Quantum Many-Body Scars and Weak Breaking of Ergodicity

Maksym Serbyn1, Dmitry A. Abanin2, and Zlatko Papic31IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria2Department of Theoretical Physics, University of Geneva,24 quai Ernest-Ansermet, 1211 Geneva, Switzerland and

3School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom(Dated: November 20, 2020)

Recent discovery of persistent revivals in quantum simulators based on Rydberg atoms have pointed to theexistence of a new type of dynamical behavior that challenged the conventional paradigms of integrability andthermalization. This novel collective effect has been named quantum many-body scars by analogy with weakergodicity breaking of a single particle inside a stadium billiard. In this overview, we provide a pedagogicalintroduction to quantum many-body scars and highlight the newly emerged connections with the semiclassicalquantization of many-body systems. We discuss the relation between scars and more general routes towardsweak violations of ergodicity due to “embedded” algebras and non-thermal eigenstates, and highlight possibleapplications of scars in quantum technology.

I. INTRODUCTION

A fundamental challenge in physics is to predict how sys-tems evolve in time when taken out of their equilibrium state.The behavior of familiar classical systems, e.g., the clock pen-dulum, is perfectly periodic and thus possible to predict at latetimes. This is an example of integrable behavior. Integrablesystems, however, are special; under perturbations, their peri-odic behavior gradually disappears and ultimately gives wayto chaos. The microscopic chaotic motion allows an ensembleof coupled systems to reach the state of thermal equilibrium,which can be effectively described using the textbook meth-ods of statistical mechanics.

While many open questions remain in the field of classi-cal chaotic systems, the non-equilibrium quantum matter isanother frontier of modern research. The intense focus on un-derstanding routes to thermal equilibrium in isolated quantumsystems with many degrees of freedom is partly due to theadvent of synthetic systems based on atoms, ions, or super-conducting circuits as elementary building blocks [1, 2]. Therelaxation times of these systems are long compared to, e.g.,those of electrons in solids, and experiments allow to mon-itor quantum dynamics at the single-atom level, thus open-ing a window to non-equilibrium quantum phenomena. Thesecapabilities have renewed the interest in foundational ques-tions of quantum physics, such as the emergence of statistical-mechanics description in an isolated many-body system. Pro-vided parts of the system are able to act as heat reservoirsfor its other parts, an initial non-equilibrium state relaxes tothermal equilibrium, with a well-defined effective tempera-ture. Such systems are called thermal or quantum-ergodic.

Surprisingly, not all quantum systems are ergodic. Forexample, finely tuned one-dimensional systems [3] may failto thermalize due to their rich symmetry structure knownas quantum integrability. However, when such systems areweakly perturbed, ergodicity is restored. On the other hand,disorder in an interacting system may induce many-body lo-calization (MBL) [4, 5], a close relative of the celebrated An-derson localization for non-interacting particles. Similar tointegrable systems, MBL systems have a macroscopic num-ber of conservation laws, which prevent thermalization. These

conservation laws, however, are robust with respect to pertur-bations, and MBL is an example of a phase of matter whichembodies strong ergodicity breaking.

The behavior of quantum-ergodic systems is governed bythe so-called Eigenstate Thermalization Hypothesis (ETH) [6,7], a powerful conjecture which explains the process of ther-malization at the level of the system’s energy eigenstates. TheETH states that individual eigenstates of quantum-ergodicsystems act as thermal ensembles, thus the system’s relax-ation does not depend strongly on the initial conditions. How-ever, in 2018, an experiment on a new family of Rydberg-atomquantum simulators revealed an unforeseen dynamical behav-ior [8]. The experiment observed significant qualitative differ-ences in the dynamics, depending on the choice of the initialstate: while certain initial states showed relaxation to thermalensembles, as expected in an ergodic system, other states ex-hibited periodic revivals. Such revivals were surprising giventhat the system did not have any conserved quantities otherthan total energy and it was free of disorder, thus ruling outintegrability and localization as possible explanations. Whatmade a particular initial state find its way back in a Hilbertspace with dimension in excess of 4 × 1010, for the experi-mental system of 51 Rydberg atoms?

A flurry of subsequent theoretical works have addressed thepuzzle posed by the experiment: the nature and origin of thisnew regime of ergodicity breaking, intermediate between ther-malization and strong ergodicity breaking. The key to under-standing this new behavior was the discovery of anomalous,non-thermal eigenstates in the highly excited energy spectrumof the Rydberg atom system [9]. These eigenstates provide anexample of what we will refer to as weak violation of ETH:despite being strongly non-thermal, the anomalous eigenstatescomprise a vanishing fraction of the Hilbert space and they areimmersed in a much larger sea of thermal eigenstates.

One may be puzzled as to why non-thermal eigenstates inthe highly-excited spectrum would bear any importance. Af-ter all, the energy level spacing between highly-excited eigen-states in a many-body system decreases exponentially withthe number of atoms, and preparing the system in a particulareigenstate is challenging, if not impossible. Therefore, sucheigenstates might be expected to remain “invisible” to any ex-

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periment performed in the lab. Nevertheless, it was shownthat these non-thermal eigenstates indeed underpin the real-time dynamics observed in experiment, inspiring new con-nections between quantum many-body dynamics and stud-ies of classical chaos [10]. By analogy with chaotic stadiumbilliards, which also host non-thermal eigenstates whose er-godicity breaking has been visualized as “scars” of classicalperiodic orbits [11], the non-thermal eigenstates in the Ry-dberg atom chains have been dubbed “quantum many-bodyscars” (QMBSs).

The discovery of QMBSs has triggered broader investiga-tions of weak ergodicity breaking in a variety of quantum sys-tems, as opposed to the strong ergodicity breaking by integra-bility and disorder. These studies have unearthed a rich land-scape of many-body scarred models with universal algebraicstructures, shedding new light on the well-known condensedmatter models such as the Hubbard model and the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. In this review, we sur-vey the recent progress in weak ergodicity breaking phenom-ena by highlighting relations between different realizations ofnon-thermal dynamics, schematically illustrated in Fig. 1. Aswe explain in detail below, known scarred models are charac-terized by a subspace which is decoupled from the rest of theenergy spectrum and cannot be attributed to a symmetry of thesystem. The origin of this subspace can vary depending on themodel, and a few common mechanisms, highlighted in Fig. 1,will be presented below. Furthermore, we discuss novel the-oretical approaches that aim to bridge quantum and classicalchaos in many-body systems, and the representation of scarreddynamics in terms of tensor networks. Finally, we outlinepromising directions for future research, putting an emphasison emerging connections between weak ergodicity breakingand other fields. We mention other routes to weak ergodicitybreaking, such as mesons in theories with confinement, the ef-fects of additional conservation laws that fracture the Hilbertspace, and frustration-driven glassy behavior, whose relationto QMBS yet remains to be fully understood. We concludewith a brief discussion of potential applications of QMBS inquantum sensing and metrology.

II. PHENOMENOLOGY OF QUANTUM SCARRING

In experiment, quantum thermalization is convenientlyprobed by quenching the system: one prepares a non-equilibrium initial state |ψ(0)〉, which is typically short-rangecorrelated (e.g., a product state of particles), and monitors itsfate after time t. For cold atoms and trapped ions, which arewell-isolated from any thermal bath, the system can be as-sumed to evolve according to the Schrodinger equation forthe system’s Hamiltonian H . The process of thermalizationis monitored by measuring the time evolution of local observ-ables, 〈O(t)〉, as the dynamics explores progressively largerparts of the system’s energy spectrum. The results of suchmeasurements are remarkably different in typical quantum-ergodic systems compared to non-ergodic localized systems.In ergodic systems, following a brief initial transient, local ob-servables relax to their thermal value and stay near that value

Krylov restricted thermalization

1D limit of FQHE,fracton models,

bosons in optical lattices

Projector embedding

topological lattice models,frustrated magnets,

lattice supersymmetry

Many-body scars and weak ETH breakdown

Quantum technologyRydberg atom

quantum simulators,quantum metrology & control

Quantum-classical correspondence

quantum many-body chaos,quantum KAM theorem

Spectrum generating algebra

AKLT model, PXP model, spin-1 XY magnets,

extended Hubbard model

FIG. 1. Top: Examples of physical systems where weak ergodic-ity breakdown has recently been discovered. These systems can beclassified according to three main mechanisms of many-body scar-ring, reviewed below. Bottom: Studies of many-body scarring havesparked interest in fundamental problems in the field of quantummany-body chaos and potential applications in quantum technology.

at later times. This behavior is reminiscent of classical chaoticsystems, which effectively “forget” their initial condition. Incontrast, in MBL systems local observables reach a station-ary value which is non-thermal, retaining the memory of theinitial state – a hallmark of broken ergodicity.

