This is, however, penalised by the Fermi-Hubbard on-site interaction and
therefore, we expect a slowing down of the dipole-conserving dynamics. The
additional spin-exchange $\hat{T}_{XY}$ increases the connectivity, but cannot
fully connect the whole dipole symmetry sector and the system remains
fragmented.

In experiment, ergodicity breaking was probed by quenching the system from an
initial state chosen to be a charge-density wave of singlons on even sites
(filling factor $\nu{=}1/2$), which is prepared using a bichromatic optical
superlattice. The initial state can be described as an incoherent mixture of
site-localised particles with random spin configuration. The subsequent
evolution is monitored by extracting the spin-resolved imbalance
$\mathcal{I}^{\sigma}=(N_{e}^{\sigma}-N_{o}^{\sigma})/N^{\sigma}$, where
$N_{e(o)}^{\sigma}$ denotes the total number of spin-$\sigma$ atoms on even
(odd) sites and $N^{\sigma}=N_{e}^{\sigma}+N_{o}^{\sigma}$. A non-zero steady-
state imbalance signals a memory of the initial state, where
$\mathcal{I}^{\sigma}(t{=}0)=1$. In the regime $\Delta\gg J,|U|$, the
effective Hamiltonian is dipole-conserving up to third order in $J/\Delta$,
which was argued Scherg  _et al._ (2021) to be responsible for the non-ergodic
behaviour of imbalance $\mathcal{I}$, which did not decay to its thermal value
at moderate times. At much later times, higher-order processes, beyond the
ones in Eq. (47), are expected to reconnect the different Krylov sectors.

Tilted Fermi-Hubbard model is a promising platform to investigate a range of
ergodicity breaking beyond fragmentation. Although experimentally challenging
due to finite evolution times, it would be interesting to reconcile the
phenomenon of Stark MBL and Hilbert-space fragmentation, by studying the
impact of weak disorder or residual harmonic confinement on the long-time
dynamics. Moreover, it would be interesting to explore the connection between
lattice gauge theories and Hilbert-space fragmentation, which could be
addressed experimentally in a similar system Yang  _et al._ (2020a). Finally,
recent work Desaules  _et al._ (2021) has proposed that quantum many-body
scars could also be realised in the same model, albeit at a different filling
factor compared to Ref. Scherg  _et al._ (2021) – see Box 9.

Box 9 $|$ Proposal for scars in the Fermi-Hubbard model

In the regime $\Delta{\approx}U{\gg}J$, the tilted Fermi-Hubbard model in Eq.
(46) was argued to host many-body scars at filling factor $\nu{=}1$ Desaules
_et al._ (2021). In this case the sum of the dipole moment and the number of
doublons is effectively conserved, giving rise to the effective Hamiltonian

$\displaystyle H_{\rm eff}$ $\displaystyle=$
$\displaystyle-J\sum_{j,\sigma}\hat{c}^{\dagger}_{j,\sigma}\hat{c}_{j+1,\sigma}\hat{n}_{j,\overline{\sigma}}(1-\hat{n}_{j+1,\overline{\sigma}})+\mathrm{h.c.},$
(48)

where hopping to the left (which decreases the total dipole moment by 1) is
only allowed if it increases the number of doublons by the same amount
($\bar{\sigma}$ is opossite spin from $\sigma$).

The action of the Hamiltonian in Eq. (48) within the $\nu{=}1$ sector
fragments the Hilbert space beyond the simple conservation of $U{+}\Delta$.
The largest connected component of the Hilbert space is the one containing the
state with alternating $\uparrow$ and $\downarrow$ fermions. In such large
sectors, after resolving all the symmetries, the model can be shown to be non-
integrable Desaules  _et al._ (2021).

The proposal for many-body scars in the effective Hamiltonian in Eq. (48) is
based on the existence of a regular subgraph which has the form of the
hypergrid – a Cartesian product of line graphs (in our case, of length 3),
i.e., the hypergrid is isomorphic to an adjacency graph of a free spin-1
paramagnet, as illustrated below.

Figure B9: Adjacency graph of the model in Eq. (48) for $L{=}6$. Red vertices
denote the states belonging to the hypergrid, with the black vertices
corresponding to $\ket{{-}{+}}$, $\ket{{+}{-}}$ states defined in the text.
Green vertices are the isolated states $\ket{\downarrow 2\uparrow}$,
$\ket{\uparrow 2\downarrow}$ which live on the tails of the graph.

