows) into the spectrally nearby generic states. These processes give significant contribution [comparable to the main channel in (a)] to the order parameter dynamics at finite times.

Figure 9

Comparison of contributions to the magnetization $[{\tilde{\mathcal{M}}}_z(\pi ,t)]$ when evaluated exactly numerically (dots) and analytically (solid lines) for $L=14$ and $W=0.1{\omega }_{\mathrm{scar}}$. The matrix elements ${\rho }_{l{l}^{\prime }}$ are calculated numerically using the eigenstates of the clean system. (a) Contribution from optimized scar states, $J=1$ tower, with the analytic curve obtained using Eq. (21). (b) Contribution from states outside the main scar subspace, decay via $J\ne 1$ towers as shown in Fig. 8, with the corresponding analytic curve obtained from Eq. (24). (c) Contribution from all states in the spectrum, with the corresponding analytic curve. In (a), we also show the net magnetization by gray dots (the connecting gray line is a guide for the eye) to illustrate that the contribution from the optimized scar states is not sufficient to account for the observed dynamics.

Figure 10

Scar resonances in the spectrum. We plot the distribution of optimized scar states in the basis of disordered eigenstates for $L=12$ and disorder strengths for $W/{\omega }_{\mathrm{scar}}=0.10,0.19,0.34,0.63,0.17$ from top to bottom, respectively. To make it easier to visualize the shape of these distributions, we binned the energy axis and summed the probabilities inside each bin. As can be seen, for sufficiently weak disorder, the scar states are broadened in the disordered setting but remain distinguishable. Eventually, at a disorder strength $W_c\approx 0.63{\omega }_{\mathrm{scar}}$ (indicated here by the green frame), the width of these states increases so much that they can no longer be spectrally resolved.

Figure 11

Spatiotemporal correlations for $L=14$ and $W=0.1{\omega }_{\mathrm{scar}}$ starting from clean scar eigenstate at zero energy as the initial state. The plots in (a)–(c) correspond to the different contributions to the net temporal correlator $\mathrm{Re}\left[{\tilde{C}}_z(t)\right]$. The solid lines correspond to the analytic result Eq. (27) whereas the dots are numerically exact values. (d) Spatial resolution of the correlator $[|{\mathcal{C}}_z(r_0,r_0+R,t)|]$ (with $r_0$ an odd site) that underlies the oscillations found in the temporal correlator.

Figure 12

Spatiotemporal correlations for $L=14$ and $W=0.1{\omega }_{\mathrm{scar}}$ starting from the $|\mathrm{\Psi }⟩=|{\mathbb{Z}}_2⟩$ state. The plots in (a)–(c) correspond to the different contributions to the temporal correlator $\mathrm{Re}\left[{\tilde{C}}_y(t)\right]$. The solid lines correspond to the analytic result Eq. (27) whereas the dots are numerically exact values. (d) Spatial resolution of the correlator $[|{\mathcal{C}}_y(r_0,r_0+R,t)|]$ (with $r_0$ an odd site) that underlies the oscillations found in the temporal correlator.

Figure 13

(a) Contour plot of $[|{\tilde{\mathcal{C}}}_y^{\mathrm{Ryd}}({\omega }_{\mathrm{scar}})|]$, normalized by the maximum value ${\tilde{C}}_y^{\max }$ in this parameter space. The dashed line corresponds to the location of the peak in ${\overline{\mathrm{\Lambda }}}_{\omega }^{(q=6)}$ for $L=12$. (b) Contour plot of the mean energy-level spacing ratio $[⟨r_n⟩]$ that varies between the Poisson value ($[⟨r_n⟩]=0.38$) and the ergodic value ($[⟨r_n⟩]=0.6$). The dashed line corresponds to approximately the value $[⟨r_n⟩]=0.45$, which we use to estimate the boundary between thermal and the MBL phases.