Emergent SU(2) dynamics and perfect quantum many-body scars

TL;DR

This paper demonstrates how a weak deformation of the Rydberg Hamiltonian can produce near-perfect quantum many-body revivals by supporting emergent SU(2) dynamics and nonergodic scar states, advancing understanding of quantum scars.

Contribution

It introduces a weak quasi-local deformation that enhances revivals and reveals an emergent SU(2) structure supporting quantum many-body scars, along with a toy model illustrating their origin.

Findings

Deformation makes revivals nearly perfect.

Emergent SU(2) dynamics underpins scar states.

Constructed a toy model with exact scars.

Abstract

Motivated by recent experimental observations of coherent many-body revivals in a constrained Rydberg atom chain, we construct a weak quasi-local deformation of the Rydberg blockade Hamiltonian, which makes the revivals virtually perfect. Our analysis suggests the existence of an underlying non-integrable Hamiltonian which supports an emergent SU(2)-spin dynamics within a small subspace of the many-body Hilbert space. We show that such perfect dynamics necessitates the existence of atypical, nonergodic energy eigenstates - quantum many-body scars. Furthermore, using these insights, we construct a toy model that hosts exact quantum many-body scars, providing an intuitive explanation of their origin. Our results offer specific routes to enhancing coherent many-body revivals, and provide a step towards establishing the stability of quantum many-body scars in the thermodynamic limit.

Figures

a) Bernien *et al.* ([2017](#bib.bib12)); Turner *et al.* ([2018a](#bib.bib13)); Ho *et al.* ([2018](#bib.bib14)); Turner *et al.* ([2018b](#bib.bib15)); Khemani *et al.* ([2018](#bib.bib16)). This is accompanied by a generally linear growth of the bipartite entanglement entropy (Fig. 1b), which is slower compared to other thermalizing initial states. Such dynamics arise due to the existence of a band of special, nonthermal ‘quantum many-body scarred’ eigenstates that are approximately equally spaced in energy, and have large overlaps with $|\mathbb{Z}_{2}\rangle$. Furthermore, these special e

a) Bernien *et al.* ([2017](#bib.bib12)); Turner *et al.* ([2018a](#bib.bib13)); Ho *et al.* ([2018](#bib.bib14)); Turner *et al.* ([2018b](#bib.bib15)); Khemani *et al.* ([2018](#bib.bib16)). This is accompanied by a generally linear growth of the bipartite entanglement entropy (Fig. 1b), which is slower compared to other thermalizing initial states. Such dynamics arise due to the existence of a band of special, nonthermal ‘quantum many-body scarred’ eigenstates that are approximately equally spaced in energy, and have large overlaps with $|\mathbb{Z}_{2}\rangle$. Furthermore, these special e

Equations

$$H_0$$

$$\delta H_2=-\sum _i{h}_2\mathcal{C}{\sigma }_i^x\mathcal{C}({\sigma }_{i+2}^z+{\sigma }_{i-2}^z)$$

$$\delta H_R=-\sum _i\sum _{d=2}^R{h}_d\mathcal{C}{\sigma }_i^x\mathcal{C}\left({\sigma }_{i-d}^z+{\sigma }_{i+d}^z\right),$$

$$h_d^{\text{ansatz}}=h_0{\left({\varphi }^{(d-1)}-{\varphi }^{-(d-1)}\right)}^{-2},$$

$$P_{\mathcal{K}}[H^z,H^{\pm }]P_{\mathcal{K}}\approx \pm \Delta P_{\mathcal{K}}{H}^{\pm }{P}_{\mathcal{K}},$$

$$(1-h)(1-h-16\sum _{n=1}^{\infty }{h}_{2n})=16\sum _{n=1}^{\infty }{h}_{2n}^2,$$

$$H_{\text{toy}}=\frac{\Omega }{2}\sum _i{\sigma }_i^x+\sum _i{V}_{i-1,i+2}{P}_{i,i+1},$$

Full text

Remarkable experimental advances have recently enabled studies of nonequilibrium dynamics of isolated, strongly interacting quantum systems Bloch et al. (2012); Ladd et al. (2010); Polkovnikov et al. (2011). In such systems, it is commonly believed that a generic state initialized far from equilibrium eventually thermalizes, whereupon any initial local information becomes unrecoverable Deutsch (1991); Srednicki (1994); Rigol et al. (2008). While this process of thermalization provides the basis of statistical mechanics, it also poses challenges for building large-scale quantum devices. Hence, it is of fundamental interest to understand mechanisms to evade thermalization. Two well-studied possibilities include many-body localization in strongly disordered systems, and fine-tuned integrable systems Nandkishore and Huse (2015); Abanin et al. (2018); Sutherland (2004).