The mapping to the hypergrid follows from analysing pairs of sites, which can
be in states $``-"\equiv({\downarrow}{\uparrow})$,
$``2"\equiv({\updownarrow}{0})$ or $``+"\equiv({\uparrow}{\downarrow})$,
leading to a three level system. Note that the configuration
$({0}{\updownarrow})$ is omitted, as doublons can only be formed by hopping to
the left. Inside the hypergrid, there are two states for which the cell
alternates between $-$ and $+$. These are the state
$\ket{{-}{+}}\equiv\ket{{-}{+}{-}{+}\ldots}=\ket{\downarrow\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\ldots}$
and its spin-inverted partner,
$\ket{{+}{-}}=\ket{{+}{-}{+}{-}\ldots}\equiv\ket{\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\downarrow\uparrow\ldots}$.
The states $\ket{{-}{+}}$ and $\ket{{+}{-}}$ are shown in black colour in
figure below. These two states are the only corners of the hypergrid (state
with only $+$ and $-$ cells) with no edges going out of it. When the system is
initialised in either of these states, it was numerically shown to exhibit
persistent oscillations in the quench dynamics and other scarring
phenomenology similar to the PXP model discussed in Sec. IV.

##  VIII Conclusions

The goal of this chapter was to present a pedagogical introduction to the new
kind of dynamical behaviour in many-body systems – weak ergodicity breaking.
While we refrained from giving a formal definition, weak ergodicity breaking
was intuitively introduced as a strong dependence of relaxation dynamics on
the system’s initial configuration. We contrasted this behaviour against
conventional ergodic systems, by showing that certain many-body states can be
long-lived and they exhibit parametrically slow relaxation, unlike other
initial configurations that quickly reach thermal equilibrium. Many recent
studies discussed above, theoretical as well as experimental, have revealed
that a number of familiar physical systems do not conform to the expectations
of the ETH in its strong form, in that such systems can host non-thermal
eigenstates and exhibit long-time coherent dynamics. The common pattern that
arises in these diverse physical systems is an emergent non-thermalising
subspace, which is dynamically decoupled from the rest of the many-body
Hilbert space. Quantum entanglement, in particular, has been the vital tool
for identifying such non-thermal subspaces in different models.

At this stage, a complete classification of non-thermal subspaces and their
underlying mechanisms is still lacking. In particular, more work is needed to
understand the connection between many-body scarring and two broad classes of
weak ergodicity breaking phenomena: theories with confinement Nandkishore and
Sondhi (2017); Kormos  _et al._ (2017); Robinson  _et al._ (2019); Yang  _et
al._ (2020b); Castro-Alvaredo  _et al._ (2020) and lattice gauge theories
Magnifico  _et al._ (2020); Chanda  _et al._ (2020); Borla  _et al._ (2020).
The latter, in particular the one-dimensional quantum link model – which has
recently been realised in a Bose-Hubbard quantum simulator Yang  _et al._
(2020a) – have intriguing connections with the PXP model Surace  _et al._
(2020b). It would be interesting to explore possible connections with PXP in
higher dimensions Celi  _et al._ (2020), as well as to understand the relation
with kinetic constraints, which can generally lead to slow glassy-like
dynamics Pancotti  _et al._ (2020); Roy and Lazarides (2020); Lan  _et al._
(2018); Feldmeier  _et al._ (2019); Hart  _et al._ (2020).

On the theory side, many questions remain about the quantum-classical
correspondence in the physical systems discussed above. The TDVP variational
approach complements recent efforts in understanding parallels between
classical chaos measures, on the one hand, and thermalising quantum dynamics
and its underlying transport coefficients on the other hand Leviatan  _et al._
(2017); Hallam  _et al._ (2019). While many-body scarred models are quantum
chaotic, their properties deviate from other chaotic models such as the
Sachdev-Ye-Kitaev (SYK) model Sachdev and Ye (1993); Kitaev (2015), which has
attracted much attention as the fastest scrambler of quantum information
Maldacena  _et al._ (2016). Such deviations could be identified and studied
more systematically using the variational approach and its resulting mixed
phase portraits, which appear to be a generic feature of local Hamiltonians.
One of the outstanding challenges is to bring together this approach to real-
time dynamics with approaches that target the many-body eigenstates, possibly
using quantum entanglement and other quantum information techniques as a way
of linking quantum revivals and eigenstate properties Alhambra  _et al._
(2020).