It is important to emphasize that the dynamical behaviorof ergodic or non-ergodic systems mentioned above has beenverified to hold for quenches from any physical initial states.This typicality was recently found to break down in experi-ments on arrays of Rydberg atoms [8]. The building blockof these experiments is an individual Rydberg atom, whichmay be viewed as an effective two level system, where thetwo states |◦〉, |•〉 correspond, respectively, to an atom in theground state and an atom in the so-called Rydberg state. Whensubject to a microwave field, each atom undergoes Rabi os-cillations, |◦〉 ↔ |•〉, freely flipping between its two states.However, when assembled in an array, the atoms in |•〉 statesinteract via repulsive van der Waals force, whose strength isstrongly dependent on the distance between the atoms. Bytuning the inter-atom distance one can achieve the regime ofthe Rydberg blockade [12] where the excitations of neighbor-ing atoms, e.g., | · · · •• · · · 〉, are energetically prohibited. Thismakes the system kinetically constrained as each atom is onlyallowed to flip if all of its neighbors are in |◦〉 state.

The dynamics of large one-dimensional arrays of Rydbergatoms has been probed by studying quenches from a period-2 density wave initial state, |Z2〉 ≡ |•◦•◦ . . .〉, which is thestate containing the maximal possible number of Rydberg ex-citations allowed by the blockade. The dynamics of domainwall density, where domain walls are defined as adjacent |◦◦〉

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|ψ|2

FIG. 2. Weak breakdown of thermalization. a: Observation ofpersistent revivals in the domain-wall density when Rydberg-atomquantum simulators are quenched from |Z2〉 initial state [8]. Theoscillations were found to persist in large chains up to 51 atoms,and they were reproduced in the dynamical simulations using matrixproduct state (MPS) methods. b: Weak breakdown of thermalizationin Rydberg atom arrays was theoretically attributed to atypical eigen-states – quantum many-body scars – which are distinguished by theiranomalously enhanced overlaps with |Z2〉 state [9]. Each dot rep-resents an eigenstate of energy E in the PXP model in Eq. (1) withL = 32 atoms. Color scheme illustrates the density of points.

or |••〉 configurations (the latter is excluded in the regime ofperfect blockade), revealed long-time oscillations shown inFig. 2a. Such robust oscillations were surprising, given thatthe state |Z2〉 is not a low-energy state, but rather correspondsto an infinite-temperature ensemble for the atoms in the Ry-dberg blockade. Thus the quench dynamics is not a priorilimited to a small part of the energy spectrum, and cannotbe attributed to weakly interacting low-energy excitations inthe system. Moreover, the oscillations could not be explainedby any known conservation law in the system, and their fre-quency did not coincide with the bare Rabi frequency, sig-nalling the importance of many-body effects.

The experimental findings have been corroborated by theo-retical studies of the idealized model of atoms in the Rydbergblockade – the so-called “PXP” model [13],

HPXP =∑i

Pi−1σxi Pi+1. (1)

Here σxi = |◦〉i〈•|i + |•〉i〈◦|i denotes the Pauli x matrix re-

sponsible for Rabi oscillations of an individual atom on site i,and projectors onto the ground state, Pi = |◦〉i〈◦|i, enforcethe Rydberg blockade and make the model interacting. Nu-merical studies have observed that the PXP energy levels re-pel, thus confirming that the model is non-integrable [9]. Fur-ther, simulations of the quench dynamics have demonstratedrevivals of the wave function and local observables when thesystem is quenched from |Z2〉 state [14, 15] as well as theperiod-3 density wave state, |Z3〉 ≡ |•◦◦•◦◦ . . .〉. In con-trast, other initial states, such as |0〉 = |◦◦◦ . . .〉 state andperiod-4 density wave state, showed fast relaxation without re-vivals [16]. Similar phenomenology has also been identified

in the two-dimensional PXP model [17, 18], and in relatedmodels of transverse Ising ladders [19] and the periodicallydriven PXP model [20–22]. This special behavior of the PXPmodel was found to be robust to certain perturbations [23] in-cluding disorder [24], while other perturbations destroy it bymaking the PXP model integrable [25], frustration-free [26]or thermalizing [16].

An important step in understanding the origin of the ob-served oscillations was made by theoretical studies of many-body eigenstates of the PXP model. Figure 2b reveals anumber of “atypical” eigenstates in the spectrum of the PXPmodel, which are distinguished by their high overlap with thesame |Z2〉 state used for initiating the quench in Figure 2a. Fora quantum-ergodic system, the ETH stipulates that these over-laps should be a smooth function of energy; by contrast, thestrong enhancement of overlaps of certain eigenstates com-pared to others at the same energy E, seen in Figure 2b, sig-nals a breakdown of the ETH. This is a weak violation ofthe ETH since the number of special eigenstates was foundto scale linearly with the number of atoms, in contrast toan exponentially larger number of thermalizing eigenstates.Moreover, the special eigenstates are “embedded” at roughlyequidistant energies in an otherwise thermalizing spectrum ofthe PXP model, thus accounting for the dynamical revivalsin Figure 2a with a single dominant frequency. We notethat various schemes for constructing some of these atypicaleigenstates in the PXP model, both exact [27] and approxi-mate [16, 28], have been devised.

The core phenomenology of the PXP model – the smallnumber of ETH-violating eigenstates within the thermalizingspectrum and the presence of many-body revivals and slow re-laxation in quenches from specific initial states – bears paral-lels with the physics of a single particle confined to a stadium-shaped billiard. In the latter case, the dynamics of a quantumwave packet is known to be sensitive to the initial conditions,which was attributed to the existence of unstable periodic or-bits when the billiard is made classical by sending ~→ 0 (seeBox 1). Initialization of a quantum wave packet on or nearsuch an orbit leads to dynamical recurrences where the par-ticle tends to cluster around the orbit, a phenomenon calledquantum scarring [11]. Moreover, the periodic orbit leavesan imprint on the eigenfunctions of a particle, which exhibitanomalous concentration in the vicinity of the periodic orbit,rather than being spread uniformly across the billiard.

While the analogy between revivals in the PXP model andscars in billiards is indeed suggestive, making it more precisehas required new theoretical insights. As we discuss below,these approaches have followed two complementary strands.The first approach, discussed in Section III, focuses on con-structing non-thermal eigenstates in many-body quantum sys-tems. This approach has been particularly fruitful in identi-fying a large family of scarred many-body systems, some ofwhich were highlighted in Fig. 1. An alternative perspective,which focuses on parallels between quantum many-body dy-namics and classical dynamical systems, will be discussed inSection IV.

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Box 1 | Scars in quantum billiards

What can we learn about the behavior of a quantum system from looking at its classical counterpart? The Bohr-Sommerfeldquantization demonstrates that classical integrable systems, such as harmonic oscillator or hydrogen atom, have very specialquantum spectrum and eigenstates. For classically chaotic systems, such as the Bunimovich stadium in Fig. B1 below, this quan-tization method does not work. Typically, quantum counterparts of classically chaotic systems display level repulsion in theirspectrum and their eigenstates look random. However, short unstable periodic orbits may leave a strong imprint on the system’squantum dynamics and eigenstate properties – this influence of classical periodic orbits is called “quantum scarring” [29]: Aquantum eigenstate of a classically chaotic system has a ‘scar’ of a periodic orbit if its density on the classical invariant mani-folds near the periodic orbit is enhanced over the statistically expected density. Scarring is a weak form of ergodicity breakingsince the individual eigenstates of the billiard are “almost always” ergodic as the phase space area affected by scarring vanishesin the semiclassical limit ~→ 0, shrinking around the periodic orbit.

How does one detect scars? Most simply, by visualizing the wave function probability density, as in the case of Bunimovichstadium in Fig. B1c-d. The probability density can be directly compared with the corresponding classical orbits in Fig. B1a-b.Moreover, scars leave an imprint on the dynamics: a wave packet launched in the vicinity of an unstable periodic orbit will tendto cluster around the orbit at later times, displaying larger return probability than a wave packet launched elsewhere in phasespace. This is seen in the autocorrelation function in Fig. B1e. Furthermore, such a wave packet can be expanded over a smallnumber of eigenstates that have approximately similar energy spacing – see Fig. B1f – in contrast to an arbitrary wave packet.