Recently, quench experiments with Rydberg atom arrays Schauß et al. (2012); Labuhn et al. (2016); Bernien et al. (2017) have discovered non-thermalizing dynamics of a new kind Bernien et al. (2017). Initialized in a high-energy Néel state, the system exhibited unexpectedly long-lived, periodic revivals, failing to thermalize on experimentally accessible timescales; in contrast, other high-energy product states exhibited thermalizing dynamics consistent with conventional expectations.

These surprising observations have stimulated strong theoretical interest Turner et al. (2018a); Ho et al. (2018); Turner et al. (2018b); Khemani et al. (2018); Lin and Motrunich (2018). Ref. Turner et al. (2018a) showed that the oscillatory dynamics stems from a small number of exceptional, nonthermal many-body eigenstates which are embedded in a sea of thermal eigenstates, that generically obey the eigenstate thermalization hypothesis (ETH) Deutsch (1991); Srednicki (1994); Rigol et al. (2008). These atypical, ergodicity-breaking eigenstates were named ‘quantum many-body scars’ in analogy to quantum scars in single-particle quantum systems, which are similarly nonergodic wavefunctions that concentrate along the unstable, periodic trajectories of the counterpart classical system Heller (1984). Ref. Ho et al. (2018) firmed up this analogy by showing that the long-lived many-body revivals were also closely related to an unstable periodic orbit in a variational, “semiclassical” description of the quantum many-body dynamics.

Despite much theoretical effort, several key questions regarding the nature of quantum many-body scars remain open. In particular, owing to the slow decay, the ultimate fate of the revivals at very long times in the thermodynamic limit is not fully understood. Another outstanding challenge is to understand the physical mechanism protecting scars in the Rygberg chain and beyond. Ref. Khemani et al. (2018) conjectured that the observed revivals can arise due to proximity to a putative integrable point, where the whole spectrum (not just scarred eigenstates) become nonthermal. In particular, Ref. Khemani et al. (2018) demonstrated the existence of a nontrivial deformation of the Rydberg blockade Hamiltonian that results in a substantial modification of the many-body level statistics, that could be interpreted as proximity to integrability. Moreover, earlier works Moudgalya et al. (2017, 2018) have demonstrated the coexistence of ETH-violating states in a generically ergodic spectrum, by explicitly constructing exact many-body eigenstates of non-integrable AKLT model that feature low entanglement at arbitrary energy densities

In the present work, we demonstrate that the periodic many-body revivals become extremely stable with a suitable weak, quasi-local deformation of the effective model describing the experiment Bernien et al. (2017), with the return probability of the Néel many-body state approaching unity within $10^{-6}$ in systems with more than 30 particles. Remarkably, despite such manifestly nonergodic dynamics and the strongly nonthermal character of the associated scarred eigenstates, the bulk of the spectrum remains well-thermal, in contrast to the special point in Ref. Khemani et al. (2018). Rather than being integrable, the revival dynamics can be understood as the coherent rotation of an emergent, large SU(2)-spin that lives within a special subspace of the many-body Hilbert space.

Our results strongly suggest the existence of a “parent” Hamiltonian with perfect oscillatory dynamics. We prove that, in generic settings, such perfect revivals impose strong constraints on the structure of energy eigenstates, necessitating the presence of some eigenstates violating ETH. This result directly relates observable nonequilibrium dynamics to properties of energy eigenstates, and parallels the mechanism behind quantum scarring in single-particle chaos theory Heller (1984). Finally, guided by the emergent SU(2)-spin structure, we construct a solvable toy model that explicitly hosts the phenomenology of quantum many-body scars, which provides an intuitive explanation of their origin in the constrained model.

Model and revivals. – The 1D chains of Rydberg atoms in the experiments Bernien et al. (2017) are well-described by a kinetically constrained Fredrickson and Andersen (1984); Palmer et al. (1984) spin-$1/2$ chain with the Hamiltonian

[TABLE]

where $\sigma_{i}^{\mu}$ ($\mu\in\{x,y,z\}$) are standard Pauli operators at site $i$, and $\mathcal{C}=\prod_{i}[1-(1+\sigma^{z}_{i})(1+\sigma^{z}_{i+1})/4]$ is a global projector constraining the Hilbert space to spin configurations without two adjacent up-spins, $|\uparrow\uparrow\rangle$, corresponding to the regime of a strong nearest neighbor Rydberg blockade Jaksch et al. (2000) in the experiments Bernien et al. (2017). The dynamics is such that a spin may flip only when both of its neighbors are in the $|\downarrow\rangle$ state, and the model is thus strongly interacting  Sun and Robicheaux (2008); Olmos et al. (2009, 2012). For simplicity we assume periodic boundary conditions.