Finally, there is strong experimental and practical interest in weak
ergodicity breaking. For example, many-body scarred revivals provide a
mechanism for maintaining coherence, despite the presence of interactions
which normally scramble local quantum information. In particular, scars in
Rydberg chains have already been utilised for the preparation of specific
entangled states Omran  _et al._ (2019). This application made use of quantum
control based on the variational TDVP approach and its identification of
entangled periodic trajectories that simultaneously have small quantum
leakage. Thus, scars may have a wider range of applications, for example in
protected state transfer on quantum networks or in quantum sensing Dooley
(2021). Such applications require deeper theoretical understanding of the
effects that protect the coherence of scars, as well as the development of
general experimental techniques for creating them on demand, e.g., using
periodic driving in Rydberg arrays Bluvstein  _et al._ (2021), pumping
protocols in dipolar Bose gases Kao  _et al._ (2021) or by tilting the Fermi-
Hubbard model Scherg  _et al._ (2021).

##  IX Acknowledgements

I would like to thank my collaborators Kieran Bull, Soonwon Choi, Jean-Yves
Desaules, Wen Wei Ho, Ana Hudomal, Mikhail Lukin, Ivar Martin, Hannes Pichler,
Nicolas Regnault, Ivana Vasić, and in particular Dmitry Abanin, Alexios
Michailidis, Maksym Serbyn and Christopher Turner. Moreover, I am grateful to
Maksym Serbyn for designing Fig. 1, and to Alexios Michailidis for help with
replotting Fig. 5(b). I would also like to acknowledge useful discussions with
B. Andrei Bernevig, Paul Fendley, Thomas Iadecola and Lesik Motrunich. This
work has been supported by the Leverhulme Trust Research Leadership Award
RL-2019-015. This research was supported in part by the National Science
Foundation under Grant No. NSF PHY-1748958.