What is the significance of quantum scars? First, scars provide a counterexample to the intuitive expectation that everyeigenstate of a classically chaotic system should locally look like random superposition of plane waves [30]. Second, alsocounterintuitively, a scarred quantum system appears to bear a stronger influence of the periodic orbit compared to its classicalcounterpart, since in the latter case there is no enhancement of density along the periodic orbit in the long-time limit. Finally,scars play a role in many physical systems, including microwave cavities [31], semiconductor quantum wells [32], and thehydrogen atom in a magnetic field [33].

a c e

Ei − Egs

b d f

Figure B1 | Scars in a stadium billiard. a: Classical particle initialized away from an unstable periodic trajectory displays chaotic motion.b: In contrast, when launched near an unstable periodic trajectory shown in red, the particle spends a long time in its vicinity before escaping.c: The probability density of a typical highly excited eigenstate of the billiard resembles a collection of random plane waves. d: Probabilitydensity of a quantum-scarred eigenstate looks very different from a collection of plane waves, instead being strongly concentrated near theperiodic trajectory. The eigenstates of the billiard are obtained by solving the Schrodinger equation with the wave function vanishing at theboundary. e-f: Autocorrelation function of a Gaussian wave packet launched vertically at the center of billiard shows pronounced revivals(blue line) and the expansion of the initial state over the billiard eigenstates reveals a periodic sequence of peaks. In contrast, the wave packetlaunched at 45◦ angle does not revive and features a continuum of frequencies (red lines).

III. MECHANISMS OF WEAK ERGODICITY BREAKING

In recent works, non-thermal eigenstates have been theoret-ically identified in a number of non-integrable quantum mod-els, revealing a plethora of scarred many-body systems. Aunifying property of all these systems is the emergence ofa decoupled subspace within the many-body Hilbert space,not related to any global symmetry of the full Hamiltonian,

spanned by the non-thermal eigenstates. Formally, the Hamil-tonian of such systems is effectively decomposed as

H ≈ Hscar

⊕Hthermal, (2)

where Hscar is the scarred subspace which is (approximately,or even exactly) decoupled from the thermalizing subspace,Hthermal. The eigenstates that inhabit the subspace Hscar vi-olate the ETH and have different properties compared to the

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(i) (ii) (iii)

DOS DOS DOS

FIG. 3. Embeddings of scarred eigenstates in a many-body system. The Hamiltonian matrix H , acting on the full Hilbert space, isschematically split into the non-ergodic subspace (blue box) and the remaining thermal subspace (red box). Panels (i)-(iii) illustrate possiblemechanisms underlying the emergence of the decoupled subspace. Density of states (DOS) schematically shows the positions of scarred eigen-states (blue dots) within the continuum of thermalizing states. (i) Exact scars due to a spectrum generating algebra and resulting eigenstateswhich are equidistant in energy. (ii) Exact Krylov subspace represented by a tridiagonal matrix. (iii) General subspace resulting from projectorembedding. In any of these cases, the scarred subspace need not be exactly decoupled from the thermalizing bulk, like in the PXP model.

majority of thermal eigenstates in Hthermal. Below we eluci-date and contrast three possible mechanisms (i)-(iii) that leadto such a decoupled subspace, whose summary is given inFig 3. As we explain next, these mechanisms are realized bydiverse physical systems illustrated in Fig. 4.

(i) Spectrum-generating algebra. A family of highly-excited non-thermal eigenstates can be constructed system-atically using the “spectrum generating algebras”, originallyintroduced in the context of high energy physics [34] andsubsequently applied to the Hubbard model [35, 36]. Thefirst exact construction of many-body scarred eigenstates us-ing this method has been achieved in the Affleck-Kennedy-Lieb-Tasaki (AKLT) model [37], extending earlier work ofRef. [38]. In the AKLT model, as well as in several othernon-integrable spin chain models [39–42], the spectrum gen-erating algebra allows to exactly construct scarred eigenstates,which can physically be interpreted as quasiparticle conden-sates built on top of the system’s ground state.

The spectrum generating algebra is defined by a local op-erator Q† which obeys the relation ([H, Q†] − ωQ†)W = 0,where ω is some energy scale and W is a linear subspace ofthe full Hilbert space left invariant under the action of Q†.In the Hubbard model, the subspace W coincides with thefull Hilbert space and Q† assumes the role of an SU(2) “eta-pairing” symmetry [43]. The scenario relevant for many-body scarring is when W is not the full Hilbert space and Q†

is not associated with a symmetry of the Hamiltonian [44],e.g., as realized in the extended Hubbard model [45, 46].Then, starting from some eigenstate |ψ0〉 of H , the towerof states, (Q†)n|ψ0〉, are all eigenstates of H with ener-gies E0 + nω. The eigenstates (Q†)n|ψ0〉 are candidatesfor many-body scars because their properties can be stronglynon-thermal, provided |ψ0〉 is sufficiently “simple” (e.g., theground state of H if the latter is gapped) and Q† is a local op-erator (e.g., a sum of on-site spin flips in quantum spin chainmodels).

Often, Q† is interpreted as creating a wave packet corre-sponding to a “quasiparticle” excitation (e.g., a magnon), andrepeated applications of Q† create a condensate of such quasi-particles. In a class of frustration-free models which includethe AKLT [47], the quasiparticle condensates are non-thermal

and violate the ETH. A convenient diagnostic for the ETHviolation and identifying scarred eigenstates is the entangle-ment entropy S, i.e., the von Neumann entropy of the reduceddensity matrix describing a subsystem of the full system in agiven eigenstate. In ETH systems, the entanglement entropyof eigenstates scales extensively with the size of the subsys-tem. By contrast, scarred eigenstates have much lower entan-glement, thus they can be detected as entropy-outliers, as seenin Fig. 4a for the AKLT model. Another important clue whenlooking for scarred eigenstates is that they appear at energieswhich are rational multiples of ω.

In related approaches, scarred subspace may be conve-niently engineered if H has a certain symmetry, such as ina spin-1 XY magnet [39]. This model also hosts perfectwave function revivals from certain easily preparable initialstates, as illustrated by its quantum fidelity, |〈ψ0|ψ(t)〉|2, inFig. 4b. Further variations of the algebraic construction havebeen given for spin-1 magnets [41], a spin-1/2 model withemergent kinetic constraints [40], a spin chain with the dy-namical Onsager algebra [42], and a two-dimensional frus-trated spin-1/2 kagome antiferromagnet [52]. Recent workshave provided a more general framework for the constructionof non-thermalizing subspaces using group theory [53–55]. Ithas also been pointed out that spectrum generating algebracan arise in open quantum systems, e.g., in the presence ofdissipation or periodic driving [56, 57].

Finally, the PXP model in Eq. (1) also realizes an embeddedalgebra, in this case an approximate su(2) spin algebra [58].The spin raising operator H+ creates a Rydberg excitationanywhere on the even sublattice and removes an excitationanywhere on the odd sublattice, while respecting the Rydbergconstraint. Similarly, the spin lowering operatorH− performsthe same process with the sublattices exchanged. The rea-son for this choice of H± is that their commutator definesthe z-projection of spin, Hz ≡ 1

2 [H+, H−], for which |Z2〉

plays the role of the extremal weight state. In principle, us-ing this construction, one could form the Casimir operator forthe su(2) algebra and obtain the tower of non-thermal eigen-states by acting repeatedly withH+ on the ground state of theCasimir, mirroring the general procedure outlined previously.However, this procedure is not analytically tractable due to thefact that the ground state of the Casimir operator is not known

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FIG. 4. Mechanisms and physical realizations of weak ergodicity breaking. a-b: Many-body scars due to spectrum generating algebras. a:Exact scars in the AKLT model distinguished by their low entanglement entropy S [37]. b: Perfect fidelity revival due to exact scars in a spin-1XY magnet [39]. c-d: Krylov subspaces in fracton-like models. The model of the fractional quantum Hall effect on a thin cylinder [48] supportsscarred eigenstates and revivals from a product state (c), while local observables in a spin-1 model with dipole conservation symmetry [49]weakly violate the ETH (d). e-f: Projector embedding of scarred states into thermalizing spectra. e: Geometrical frustration on a 2D square-kagome lattice (inset) allows to embed weakly-entangled states of localized magnons (circled) [50]. f: Deformations of topological models,such as the 1D cluster model, allows to embed scarred states (circled) [51]. The same method applies to 2D toric code and 3D X-cube models.

and the algebra of the PXP model is only approximate (re-cent works, however, have shown that weak deformations ofthe PXP model make the algebra and dynamical revivals pro-gressively more accurate [58–60]). Instead, different schemeshave been used to approximate scarred eigenstates in the PXPmodel starting from |Z2〉 product state [16, 61].