The model in Eq. (1) exhibits unexpected, long-lived periodic revivals when initialized in the Néel state \mbox{|\mathbb{Z}{2}\rangle}=\mbox{|\uparrow\downarrow\uparrow\downarrow\cdots\rangle}. Despite its large energy density (formally corresponding to an infinite temperature), the quench dynamics from this initial state exhibits large recurrences of the Loschmidt echo g_{0}(t)\equiv|\mbox{\langle\mathbb{Z}{2}|}e^{-iH_{0}t}\mbox{|\mathbb{Z}{2}\rangle}|^{2} at multiples of a period $\tau$ with a slow overall decay (Fig. 1a)  Bernien et al. (2017); Turner et al. (2018a); Ho et al. (2018); Turner et al. (2018b); Khemani et al. (2018). This is accompanied by a generally linear growth of the bipartite entanglement entropy (Fig. 1b), which is slower compared to other thermalizing initial states. Such dynamics arise due to the existence of a band of special, nonthermal ‘quantum many-body scarred’ eigenstates that are approximately equally spaced in energy, and have large overlaps with $|\mathbb{Z}_{2}\rangle$. Furthermore, these special eigenstates can be approximately constructed using an analytical framework dubbed the forward scattering approximation (FSA) Turner et al. (2018a, b). In essence, FSA relies on decomposing the Hamiltonian into a “raising” and “lowering” part, $H_{0}=H^{+}_{0}+H^{-}_{0}$, with $H^{\pm}_{0}=\sum_{i\in\text{even}}\mathcal{C}\sigma_{i}^{\pm}\mathcal{C}+\sum_{i\in\text{odd}}\mathcal{C}\sigma_{i}^{\mp}\mathcal{C}$. Then, one can recursively define $N+1$ vectors $|k\rangle_{0}=\beta_{k}H^{+}_{0}|k-1\rangle_{0}$ starting from \mbox{|0\rangle}_{0}=\mbox{|\mathbb{Z}{2}\rangle}, where $k\in\{0,1,2,\dots,N\}$ and $\beta_{k}$ is the normalization coefficient. It has been shown that eigenstates belonging to the special band are predominantly supported by these FSA vectors spanning the subspace $\mathcal{K}$ Turner et al. (2018a, b).

Stabilizing revivals.—In order to stabilize the revivals of $|\mathbb{Z}_{2}\rangle$, we considered various perturbations that preserve the particle-hole and time-reversal symmetry of the system (thus, pinning the energy of $|\mathbb{Z}_{2}\rangle$). Generically, most peturbations weaken the revivals. However, we find that the following range-4 deformation

[TABLE]

with $h_{2}\approx 0.05$ (derived below), significantly improves the fidelities of the revivals. We note that this form of perturbation has been previously considered in Ref. Khemani et al. (2018), which numerically found that at $h_{2}\approx 0.024$, the entire spectrum becomes least thermal 111Our basis convention is different from Ref. Khemani et al. (2018), which accounts for a trivial difference in sign of the optimal $h_{2}$.. In contrast, our value of $h_{2}$ is approximately twice larger, and the spectrum remains thermal, aside from the scarred eigenstates (see below).

Our key observation is that $\delta H_{2}$ partially cancels the errors arising in the FSA analysis. More specifically, the precision of FSA, and therefore the stability and magnitude of revivals, relies on the dynamics of $|\mathbb{Z}_{2}\rangle$ generated by $H^{\pm}_{0}$ being (nearly) closed in the subspace $\mathcal{K}$. This condition would be exactly achieved if the vectors $|k\rangle$ were eigenstates of the operator $H^{z}_{0}\equiv[H_{0}^{+},H_{0}^{-}]$, but is not satisfied for $2\leq k\leq N-2$. We find that this error can be reduced by adding $\delta H_{2}$ to the Hamiltonian and properly redefining the raising (lowing) operators, $H_{2}^{\pm}$, and the subspace $\mathcal{K}$ by replacing $\sigma_{i}^{\pm}\mapsto\sigma_{i}^{\pm}\left(1+h_{2}(\sigma_{i+2}^{z}+\sigma_{i-2}^{z})\right)$. For example, one can analytically show that the component of H_{2}^{z}\mbox{|2\rangle} perpendicular to $|2\rangle$ is minimized when $h_{2}=1/2-1/\sqrt{5}\approx 0.053$ SOM . Surprisingly, this perturbation strongly improves many-body revivals, leading to fidelity $g(\tau)\approx 0.998$ at its first maximum for a system size $N=32$. Furthermore, this deformation significantly slows down the growth of bipartite entanglement entropy SOM .