## References

  * Bloch  _et al._ (2012) Immanuel Bloch, Jean Dalibard, and Sylvain Nascimbène, “Quantum simulations with ultracold quantum gases,” Nature Physics 8, 267–276 (2012).
  * Georgescu  _et al._ (2014) I. M. Georgescu, S. Ashhab, and Franco Nori, “Quantum simulation,” Rev. Mod. Phys. 86, 153–185 (2014).
  * Deutsch (1991) J. M. Deutsch, “Quantum statistical mechanics in a closed system,” Phys. Rev. A 43, 2046–2049 (1991).
  * Srednicki (1994) Mark Srednicki, “Chaos and quantum thermalization,” Phys. Rev. E 50, 888–901 (1994).
  * Gornyi  _et al._ (2005) I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, “Interacting electrons in disordered wires: Anderson localization and low-$t$ transport,” Phys. Rev. Lett. 95, 206603 (2005).
  * Basko  _et al._ (2006) D.M. Basko, I.L. Aleiner, and B.L. Altshuler, “Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states,” Annals of Physics 321, 1126 – 1205 (2006).
  * Pal and Huse (2010) Arijeet Pal and David A. Huse, “Many-body localization phase transition,” Phys. Rev. B 82, 174411 (2010).
  * Calabrese and Cardy (2006) Pasquale Calabrese and John Cardy, “Time dependence of correlation functions following a quantum quench,” Phys. Rev. Lett. 96, 136801 (2006).
  * Žnidarič  _et al._ (2008) M. Žnidarič, T. Prosen, and P. Prelovšek, “Many-body localization in the heisenberg xxz magnet in a random field,” Phys. Rev. B 77, 064426 (2008).
  * Bardarson  _et al._ (2012) Jens H. Bardarson, Frank Pollmann, and Joel E. Moore, “Unbounded growth of entanglement in models of many-body localization,” Phys. Rev. Lett. 109, 017202 (2012).
  * Serbyn  _et al._ (2013) Maksym Serbyn, Z. Papić, and Dmitry A. Abanin, “Universal slow growth of entanglement in interacting strongly disordered systems,” Phys. Rev. Lett. 110, 260601 (2013).
  * Bernien  _et al._ (2017) Hannes Bernien, Sylvain Schwartz, Alexander Keesling, Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S. Zibrov, Manuel Endres, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551, 579–584 (2017).
  * Turner  _et al._ (2018a) C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, “Weak ergodicity breaking from quantum many-body scars,” Nature Physics 14, 745–749 (2018a).
  * Ho  _et al._ (2019) 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 approach,” Phys. Rev. Lett. 122, 040603 (2019).
  * Heller (1984) Eric J. Heller, “Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits,” Phys. Rev. Lett. 53, 1515–1518 (1984).
  * Moudgalya  _et al._ (2018a) Sanjay Moudgalya, Stephan Rachel, B. Andrei Bernevig, and Nicolas Regnault, “Exact excited states of nonintegrable models,” Phys. Rev. B 98, 235155 (2018a).
  * Moudgalya  _et al._ (2018b) Sanjay Moudgalya, Nicolas Regnault, and B. Andrei Bernevig, “Entanglement of exact excited states of Affleck-Kennedy-Lieb-Tasaki models: Exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis,” Phys. Rev. B 98, 235156 (2018b).
  * Khemani  _et al._ (2020) Vedika Khemani, Michael Hermele, and Rahul Nandkishore, “Localization from Hilbert space shattering: From theory to physical realizations,” Phys. Rev. B 101, 174204 (2020).
  * Sala  _et al._ (2020) Pablo Sala, Tibor Rakovszky, Ruben Verresen, Michael Knap, and Frank Pollmann, “Ergodicity breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians,” Phys. Rev. X 10, 011047 (2020).
  * Shiraishi and Mori (2017) Naoto Shiraishi and Takashi Mori, “Systematic construction of counterexamples to the eigenstate thermalization hypothesis,” Phys. Rev. Lett. 119, 030601 (2017).
  * Scherg  _et al._ (2021) Sebastian Scherg, Thomas Kohlert, Pablo Sala, Frank Pollmann, Bharath Hebbe Madhusudhana, Immanuel Bloch, and Monika Aidelsburger, “Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains,” Nature Communications 12, 4490 (2021).
  * Serbyn  _et al._ (2021) Maksym Serbyn, Dmitry A. Abanin, and Zlatko Papić, “Quantum many-body scars and weak breaking of ergodicity,” Nature Physics 17, 675–685 (2021).
  * Nandkishore and Huse (2015) Rahul Nandkishore and David A. Huse, “Many-body localization and thermalization in quantum statistical mechanics,” Annual Review of Condensed Matter Physics 6, 15–38 (2015).