(ii) Krylov restricted thermalization. A spectrum gener-ating algebra relies on the tower operator Q† to construct thenon-thermalizing subspace. In practice, finding such opera-tors is likely to be limited to cases where they can be expressedin a simple local form. We next describe a related mechanismfor producing exactly embedded subspaces, which does notrequire a priori knowledge of Q†.

For a Hamiltonian H and some arbitrary vector in theHilbert space, |ψ0〉, the Krylov subspace, K, is defined as aset of all vectors obtained by repeated action of H on |ψ0〉,i.e., K = span{|ψ0〉, H|ψ0〉, H2|ψ0〉, . . .}. In numerical lin-ear algebra, subspace K is routinely used in iterative methodsfor finding extremal eigenvalues of a large matrix H , such asthe Arnoldi and Lanczos algorithms. A special case of partic-ular interest is when K happens to be finite, i.e., the sequenceof Krylov vectors terminates after n+1 steps: Hn+1|ψ0〉 = 0.This causes a dynamical “fracture” of the Hilbert space in thesense that the Schrodinger dynamics initialized in any state|ψ〉 ∈ K must remain within the same subspace at any latertime, exp(− i

~ tH)|ψ〉 ∈ K, hence the name “Krylov restrictedthermalization” [62]. Note that K can be as small as one-dimensional or exponentially large in system size, and it canbe either integrable or thermalizing. By performing a Gram-Schmidt orthogonalization, we can transformK into a tridiag-

onal matrix, which will be perfectly decoupled from the restof the spectrum of H .

We distinguish the case of the Krylov subspace from theprevious case of the algebra subspace because, in general, itcan be difficult to analytically diagonalize the tridiagonal ma-trix and construct the corresponding Q†. Note that a tridiag-onal Krylov subspace does not guarantee revivals, even if theroot state |ψ0〉 is experimentally preparable. This is becausegeneral tridiagonal matrices do not support revivals, unlesstheir matrix elements are tuned to special values. Moreover,depending on the model, the Krylov subspace may only beapproximately decoupled, i.e., there may exist small matrixelements that connect it with the thermalizing bulk.

Realizations of Krylov subspaces occur in models of thefractional quantum Hall effect in a quasi one-dimensionallimit [48] and in models of bosons with constrained hoppingon optical lattices [63, 64]. In the former case, shown inFig. 4c, K is thermalizing but also supports scarred eigen-states and revivals from a particular density-wave productstate. Scarred towers in overlaps with a density-wave statein Fig. 4c signal a weak ETH violation due to anomalous con-centration of eigenstates in the Hilbert space, analogous to thecase of PXP model in Fig. 2b.

More generally, the Krylov fracture was shown to bepresent in both Hamiltonian models [49] and random-circuits [65, 66] which feature a conservation of dipole mo-ment – a symmetry characteristic of the so-called fracton topo-logical phases. An example of weak breaking of the ETH in aone-dimensional fracton-like model [49] is shown in Fig. 4d,which shows the matrix elements of a local spin observable

7

in the system’s eigenstates. Once again, the large spread ofthese matrix elements and their nonmonotonic behavior as afunction of energy indicate a weak violation of the ETH.

(iii) Projector embedding. At the highest level of abstrac-tion, we can ask if one could embed an arbitrary subspace intothe spectrum of a thermalizing system? For concreteness, as-sume we are given an arbitrary set of states |ψi〉 that span ourtarget scar subspace Hscar = span{|ψi〉}, which we wish toembed into a thermalizing Hamiltonian H as in Eq. (2). Thiscan be achieved via the “projector embedding” constructiondue to Shiraishi and Mori [67], which has proven to be a ver-satile method for constructing scarred states in diverse modelsranging from lattice supersymmetry [68] to flat bands [69].

Projector embedding assumes that our target states |ψi〉, be-ing non-thermal, are annihilated by local projectors, Pi|ψj〉 =0, for any i ranging over lattice sites 1, 2, . . . , L. Next, con-sider a lattice Hamiltonian of the form H =

∑Li=1 PihiPi +

H ′, where hi are arbitrary operators that are have support ona finite number of sites around i, and [H ′, Pi] = 0 for alli. It follows that PiH|ψj〉 = PiH

′|ψj〉 = H ′Pi|ψj〉 = 0,thus [H,

∑i Pi] = 0. Therefore, H takes the desired block

diagonal form in Eq. (2). The target eigenstates can in prin-ciple be embedded at arbitrarily high energies for a suitablychosen H ′, which also ensures that the model is overall non-integrable [67]. However, there is no guarantee that the em-bedded states must be equidistant in energy and they may evenbe degenerate, such that this scheme could produce modelswhich do not exhibit wavefunction revivals.

Physical mechanisms such as geometrical frustration incorrelated materials can implement the projector embed-ding. Fig. 4e shows an example of a 2D square-kagomelattice model, where scarred states representing localizedmagnons can be embedded into the highly-excited energyspectrum [50]. Moreover, projector embedding naturallylends itself to physical realization in various topologically-ordered lattice models [70], which are themselves defined interms of projectors. Deformations of such models [51], whichinclude the 2D toric code model and 3D X-cube model, canbe used to embed scarred states, as shown in Fig. 4f.

In summary, we have presented three mechanisms (i)-(iii)that can give rise to non-thermal eigenstates in the spectrumof an otherwise thermalizing model. We emphasize that thesemechanisms are not mutually exclusive, they rather under-score how weak ergodicity breaking is manifested in vari-ous models. The connections between different mechanismsare most saliently illustrated by the PXP model. The PXPmodel, with its generalizations to higher spins [10] and quan-tum clock models [71], falls into category (i), yet its algebra isonly approximate, and the subspace has weak couplings to thethermal bulk. In addition, the PXP model also realizes an ap-proximate Krylov subspace built upon |Z2〉 state (the subspacebecomes exact when generated byH+). Finally, a single PXPeigenstate at zero energy in the middle of the spectrum canbe viewed as projector-embedded AKLT ground state [72].While similar connections between scarring mechanisms maybe anticipated in other models mentioned above, their demon-stration remains an open problem.

IV. SCARS AND PERIODIC ORBITS IN MANY-BODYSYSTEMS

Thus far we have surveyed a growing landscape of mod-els which host ETH-violating scarred eigenstates. While theterm “quantum many-body scar” is often used to denote gen-eral non-thermal eigenstates in a non-integrable system, acomplete analogy with scars in quantum billiards necessitatessome notion of a classical trajectory underlying the atypicaleigenstates. This directly relates to the important problem inthe field of quantum chaos: finding the classical counterpartof a many-body quantum system. Two commonly used meth-ods – mean-field theory and large-N limit – are not expectedto work for the majority of quantum models discussed in Sec-tion III, as these models have small on-site Hilbert space, thusare far away from the mean-field limit. Progress on this ques-tion has recently been achieved using a variational frameworkbased on matrix product states (MPS). Below we briefly out-line this method and discuss the new physical insights it al-lowed to obtain about the PXP model – the only availableexample at this stage. Afterwards we discuss the prospectsof finding periodic trajectories in other scarred models andbroader applications of the method beyond many-body scars.

Optimal projection of quantum dynamics onto a given vari-ational manifold can be achieved by the time-dependent vari-ational principle (TDVP) pioneered by Dirac [73]. The vari-ational principle determines the best direction of evolutionwithin the manifold, such that the difference between exactand projected dynamics is minimal – see Box 2 for details. Ithas been realized [74] that TDVP can be turned into a power-ful computational tool for many-body systems when the man-ifold is parametrized by a family of MPS states. The advan-tage of MPS parametrization is that it naturally extends meanfield-like product states by incorporating non-trivial but finiterange correlations. In particular, the results of unitary timeevolution from the initial product state can be efficiently rep-resented as an MPS state. This MPS framework has recentlybeen extended into a path integral over “entanglement”-baseddegrees of freedom [75], and it has been used to calculate thediffusion coefficient in quantum many-body systems [76].