The dramatic increase in revival fidelities owing to $\delta H_{2}$ suggests that it might be possible to further enhance the oscillations, making them perfect. Extending our analytical considerations, it is natural to consider longer-range perturbations of the form

[TABLE]

which describe additional interactions between pairs of spins separated by a distance $d$, with strengths $\{h_{d}\}$. We numerically optimize $\{h_{d}\}$ by maximizing the fidelity $g(t)$ under $H=H_{0}+\delta H_{R}$ at its first revival, whose results are summarized in Fig. 1c for a system size $N=20$ with $R=10$. In SOM , we show that qualitatively similar results are obtained from other optimization methods, e.g. minimizing errors in FSA, etc. We find that the optimized $h_{d}$ decay exponentially at large $d$, and can be intriguingly very well approximated by the analytical expression

[TABLE]

where $\phi=\left(1+\sqrt{5}\right)/2$ is the golden ratio, and $h_{0}$ is a single parameter determining the overall strength. Henceforth, we will use $h_{d}$ from Eq. (4) truncated at the maximum distance $N/2$, which allows us to perform a meaningful finite-size scaling analysis. Numerical optimization of the ansatz yields $h_{0}\approx 0.051$. Below, we will derive this value from certain algebraic relations of $H^{\pm},H^{z}$ within the subspace $\mathcal{K}$.

Dynamics under the Hamiltonian $H=H_{0}+\delta H_{R}$ makes the $|\mathbb{Z}_{2}\rangle$ revivals even more stable, with $1-g(\tau)\approx 10^{-6}$ for $N=32$ at the first revival (Fig. 1a). Simultaneously, we observe that the linear growth of the bipartitle entanglement entropy is significantly reduced, and is barely discernible (Fig. 1b). The scaling analysis in SOM suggests that the average rate of local thermalization, defined by the decay of $g(t)^{1/N}$, at late times vanishes in the thermodynamic limit.

Dynamics constrains eigenstate properties.—The possible existence of a parent Hamiltonian leading to perfect oscillatory dynamics, strongly and quantifiably constrains the nonergodic nature of the quantum many-body scars. Specifically, we can appeal to the following general relation, whose proof is simple and given in SOM :

*Lemma: Consider a generic many-body Hamiltonian $H$ with extensive energy, $||H||=O(N)$. If an initial state $|\Psi_{0}\rangle$ under time evolution perfectly comes back to itself after some time $\tau>0$, independent of the system size $N$, i.e. |\mbox{\langle\Psi_{0}|}e^{-iH\tau}\mbox{|\Psi_{0}\rangle}|=1, then $|\Psi_{0}\rangle$ can be decomposed into $O(N)$ energy eigenstates, and at least one of them, $|\epsilon\rangle$, has a large overlap $|\langle\epsilon|\Psi_{0}\rangle|^{2}\geq O(1/N)$. *

If the periodic revival occurs for a physical state $|\Psi_{0}\rangle$ with a finite energy density (that obey the cluster decomposition, so that the energy variance goes as $N$), such as $|\mathbb{Z}_{2}\rangle$ in our case, this Lemma dictates the presence of a high energy eigenstate with a large overlap $\sim 1/N$ with a low-entangled state. This constitutes a violation of the ergodic scenario, where a high energy eigenstate can be viewed as a random vector in the exponentially large Hilbert space.

In accordance with this result, the decomposition of the Néel state $|\mathbb{Z}_{2}\rangle$ in the energy eigenbasis of the perturbed Hamiltonian $H$ can seen to be dominated by $N+1$ special eigenstates (Fig. 2a), which are much better separated from the bulk than in the case of unperturbed Hamiltonian $H_{0}$. We also confirm that these eigenstates exhibit nonergodic properties, such as the logarithmic scaling of entanglement entropy, and can be constructed by a straightforward extension of FSA with significantly improved accuracy Turner et al. (2018b); SOM .