  * Gogolin and Eisert (2016) Christian Gogolin and Jens Eisert, “Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems,” Rep. Prog. Phys. 79, 056001 (2016).
  * Mori  _et al._ (2018) Takashi Mori, Tatsuhiko N Ikeda, Eriko Kaminishi, and Masahito Ueda, “Thermalization and prethermalization in isolated quantum systems: a theoretical overview,” Journal of Physics B: Atomic, Molecular and Optical Physics 51, 112001 (2018).
  * Garrahan  _et al._ (2011) Juan P Garrahan, Peter Sollich, and Cristina Toninelli, “Kinetically constrained models,” Dynamical heterogeneities in glasses, colloids, and granular media 150, 111–137 (2011).
  * Morampudi  _et al._ (2020) S. C. Morampudi, A. Chandran, and C. R. Laumann, “Universal entanglement of typical states in constrained systems,” Phys. Rev. Lett. 124, 050602 (2020).
  * Page (1993) Don N. Page, “Average entropy of a subsystem,” Phys. Rev. Lett. 71, 1291–1294 (1993).
  * Ueda (2020) Masahito Ueda, “Quantum equilibration, thermalization and prethermalization in ultracold atoms,” Nature Reviews Physics 2, 669–681 (2020).
  * Rigol  _et al._ (2008) Marcos Rigol, Vanja Dunjko, and Maxim Olshanii, “Thermalization and its mechanism for generic isolated quantum systems,” Nature 452, 854–858 (2008).
  * Sutherland (2004) B. Sutherland,  _Beautiful Models: 70 Years of Exactly Solved Quantum Many-body Problems_ (World Scientific, 2004).
  * Abanin  _et al._ (2019) Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn, “Colloquium: Many-body localization, thermalization, and entanglement,” Rev. Mod. Phys. 91, 021001 (2019).
  * Pérez-García  _et al._ (2007) David Pérez-García, Frank Verstraete, Michael M. Wolf, and J. Ignacio Cirac, “Matrix product state representations,” Quantum Inf. Comput. 7, 401–430 (2007).
  * Eisert  _et al._ (2010) J. Eisert, M. Cramer, and M. B. Plenio, “Colloquium: Area laws for the entanglement entropy,” Rev. Mod. Phys. 82, 277–306 (2010).
  * Verstraete  _et al._ (2008) F. Verstraete, V. Murg, and J.I. Cirac, “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,” Advances in Physics 57, 143–224 (2008).
  * Affleck  _et al._ (1987) Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki, “Rigorous results on valence-bond ground states in antiferromagnets,” Phys. Rev. Lett. 59, 799–802 (1987).
  * Affleck  _et al._ (1988) Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki, “Valence bond ground states in isotropic quantum antiferromagnets,” Communications in Mathematical Physics 115, 477–528 (1988).
  * Verstraete and Cirac (2006) F. Verstraete and J. I. Cirac, “Matrix product states represent ground states faithfully,” Phys. Rev. B 73, 094423 (2006).
  * Hastings (2007) M. B. Hastings, “An area law for one-dimensional quantum systems,” Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007).
  * Haegeman  _et al._ (2013) Jutho Haegeman, Tobias J. Osborne, and Frank Verstraete, “Post-matrix product state methods: To tangent space and beyond,” Phys. Rev. B 88, 075133 (2013).
  * Vanderstraeten  _et al._ (2015) Laurens Vanderstraeten, Michaël Mariën, Frank Verstraete, and Jutho Haegeman, “Excitations and the tangent space of projected entangled-pair states,” Phys. Rev. B 92, 201111 (2015).
  * Vanderstraeten  _et al._ (2019) Laurens Vanderstraeten, Jutho Haegeman, and Frank Verstraete, “Tangent-space methods for uniform matrix product states,” SciPost Phys. Lect. Notes , 7 (2019).
  * Arovas  _et al._ (1988) Daniel P. Arovas, Assa Auerbach, and F. D. M. Haldane, “Extended Heisenberg models of antiferromagnetism: Analogies to the fractional quantum Hall effect,” Phys. Rev. Lett. 60, 531–534 (1988).
  * Moudgalya  _et al._ (2020a) Sanjay Moudgalya, Edward O’Brien, B. Andrei Bernevig, Paul Fendley, and Nicolas Regnault, “Large classes of quantum scarred Hamiltonians from matrix product states,” Phys. Rev. B 102, 085120 (2020a).
  * Arovas (1989) Daniel P. Arovas, “Two exact excited states for the $s=1$ AKLT chain,” Physics Letters A 137, 431–433 (1989).
  * Vanderstraeten (2017) L. Vanderstraeten,  _Tensor Network States and Effective Particles for Low-Dimensional Quantum Spin Systems_, Springer Theses (Springer International Publishing, 2017).