The ability of MPS states to incorporate entanglement madeit an ideal tool to explore semiclassical dynamics of the PXPmodel, which involves non-local correlations due to the Ry-dberg blockade. The variational MPS approach is tailored tocapture dynamics from the initial |Z2〉 state [10]. This stateis a period-2 density wave, which can be described using a2-atom unit cell. The variational MPS state is parametrizedby two angles, θo and θe that describe the state of Ryd-berg atoms on odd and even sites, respectively. In partic-ular, points (θo, θe) = (0, π) and (θo, θe) = (π, 0) corre-spond to two distinct period-2 density wave product states,|Z2〉 = |•◦•◦ . . .〉 and its partner shifted by one lattice site,|Z′2〉 = |◦•◦• . . .〉. By projecting quantum dynamics ontothis manifold using TDVP, one obtains a system of classicalnon-linear equations which govern the evolution of the angles,∂tθo,e = fa,b(θo, θe). The resulting phase portrait in Fig. 5areveals a periodic trajectory which passes through |Z2〉 and|Z′2〉 states, thereby explaining the revivals in the PXP model

8

0.0 0.2 0.4 0.6 0.8 1.00.5

0.6

0.7

0.8

0.9

1.0

FIG. 5. Many-body scarring and periodic orbits in the semiclas-sical dynamics. a: Phase diagram obtained from the variationalansatz with two degrees of freedom reveals an unstable periodic tra-jectory, which is responsible for the revivals in the PXP model from|Z2〉 state [10]. Color scale shows quantum leakage which quanti-fies the instantaneous discrepancy between quantum and variationaldynamics. b: Dynamical system with three angles θa,b,c results in aPoincare section with many stable periodic trajectories (one of themdenoted by the star) in centers of regular regions surrounded by thechaotic sea [77]. This classical dynamical system describes, e.g.,quantum revivals from |Z3〉 initial state in the PXP model.

from a period-2 density wave state. The projection of quan-tum dynamics is not exact, however, and the so-called quan-tum leakage, γ2 (see Box 2), quantifies the discrepancy be-tween quantum and variational dynamics. The leakage, shownby color scale in Fig. 5a, remains low in the vicinity of theclassical trajectory, thus justifying the existence of revivals inquantum dynamics.

Finding the classical trajectory behind quantum revivalsfrom |Z2〉 state provided a complementary picture behindweak ergodicity breaking by eigenstates of the PXP model.This variational approach was also extended to capture quan-tum dynamics for PXP model from more complicated ini-tial states or in higher dimensions [77]. In particular, anMPS variational ansatz with three angles θa, a = 1, 2, 3,was used to study dynamics from period-3 density wave state,|Z3〉 = |•◦◦•◦◦ . . .〉. TDVP projection of quantum dynamicsonto such an MPS manifold gave rise to a three-dimensionaldynamical system, whose Poincare sections are shown inFig. 5b. A prominent feature of this dynamical system was theexistence of mixed phase space – the coexistence of classicalchaotic regions with regions of stable motion, the so-calledKolmogorov-Arnold-Moser (KAM) tori [78]. The KAM torihost stable periodic orbits in their centers (one such orbit is

shown by the star in Fig. 5b). This provides a practical way offinding the most non-ergodic initial state with period-3 trans-lation invariance. Furthermore, one may expect that the ro-bustness of classical KAM tori to weak deformations may beused to infer the stability of the corresponding quantum modelto local perturbations. The mapping of quantum dynamicsonto a classical dynamical system via TDVP suggests an in-triguing direction for extending the KAM theorem to quan-tum systems, in a way that complements other recent attemptsbased on weakly-broken quantum integrability [79]. Pursu-ing this direction, however, requires a better understanding ofquantum leakage, which plays a decisive role in discriminat-ing which trajectories give rise to slow thermalization in exactquantum dynamics and which do not.

The existence of classical periodic trajectories underlyingquantum revivals in the PXP model calls for exploration ofsemiclassical dynamics in other scarred models discussed inSection III. While the projector-based embedding scenario(iii) is, in general, unlikely to have underlying periodic tra-jectories, for models with the spectrum generating algebra (i)one could imagine constructing a variational subspace usingproperties ofQ†-operator. Such a construction of classical pe-riodic trajectories in a broader family of models could be usedas a finer classification scheme for models displaying weakETH violation in their eigenstate properties.

Box 2 | Variational method for quantum many-body scars

Optimal projection of quantum dynamics onto a given vari-ational manifoldM, parametrized as |ψ(z)〉, can be achievedby the time-dependent variational principle (TDVP). Thevariational principle determines the best direction of evolu-tion within the manifold, z∂z|ψ〉, such that the differencebetween exact and projected dynamics is minimal, γ2 =

|z∂z|ψ〉+ iH|ψ〉|2. In order to capture quantum many-bodydynamics, the manifold can be parametrized by matrix prod-uct states (MPS), whose bond dimension controls the amountof entanglement generated during time evolution [74]. Semi-classical dynamics emerges naturally in this approach whenthe bond dimension of the MPS is kept small, allowing to sys-tematically study corrections to the mean-field behavior.

While the TDVP approach can be applied to any quantumsystem, its application to the PXP model with MPS of bonddimension 2 results in a particularly elegant and analytically-tractable classical dynamical system [10]. In order to describecoherent many-body oscillations in the PXP model, inspiredby the mean-field picture, one may attempt to use the mani-fold of product states, . . . (cos θe|◦〉 + sin θe|•〉)(cos θo|◦〉 +sin θo|•〉) . . ., parametrized by two degrees of freedom – theangles, θe and θo. These angles describe the state of atomson even and odd sites in the chain. However, such a state vi-olates the Rydberg blockade condition. Instead, the Rydbergconstraint can be satisfied in the manifold of matrix productstates, where one assigns the tensor A• = σ+ to the |•〉 state.Property A• · A• = (σ+)2 = 0 of Pauli raising operatorsσ+ = (σx + iσy)/2 guarantees that the weight of configu-rations with any two adjacent states |••〉 vanishes. Applica-

9

tion of TDVP to such states results in the system of classicalnon-linear differential equations for θe,o. The projection ofquantum dynamics is not exact, however, and the so-calledquantum leakage, defined by γ2 above, quantifies the instan-taneous disagreement, see Fig. B2.

The mapping of quantum dynamics onto a classical dynam-ical systems provides a useful method of studying many-bodysystems without an obvious semiclassical limit. Finding suit-able manifolds and extending this approach to dynamics inother models that have non-thermal eigenstates remains anoutstanding open problem.

Figure B2 | Variational method for quantum many-body scars.Time-dependent variational principle captures the optimal projectionof quantum dynamics onto a given variational manifold. The studiesof PXP model use matrix product states (MPS) as a variational mani-fold, where each atom is described by a single angle. The presence ofthe Rydberg blockade, implemented by projector P , results in non-zero entanglement, which requires a bond dimension 2 of MPS ten-sors, Ai.

V. PERSPECTIVE

Recent studies of quantum many-body scars and relatedphenomena, described in this article, have revealed a newkind of dynamical behavior in many-body systems – weakergodicity breaking. The defining feature of weak ergodic-ity breaking is strong dependence of relaxation dynamics onthe system’s initial configuration. In contrast to conventionalergodic systems, certain many-body states are long-lived, ex-hibiting parametrically slow relaxation, as opposed to otherinitial configurations that thermalize quickly. These findingshave pointed out that ETH, in its strong form, does not applyto a large class of systems that host anomalous, non-thermaleigenstates, and exhibit long-time coherent dynamics.

The discovery of quantum many-body scars serves as a re-minder that the understanding of thermalization and chaos inquantum many-body systems is far from complete and re-quires building a more elaborate theory and the accompanyingmethodological toolbox. The eigenstate description of quan-tum many-body scars revealed a common pattern in differ-

ent families of scarred systems – an emergent decoupled sub-space within the full many-body Hilbert space. While we pre-sented a few mechanisms that could underlie the existence ofsuch a non-thermal subspace, their complete classification iscurrently lacking. In particular, more work is needed to un-derstand the precise connection between many-body scarringand two broad classes of weak ergodicity breaking phenom-ena: theories with confinement [80–83] and lattice gauge the-ories [84–86]. The latter, in particular the one-dimensionalquantum link model, which has recently been realized in aBose-Hubbard quantum simulator [87], have intriguing con-nections with the PXP model [88], and it would be interest-ing to explore possible connections in higher dimensions [89].If accomplished, the classification of non-thermal subspacescould shed light on the physical mechanisms that lead to quan-tum many-body scars. Empirically, kinetically-constrainedsystems appear to be especially likely to host quantum many-body scars, hence it would be important to understand the roleof dynamical constraints which can lead to glassy-like dynam-ics [90–94].