Importantly, while the deformed model shows very stable revivals, the bulk of the spectrum remains thermal. To illustrate this, we compute the $r$-statistics associated to the level repulsion of the energy levels $E_{i}$, $\langle r_{i}\rangle=\left\langle{\textrm{min}(\delta_{i},\delta_{i+1})}/{\textrm{max}(\delta_{i},\delta_{i+1})}\right\rangle,$ where $\delta_{i}=E_{i+1}-E_{i}$ is the level spacing and $\langle\cdot\rangle$ indicates averaging over a symmetry-resolved Hilbert space sector foo . Figure 2b shows a clear flow in system size towards $\langle r_{i}\rangle\approx 0.53$, the Wigner-Dyson value associated with quantum chaotic Hamiltonians. In contrast, the Poisson level statistics associated with the presence of integrable dynamics would correspond to $\langle r_{i}\rangle\approx 0.386$. We note that the flow of $\langle r_{i}\rangle$ towards its Wigner-Dyson value is faster than that of the unperturbed model $H_{0}$ Turner et al. (2018a), suggesting that the deformation enhances thermalization in the bulk. In addition, the probability distribution $P(s)$ of the unfolded level spacing $s$ is consistent with the Wigner-Dyson class of the Gaussian Orthogonal Ensemble.

Algebraic structure in the subspace $\mathcal{K}$.—The almost perfect fidelity revivals of the deformed Hamiltonian imply that operators $H^{\pm}$ and $H^{z}$ form a closed algebra within the subspace $\mathcal{K}$. Indeed we find numerically that

[TABLE]

where P_{\mathcal{K}}=\sum_{k}\mbox{|k\rangle}\mbox{\langle k|} is the projector onto the subspace, and $\Delta$ is a constant. As $|0\rangle=|\mathbb{Z}_{2}\rangle$ is an eigenstate of $H^{z}$, $|k\rangle$ are also approximate eigenvectors of $H^{z}$ with harmonically spaced eigenvalues $H_{k}^{z}=\langle k|H^{z}|k\rangle$ so that $\Delta=H^{z}_{k+1}-H^{z}_{k}$. Thus, upon a suitable rescaling, the operator $H^{z}$ plays the role of $S^{z}$ in the SU(2) algebra, and $H^{\pm}$ play the role of spin-raising and lowering operators within $\cal K$. As the dimensionality of the subspace $\cal K$ is $N+1$, this implies that the operators form a spin $s=N/2$ representation of the SU(2) algebra, with $|\mathbb{Z}_{2}\rangle$ and |{\mathbb{Z}^{\prime}_{2}}\rangle=\mbox{|\downarrow\uparrow\downarrow\uparrow\dots\rangle} being the lowest and highest weight states respectively. To check this, we explicitly evaluated the matrix elements \mbox{\langle k+1|}H^{+}\mbox{|k\rangle}. Figure 3a confirms that up to an overall multiplicative factor, these matrix elements of $H^{+}$ reproduce the corresponding matrix elements of the spin raising operator $S^{+}$ in this representation.

Thus, the virtually perfect oscillatory dynamics of $|\mathbb{Z}_{2}\rangle$ can be understood as a large spin $(s=N/2)$ pointing initially in an emergent “$z$-direction”, undergoing a coherent Rabi oscillation under the Hamiltonian $H=H^{+}+H^{-}$, which is akin to the $S^{x}$ operator, with period $\tau=2\pi/\sqrt{2\Delta}$. We stress that the emergence of this SU(2) structure within $\mathcal{K}$ is nontrivial, since the Hamiltonian $H$ by itself does not have any rotational symmetry.

The identification of this emergent algebra allows us to fix $h_{0}$ of our ansatz for $h_{d}$ analytically. In particular, $H^{z}_{k}$ can be explicitly calculated for $k=0,1$ in the thermodynamic limit. Imposing a harmonic spacing, i.e. $H^{z}_{k}=\Delta(k-N/2)$, leads to a nontrivial constraint SOM

[TABLE]

where $h\equiv 2\sum_{n\geq 2}h_{n}(-1)^{n}$. This fixes $h_{0}\approx 0.0506656$ in our ansatz Eq. (4), which agrees very well with the numerically optimized value. Furthermore, Eq. (6) determines the harmonic gap $\Delta=(1-h)^{2}\approx 0.835845$, and, correspondingly, the oscillation period $\tau\approx 4.85962$, which are also in excellent agreement with those from exact numerical simulations SOM .