More generally, the variational approach to scarred dynam-ics complements recent efforts in understanding parallels be-tween classical chaos measures, on the one hand, and thermal-izing quantum dynamics and its underlying transport coeffi-cients on the other hand [76, 95]. While many-body scarredmodels are quantum chaotic, their properties deviate fromother chaotic models such as the Sachdev-Ye-Kitaev (SYK)model [96, 97], which has attracted much attention as thefastest scrambler of quantum information [98]. Such devia-tions could be identified and studied more systematically us-ing the variational approach and its resulting mixed phase por-traits, which appear to be a generic feature of local Hamilto-nians. Armed with a better understanding of quantum leak-age from the variational manifold, this method could be usedto detect atypical behavior and absence of maximal scram-bling in general models that do not have an obvious large-Nlimit. One of the ultimate future challenges is to bring togetherthis approach to real-time dynamics with approaches target-ing directly the properties of many-body eigenstates. Thischallenge may be attacked using quantum information tech-niques, which, for example, allow one to link revivals andeigenstates properties [99]. Related techniques, as well thoseused to bound relaxation times from below in prethermal sys-tem, may give crucial insights into the response of QMBS toperturbations, and their dynamics at long times.

Finally, there is strong experimental and practical interestin quantum many-body scars. Many-body scarred revivalsprovide a mechanism for maintaining coherence, despite thepresence of interactions which normally scramble local quan-tum information. In particular, scars in Rydberg chains – sofar the main experimental realization of the phenomenon –have already been utilized for the preparation of specific en-tangled states [100]. This application made use of the quan-tum control based on the variational TDVP approach and itsidentification of entangled periodic trajectories that simul-taneously have small quantum leakage. This suggests thatscars may have wider range of applications, for example inprotected state transfer on quantum networks or in quantum

10

sensing. Such applications require deeper theoretical under-standing of the effects that protect the coherence of scars, aswell as the development of general experimental techniquesfor creating them on demand, e.g., using pumping protocolsin dipolar Bose gases [101] or by tilting the Fermi-Hubbardmodel [102]. Looking back at the field of single-particle scars,which have been realized in a multitude of experimental set-tings, one may also expect many-body quantum scars to berelevant to a much broader family of quantum simulators be-yond Rydberg atoms and perhaps even in solid state materials.In all these platforms, quantum many-body scars may serve asa vehicle for controlling and manipulating many-body states,which may prove useful in a range of quantum technologyapplications. At this stage, the new field of weakly broken er-godicity is still at its infancy, and experimental progress willundoubtedly yield more surprises in the near future.

VI. ACKNOWLEDGEMENTS

We are grateful to our collaborators Kieran Bull, Soon-won Choi, Jean-Yves Desaules, Wen Wei Ho, Ana Hudo-mal, Misha Lukin, Ivar Martin, Hannes Pichler, Nicolas Reg-nault, Ivana Vasic, and in particular Alexios Michailidis andChristopher Turner, without whom this work would not havebeen possible. Moreover, we benefited from useful discus-sions with Ehud Altman, B. Andrei Bernevig, Anushya Chan-dran, Paul Fendley, Vedika Khemani, and Lesik Motrunich.M.S. was supported by European Research Council (ERC)under the European Union’s Horizon 2020 research and in-novation program (Grant Agreement No. 850899). D.A.was supported by the Swiss National Science Foundation andby the European Research Council (ERC) under the Euro-pean Union’s Horizon 2020 research and innovation program.Z.P. acknowledges support by the Leverhulme Trust ResearchLeadership Award RL-2019-015.

[1] Immanuel Bloch, Jean Dalibard, and Sylvain Nascimbene,“Quantum simulations with ultracold quantum gases,” NaturePhysics 8, 267 EP – (2012), review Article.

[2] I. M. Georgescu, S. Ashhab, and Franco Nori, “Quantum sim-ulation,” Rev. Mod. Phys. 86, 153–185 (2014).

[3] Toshiya Kinoshita, Trevor Wenger, and David S. Weiss, “Aquantum Newton’s cradle,” Nature 440, 900 (2006).

[4] Rahul Nandkishore and David A Huse, “Many-body local-ization and thermalization in quantum statistical mechanics,”Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).

[5] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, andMaksym Serbyn, “Colloquium: Many-body localization, ther-malization, and entanglement,” Rev. Mod. Phys. 91, 021001(2019).

[6] J. M. Deutsch, “Quantum statistical mechanics in a closed sys-tem,” Phys. Rev. A 43, 2046 (1991).

[7] Mark Srednicki, “Chaos and quantum thermalization,” Phys.Rev. E 50, 888 (1994).

[8] Hannes Bernien, Sylvain Schwartz, Alexander Keesling,Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi,Alexander S. Zibrov, Manuel Endres, Markus Greiner, VladanVuletic, and Mikhail D. Lukin, “Probing many-body dynam-ics on a 51-atom quantum simulator,” Nature 551, 579 (2017).

[9] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn,and Z. Papic, “Weak ergodicity breaking from quantum many-body scars,” Nat. Phys. 14, 745 (2018).

[10] Wen Wei Ho, Soonwon Choi, Hannes Pichler, and Mikhail D.Lukin, “Periodic orbits, entanglement, and quantum many-body scars in constrained models: Matrix product state ap-proach,” Phys. Rev. Lett. 122, 040603 (2019).

[11] Eric J. Heller, “Bound-state eigenfunctions of classicallychaotic hamiltonian systems: Scars of periodic orbits,” Phys.Rev. Lett. 53, 1515 (1984).

[12] Henning Labuhn, Daniel Barredo, Sylvain Ravets, Sylvainde Leseleuc, Tommaso Macrı, Thierry Lahaye, and AntoineBrowaeys, “Tunable two-dimensional arrays of single Ryd-berg atoms for realizing quantum Ising models,” Nature 534,667 (2016).

[13] Igor Lesanovsky and Hosho Katsura, “Interacting Fibonaccianyons in a Rydberg gas,” Phys. Rev. A 86, 041601 (2012).

[14] B. Sun and F. Robicheaux, “Numerical study of two-body cor-relation in a 1d lattice with perfect blockade,” New Journal ofPhysics 10, 045032 (2008).

[15] B. Olmos, R. Gonzalez-Ferez, and I. Lesanovsky, “Collec-tive Rydberg excitations of an atomic gas confined in a ringlattice,” Phys. Rev. A 79, 043419 (2009).

[16] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, andZ. Papic, “Quantum scarred eigenstates in a Rydberg atomchain: Entanglement, breakdown of thermalization, and sta-bility to perturbations,” Phys. Rev. B 98, 155134 (2018).

[17] A. A. Michailidis, C. J. Turner, Z. Papic, D. A. Abanin,and M. Serbyn, “Stabilizing two-dimensional quantum scarsby deformation and synchronization,” Phys. Rev. Research 2,022065 (2020).

[18] Cheng-Ju Lin, Vladimir Calvera, and Timothy H. Hsieh,“Quantum many-body scar states in two-dimensional Rydbergatom arrays,” Phys. Rev. B 101, 220304 (2020).

[19] Bart van Voorden, Jirı Minar, and Kareljan Schoutens, “Quan-tum many-body scars in transverse field Ising ladders and be-yond,” Phys. Rev. B 101, 220305 (2020).

[20] Sho Sugiura, Tomotaka Kuwahara, and Keiji Saito, “Many-body scar state intrinsic to periodically driven system: Rigor-ous results,” arXiv preprint arXiv:1911.06092 (2019).

[21] Kaoru Mizuta, Kazuaki Takasan, and Norio Kawakami, “Ex-act Floquet quantum many-body scars under Rydberg block-ade,” Phys. Rev. Research 2, 033284 (2020).

[22] Bhaskar Mukherjee, Sourav Nandy, Arnab Sen, Diptiman Sen,and K. Sengupta, “Collapse and revival of quantum many-body scars via Floquet engineering,” Phys. Rev. B 101, 245107(2020).

[23] Cheng-Ju Lin, Anushya Chandran, and Olexei I. Motrunich,“Slow thermalization of exact quantum many-body scar statesunder perturbations,” Phys. Rev. Research 2, 033044 (2020).

[24] Ian Mondragon-Shem, Maxim G. Vavilov, and Ivar Martin,“The fate of quantum many-body scars in the presence of dis-order,” (2020), arXiv:2010.10535 [cond-mat.quant-gas].

[25] Paul Fendley, K. Sengupta, and Subir Sachdev, “Competingdensity-wave orders in a one-dimensional hard-boson model,”Phys. Rev. B 69, 075106 (2004).

11

[26] Igor Lesanovsky, “Liquid ground state, gap, and excited statesof a strongly correlated spin chain,” Phys. Rev. Lett. 108,105301 (2012).

[27] Cheng-Ju Lin and Olexei I. Motrunich, “Exact quantum many-body scar states in the Rydberg-blockaded atom chain,” Phys.Rev. Lett. 122, 173401 (2019).

[28] Thomas Iadecola, Michael Schecter, and Shenglong Xu,“Quantum many-body scars from magnon condensation,”Phys. Rev. B 100, 184312 (2019).

[29] E. J. Heller, “Wavepacket dynamics and quantum chaology,”in Chaos and quantum physics, Vol. 52 (North-Holland: Am-sterdam, 1991).

[30] M. V. Berry, in Chaotic Behaviour of Deterministic Sys-tems, Vol. 36, edited by Alain Chenciner, Anthony Patera,Daniel D Joseph, Robert McCredie May, Sheldon E New-house, Joseph Albert Libchaber, Jean-Pierre Eckmann, Jerry PGollub, Michel Henon, Michał Misiurewicz, et al. (North-Holland, 1983).

[31] S. Sridhar, “Experimental observation of scarred eigenfunc-tions of chaotic microwave cavities,” Phys. Rev. Lett. 67, 785(1991).

[32] P. B. Wilkinson, T. M. Fromhold, L. Eaves, F. W. Sheard,N. Miura, and T. Takamasu, “Observation of scarred wave-functions in a quantum well with chaotic electron dynamics,”Nature 380, 608 (1996).

[33] D. Wintgen and A. Honig, “Irregular wave functions of a hy-drogen atom in a uniform magnetic field,” Phys. Rev. Lett. 63,1467–1470 (1989).

[34] Bohm Arno, Asim Orhan Barut, et al., Dynamical groups andspectrum generating algebras, Vol. 1 (World Scientific, 1988).

[35] Chen Ning Yang, “η pairing and off-diagonal long-range orderin a Hubbard model,” Phys. Rev. Lett. 63, 2144–2147 (1989).

[36] Shoucheng Zhang, “Pseudospin symmetry and new collectivemodes of the Hubbard model,” Phys. Rev. Lett. 65, 120–122(1990).

[37] Sanjay Moudgalya, Nicolas Regnault, and B. AndreiBernevig, “Entanglement of exact excited states of Affleck-Kennedy-Lieb-Tasaki models: Exact results, many-bodyscars, and violation of the strong eigenstate thermalization hy-pothesis,” Phys. Rev. B 98, 235156 (2018).

[38] Daniel P Arovas, “Two exact excited states for the s = 1AKLT chain,” Physics Letters A 137, 431–433 (1989).

[39] Michael Schecter and Thomas Iadecola, “Weak ergodicitybreaking and quantum many-body scars in spin-1 xy mag-nets,” Phys. Rev. Lett. 123, 147201 (2019).

[40] Thomas Iadecola and Michael Schecter, “Quantum many-body scar states with emergent kinetic constraints and finite-entanglement revivals,” Phys. Rev. B 101, 024306 (2020).

[41] Sambuddha Chattopadhyay, Hannes Pichler, Mikhail D.Lukin, and Wen Wei Ho, “Quantum many-body scars fromvirtual entangled pairs,” Phys. Rev. B 101, 174308 (2020).

[42] Naoyuki Shibata, Nobuyuki Yoshioka, and Hosho Katsura,“Onsager’s scars in disordered spin chains,” Phys. Rev. Lett.124, 180604 (2020).

[43] Oskar Vafek, Nicolas Regnault, and B. Andrei Bernevig,“Entanglement of Exact Excited Eigenstates of the HubbardModel in Arbitrary Dimension,” SciPost Phys. 3, 043 (2017).

[44] Daniel K. Mark, Cheng-Ju Lin, and Olexei I. Motrunich,“Unified structure for exact towers of scar states in the Affleck-Kennedy-Lieb-Tasaki and other models,” Phys. Rev. B 101,195131 (2020).

[45] Daniel K. Mark and Olexei I. Motrunich, “η-pairing states astrue scars in an extended Hubbard model,” Phys. Rev. B 102,075132 (2020).

[46] Sanjay Moudgalya, Nicolas Regnault, and B. AndreiBernevig, “η-pairing in Hubbard models: From spectrum gen-erating algebras to quantum many-body scars,” Phys. Rev. B102, 085140 (2020).

[47] Sanjay Moudgalya, Edward O’Brien, B. Andrei Bernevig,Paul Fendley, and Nicolas Regnault, “Large classes of quan-tum scarred Hamiltonians from matrix product states,” Phys.Rev. B 102, 085120 (2020).

[48] Sanjay Moudgalya, B. Andrei Bernevig, and Nicolas Reg-nault, “Quantum Many-body Scars in a Landau Level on aThin Torus,” arXiv e-prints (2019), arXiv:1906.05292 [cond-mat.str-el].

[49] Pablo Sala, Tibor Rakovszky, Ruben Verresen, MichaelKnap, and Frank Pollmann, “Ergodicity breaking arising fromhilbert space fragmentation in dipole-conserving Hamiltoni-ans,” Phys. Rev. X 10, 011047 (2020).

[50] Paul A McClarty, Masudul Haque, Arnab Sen, and Jo-hannes Richter, “Disorder-free localization and many-bodyquantum scars from magnetic frustration,” arXiv preprintarXiv:2007.01311 (2020).

[51] Seulgi Ok, Kenny Choo, Christopher Mudry, Claudio Castel-novo, Claudio Chamon, and Titus Neupert, “Topologicalmany-body scar states in dimensions one, two, and three,”Phys. Rev. Research 1, 033144 (2019).

[52] Kyungmin Lee, Ronald Melendrez, Arijeet Pal, and Hitesh J.Changlani, “Exact three-colored quantum scars from geomet-ric frustration,” Phys. Rev. B 101, 241111 (2020).

[53] Kiryl Pakrouski, Preethi N Pallegar, Fedor K Popov, andIgor R Klebanov, “Many body scars as a group invariant sectorof Hilbert space,” arXiv preprint arXiv:2007.00845 (2020).

[54] Jie Ren, Chenguang Liang, and Chen Fang, “Quasi-symmetry groups and many-body scar dynamics,” arXivpreprint arXiv:2007.10380 (2020).

[55] Nicholas O’Dea, Fiona Burnell, Anushya Chandran, andVedika Khemani, “From tunnels to towers: quantum scarsfrom Lie algebras and q-deformed Lie algebras,” arXivpreprint arXiv:2007.16207 (2020).

[56] Berislav Buca, Joseph Tindall, and Dieter Jaksch, “Non-stationary coherent quantum many-body dynamics throughdissipation,” Nature Communications 10, 1730 (2019).

[57] J. Tindall, B. Buca, J. R. Coulthard, and D. Jaksch, “Heating-induced long-range η pairing in the Hubbard model,” Phys.Rev. Lett. 123, 030603 (2019).

[58] Soonwon Choi, Christopher J. Turner, Hannes Pichler,Wen Wei Ho, Alexios A. Michailidis, Zlatko Papic, MaksymSerbyn, Mikhail D. Lukin, and Dmitry A. Abanin, “Emergentsu(2) dynamics and perfect quantum many-body scars,” Phys.Rev. Lett. 122, 220603 (2019).

[59] Vedika Khemani, Chris R. Laumann, and Anushya Chan-dran, “Signatures of integrability in the dynamics of Rydberg-blockaded chains,” Phys. Rev. B 99, 161101 (2019).

[60] Kieran Bull, Jean-Yves Desaules, and Zlatko Papic, “Quan-tum scars as embeddings of weakly broken Lie algebra repre-sentations,” Phys. Rev. B 101, 165139 (2020).

[61] Christopher J. Turner, Jean-Yves Desaules, Kieran Bull,and Zlatko Papic, “Correspondence principle for many-bodyscars in ultracold Rydberg atoms,” (2020), arXiv:2006.13207[quant-ph].

[62] Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nico-las Regnault, and B Andrei Bernevig, “Thermalization andits absence within Krylov subspaces of a constrained Hamilto-nian,” arXiv e-prints (2019), arXiv:1910.14048 [cond-mat.str-el].

12

[63] Ana Hudomal, Ivana Vasic, Nicolas Regnault, and ZlatkoPapic, “Quantum scars of bosons with correlated hopping,”Communications Physics 3, 99 (2020).

[64] Hongzheng Zhao, Joseph Vovrosh, Florian Mintert, and Jo-hannes Knolle, “Quantum many-body scars in optical lat-tices,” Phys. Rev. Lett. 124, 160604 (2020).

[65] Shriya Pai and Michael Pretko, “Dynamical scar states indriven fracton systems,” Phys. Rev. Lett. 123, 136401 (2019).