Toy model.—The above investigations reveal that an emergent SU(2) structure within a special subspace underpins the many-body revivals. Motivated by this, we construct a (solvable) toy model that exhibits similar phenomenology: in this model, there is a band of nonthermal eigenstates supporting perfect oscillatory dynamics and exhibiting logarithmic entanglement, embedded in an otherwise thermal spectrum.

Consider a system of $N$ spin-1/2 particles on a ring. The special subspace $\mathcal{V}$ of our model is defined as the common null space of $N$ projection operators $P_{i,i+1}=(1-\vec{\sigma}_{i}\cdot\vec{\sigma}_{i+1})/4$ onto neighboring pairs of singlets. Such subspace is spanned by the $N+1$ states of the largest spin representation $s=N/2$ of the SU(2) algebra. We enumerate the basis states of $\mathcal{V}$ by eigenstates of the $S^{x}=\sum_{i}\sigma_{i}^{x}/2$ operator, $|s=N/2,S^{x}=m_{x}\rangle$ with $m_{x}\in\{-s,\dots s\}$.

Now, we take any Hamiltonian of the form

[TABLE]

where $V_{ij}$ is a generic two-spin operator acting on spins $i$ and $j$, e. g. $V_{i,j}=\sum_{\mu\nu}J_{ij}^{\mu\nu}\sigma_{i}^{\mu}\sigma_{j}^{\nu}$ with arbitrary coefficients $J_{ij}^{\mu\nu}$. Note that $H_{\text{toy}}$ does not commute with $P_{i,i+1}$ nor $S^{x}$; thus, it does not have any obvious local symmetries. However, it can be easily verified that the states \mbox{|s=N/2,S^{x}=m_{x}\rangle}\in\cal V are eigenstates with harmonically spaced energies $E=\Omega m_{x}$. On the other hand, the states in the Hilbert space that do not belong to $\mathcal{V}$ are affected by the second term in Eq. (7), and hybridize to form ergodic eigenstates SOM . Now, initializing our system, for example, in the lowest weight state $|N/2,S^{z}=-N/2\rangle$ leads to rotations of a large spin around the $x$-axis with frequency $\Omega$, whose motion remains in the subspace $\cal V$. We note that our construction is reminiscent of Shiraishi and Mori’s Shiraishi and Mori (2017) where a set of local projectors were used to embed certain nonergodic energy eigenstates into the bulk of a many-body spectrum.

Clearly, $H_{\text{toy}}$ exhibits all the features of perfect quantum many-body scarring, and is appealing as an intuitive understanding of the origin of scars in the constrained spin models. However, there remain many open questions: first, the explicit relationship between the constrained spin model Eqs. (1)-(3) and the toy model Eq. (7) is not obvious. The nonisomorphic Hilbert spaces, as well as the nontrivial entanglement dynamics in the constrained model (Fig. 1b), suggests that the mapping between these two models, if exists, cannot be strictly local. Second, it is highly desirable to find an analytic derivation of the deformation, Eq. (3), that leads to the emergent SU(2) algebra in the constrained spin model, and understand when such deformations exist for other local models. We note that this emergent algebra is reminiscent of the $\eta$-pairing symmetry that holds exactly in the Hubbard model Yang (1989), which allows to construct exact eigenstates at finite energy density with logarithmic Vafek et al. (2017) and subthermal entanglement Veness et al. (2017). The exact expression for the analogue of $\eta$-pairing operator in our case, as well as the general relations between such operators, their algebra and scars, also remain an open question.

Summary and outlook.— To summarize, we have constructed a constrained spin model which exhibits nearly perfect quantum many-body scars. The remarkably long-lived oscillatory dynamics suggests that quantum scars remain stable in the thermodynamic limit. We showed that the dynamics can be understood in terms of a large, precessing SU(2) spin, and used this intuition to introduce a family of toy models with perfect scarring. In future work, it would be highly desirable to find an analytical mapping between the toy models and the constrained spin model. Moreover, the approach developed here may be applied to stabilize other types of quantum scars, in particular the ones originating from the $|\mathbb{Z}_{3}\rangle$ state in the model (1) Turner et al. (2018b), as well as the ones found in higher-spin constrained models Ho et al. (2018). Another exciting challenge is to find models in which the MPS-based description of quantum scars trajectory becomes exact Ho et al. (2018). In a broader context, special non-thermalizing trajectories may have intriguing connections to revivals/slow thermalization in strongly rotating gravitational systems da Silva et al. (2015); DJa . To understand the origin of this non-thermalizing dynamics, it would be valuable to establish whether quantum many-body scars can emerge from a dynamics that goes through states with high entanglement.