[66] Vedika Khemani, Michael Hermele, and Rahul Nandkishore,“Localization from Hilbert space shattering: From theory tophysical realizations,” Phys. Rev. B 101, 174204 (2020).

[67] Naoto Shiraishi and Takashi Mori, “Systematic construction ofcounterexamples to the eigenstate thermalization hypothesis,”Phys. Rev. Lett. 119, 030601 (2017).

[68] Federica Maria Surace, Giuliano Giudici, and Marcello Dal-monte, “Weak-ergodicity-breaking via lattice supersymme-try,” arXiv preprint arXiv:2003.11073 (2020).

[69] Yoshihito Kuno, Tomonari Mizoguchi, and Yasuhiro Hat-sugai, “Flat band quantum scar,” (2020), arXiv:2010.02044[cond-mat.quant-gas].

[70] Julia Wildeboer, Alexander Seidel, N. S. Srivatsa, AnneE. B. Nielsen, and Onur Erten, “Topological quantum many-body scars in quantum dimer models on the kagome lattice,”(2020), arXiv:2009.00022 [cond-mat.str-el].

[71] Kieran Bull, Ivar Martin, and Z. Papic, “Systematic construc-tion of scarred many-body dynamics in 1d lattice models,”Phys. Rev. Lett. 123, 030601 (2019).

[72] Naoto Shiraishi, “Connection between quantum-many-bodyscars and the Affleck–Kennedy–Lieb–Tasaki model from theviewpoint of embedded hamiltonians,” Journal of StatisticalMechanics: Theory and Experiment 2019, 083103 (2019).

[73] P. A. M. Dirac, “Note on exchange phenomena in the Thomasatom,” Mathematical Proceedings of the Cambridge Philo-sophical Society 26, 376–385 (1930).

[74] Jutho Haegeman, J. Ignacio Cirac, Tobias J. Osborne, Iz-tok Pizorn, Henri Verschelde, and Frank Verstraete, “Time-dependent variational principle for quantum lattices,” Phys.Rev. Lett. 107, 070601 (2011).

[75] A. G. Green, C. A. Hooley, J. Keeling, and S. H. Si-mon, “Feynman path integrals over entangled states,” (2016),arXiv:1607.01778 [cond-mat.str-el].

[76] Eyal Leviatan, Frank Pollmann, Jens H. Bardarson, David A.Huse, and Ehud Altman, “Quantum thermalization dynamicswith matrix-product states,” (2017), arXiv:1702.08894 [cond-mat.stat-mech].

[77] A. A. Michailidis, C. J. Turner, Z. Papic, D. A. Abanin, andM. Serbyn, “Slow quantum thermalization and many-body re-vivals from mixed phase space,” Phys. Rev. X 10, 011055(2020).

[78] Vladimir Igorevich Arnol’d, Mathematical methods of classi-cal mechanics, Vol. 60 (Springer Science & Business Media,2013).

[79] G. P. Brandino, J.-S. Caux, and R. M. Konik, “Glimmers ofa quantum kam theorem: Insights from quantum quenches inone-dimensional bose gases,” Phys. Rev. X 5, 041043 (2015).

[80] Marton Kormos, Mario Collura, Gabor Takacs, and PasqualeCalabrese, “Real-time confinement following a quantumquench to a non-integrable model,” Nat. Phys. 13, 246 (2016).

[81] Neil J. Robinson, Andrew J. A. James, and Robert M. Konik,“Signatures of rare states and thermalization in a theory withconfinement,” Phys. Rev. B 99, 195108 (2019).

[82] Zhi-Cheng Yang, Fangli Liu, Alexey V. Gorshkov, andThomas Iadecola, “Hilbert-space fragmentation from strictconfinement,” Phys. Rev. Lett. 124, 207602 (2020).

[83] Olalla A. Castro-Alvaredo, Mate Lencses, Istvan M.Szecsenyi, and Jacopo Viti, “Entanglement oscillations near aquantum critical point,” Phys. Rev. Lett. 124, 230601 (2020).

[84] Giuseppe Magnifico, Marcello Dalmonte, Paolo Facchi, Save-rio Pascazio, Francesco V Pepe, and Elisa Ercolessi, “Realtime dynamics and confinement in the Zn Schwinger-Weyllattice model for 1+ 1 qed,” Quantum 4, 281 (2020).

[85] Titas Chanda, Jakub Zakrzewski, Maciej Lewenstein, andLuca Tagliacozzo, “Confinement and lack of thermalizationafter quenches in the bosonic Schwinger model,” Phys. Rev.Lett. 124, 180602 (2020).

[86] Umberto Borla, Ruben Verresen, Fabian Grusdt, andSergej Moroz, “Confined phases of one-dimensional spinlessfermions coupled to Z2 gauge theory,” Phys. Rev. Lett. 124,120503 (2020).

[87] Bing Yang, Hui Sun, Robert Ott, Han-Yi Wang, Torsten V.Zache, Jad C. Halimeh, Zhen-Sheng Yuan, Philipp Hauke,and Jian-Wei Pan, “Observation of gauge invariance ina 71-site Bose-Hubbard quantum simulator,” (2020),arXiv:2003.08945 [cond-mat.quant-gas].

[88] Federica M. Surace, Paolo P. Mazza, Giuliano Giudici,Alessio Lerose, Andrea Gambassi, and Marcello Dalmonte,“Lattice gauge theories and string dynamics in Rydberg atomquantum simulators,” Phys. Rev. X 10, 021041 (2020).

[89] Alessio Celi, Benoıt Vermersch, Oscar Viyuela, Hannes Pich-ler, Mikhail D. Lukin, and Peter Zoller, “Emerging two-dimensional gauge theories in Rydberg configurable arrays,”Phys. Rev. X 10, 021057 (2020).

[90] Nicola Pancotti, Giacomo Giudice, J. Ignacio Cirac, Juan P.Garrahan, and Mari Carmen Banuls, “Quantum East model:Localization, nonthermal eigenstates, and slow dynamics,”Phys. Rev. X 10, 021051 (2020).

[91] Sthitadhi Roy and Achilleas Lazarides, “Strong ergodicitybreaking due to local constraints in a quantum system,” Phys.Rev. Research 2, 023159 (2020).

[92] Zhihao Lan, Merlijn van Horssen, Stephen Powell, and Juan P.Garrahan, “Quantum slow relaxation and metastability due todynamical constraints,” Phys. Rev. Lett. 121, 040603 (2018).

[93] Johannes Feldmeier, Frank Pollmann, and Michael Knap,“Emergent glassy dynamics in a quantum dimer model,” Phys.Rev. Lett. 123, 040601 (2019).

[94] Oliver Hart, Giuseppe De Tomasi, and Claudio Castelnovo,“The random quantum comb: from compact localized states tomany-body scars,” arXiv preprint arXiv:2005.03036 (2020).

[95] A. Hallam, J. G. Morley, and A. G. Green, “The Lyapunovspectra of quantum thermalisation,” Nature Communications10, 2708 (2019).

[96] Subir Sachdev and Jinwu Ye, “Gapless spin-fluid ground statein a random quantum Heisenberg magnet,” Phys. Rev. Lett.70, 3339–3342 (1993).

[97] A Kitaev, “A simple model of quantum holography, talk atKITP,” University of California, Santa Barbara USA 7 (2015).

[98] Juan Maldacena, Stephen H. Shenker, and Douglas Stanford,“A bound on chaos,” Journal of High Energy Physics 2016,106 (2016).

[99] Alvaro M. Alhambra, Anurag Anshu, and Henrik Wilming,“Revivals imply quantum many-body scars,” Phys. Rev. B101, 205107 (2020).

[100] A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang,S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi,J. Cui, M. Rossignolo, P. Rembold, S. Montangero, T. Calarco,M. Endres, M. Greiner, V. Vuletic, and M. D. Lukin, “Gen-eration and manipulation of Schrodinger cat states in Rydbergatom arrays,” Science 365, 570–574 (2019).

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[101] Wil Kao, Kuan-Yu Li, Kuan-Yu Lin, Sarang Gopalakrishnan,and Benjamin L. Lev, “Creating quantum many-body scarsthrough topological pumping of a 1d dipolar gas,” (2020),arXiv:2002.10475 [cond-mat.quant-gas].

[102] Sebastian Scherg, Thomas Kohlert, Pablo Sala, Frank Poll-mann, Bharath H. M., Immanuel Bloch, and Monika Aidels-burger, “Observing non-ergodicity due to kinetic constraintsin tilted Fermi-Hubbard chains,” (2020), arXiv:2010.12965[cond-mat.quant-gas].