  * Dirac (1930) P. A. M. Dirac, “Note on exchange phenomena in the Thomas atom,” Mathematical Proceedings of the Cambridge Philosophical Society 26, 376–385 (1930).
  * Kramer and Saraceno (1981) P. Kramer and M. Saraceno,  _Geometry of the Time-Dependent Variational Principle in Quantum Mechanics_, Lecture Notes in Physics (Springer Berlin Heidelberg, 1981).
  * Haegeman  _et al._ (2011) Jutho Haegeman, J. Ignacio Cirac, Tobias J. Osborne, Iztok Pižorn, Henri Verschelde, and Frank Verstraete, “Time-dependent variational principle for quantum lattices,” Phys. Rev. Lett. 107, 070601 (2011).
  * Bauernfeind and Aichhorn (2020) Daniel Bauernfeind and Markus Aichhorn, “Time Dependent Variational Principle for Tree Tensor Networks,” SciPost Phys. 8, 24 (2020).
  * Strogatz (2014) S.H. Strogatz,  _Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering_, Studies in Nonlinearity (Avalon Publishing, 2014).
  * Gutzwiller (2013) Martin C Gutzwiller,  _Chaos in classical and quantum mechanics_ , Vol. 1 (Springer Science & Business Media, 2013).
  * Schecter and Iadecola (2019) Michael Schecter and Thomas Iadecola, “Weak ergodicity breaking and quantum many-body scars in spin-1 XY magnets,” Phys. Rev. Lett. 123, 147201 (2019).
  * Iadecola and Schecter (2020) Thomas Iadecola and Michael Schecter, “Quantum many-body scar states with emergent kinetic constraints and finite-entanglement revivals,” Phys. Rev. B 101, 024306 (2020).
  * Arno  _et al._ (1988) Bohm Arno, Asim Orhan Barut, _et al._ ,  _Dynamical groups and spectrum generating algebras_ , Vol. 1 (World Scientific, 1988).
  * Yang (1989) Chen Ning Yang, “$\eta$ pairing and off-diagonal long-range order in a Hubbard model,” Phys. Rev. Lett. 63, 2144–2147 (1989).
  * Zhang (1990) Shoucheng Zhang, “Pseudospin symmetry and new collective modes of the Hubbard model,” Phys. Rev. Lett. 65, 120–122 (1990).
  * Mark  _et al._ (2020) 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).
  * Vafek  _et al._ (2017) Oskar Vafek, Nicolas Regnault, and B. Andrei Bernevig, “Entanglement of Exact Excited Eigenstates of the Hubbard Model in Arbitrary Dimension,” SciPost Phys. 3, 043 (2017).
  * Mark and Motrunich (2020) Daniel K. Mark and Olexei I. Motrunich, “$\eta$-pairing states as true scars in an extended Hubbard model,” Phys. Rev. B 102, 075132 (2020).
  * Moudgalya  _et al._ (2020b) Sanjay Moudgalya, Nicolas Regnault, and B. Andrei Bernevig, “$\eta$-pairing in Hubbard models: From spectrum generating algebras to quantum many-body scars,” Phys. Rev. B 102, 085140 (2020b).
  * Iadecola and Žnidarič (2019) Thomas Iadecola and Marko Žnidarič, “Exact localized and ballistic eigenstates in disordered chaotic spin ladders and the Fermi-Hubbard model,” Phys. Rev. Lett. 123, 036403 (2019).
  * Shibata  _et al._ (2020) Naoyuki Shibata, Nobuyuki Yoshioka, and Hosho Katsura, “Onsager’s scars in disordered spin chains,” Phys. Rev. Lett. 124, 180604 (2020).
  * Chattopadhyay  _et al._ (2020) Sambuddha Chattopadhyay, Hannes Pichler, Mikhail D. Lukin, and Wen Wei Ho, “Quantum many-body scars from virtual entangled pairs,” Phys. Rev. B 101, 174308 (2020).
  * Lee  _et al._ (2020) Kyungmin Lee, Ronald Melendrez, Arijeet Pal, and Hitesh J. Changlani, “Exact three-colored quantum scars from geometric frustration,” Phys. Rev. B 101, 241111 (2020).
  * Pakrouski  _et al._ (2020) K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R. Klebanov, “Many-body scars as a group invariant sector of Hilbert space,” Phys. Rev. Lett. 125, 230602 (2020).
  * Ren  _et al._ (2020) Jie Ren, Chenguang Liang, and Chen Fang, “Quasi-symmetry groups and many-body scar dynamics,” arXiv preprint arXiv:2007.10380 (2020).
  * O’Dea  _et al._ (2020) Nicholas O’Dea, Fiona Burnell, Anushya Chandran, and Vedika Khemani, “From tunnels to towers: Quantum scars from Lie algebras and $q$-deformed Lie algebras,” Phys. Rev. Research 2, 043305 (2020).
  * Buča  _et al._ (2019) Berislav Buča, Joseph Tindall, and Dieter Jaksch, “Non-stationary coherent quantum many-body dynamics through dissipation,” Nature Communications 10, 1730 (2019).
  * Tindall  _et al._ (2019) J. Tindall, B. Buča, J. R. Coulthard, and D. Jaksch, “Heating-induced long-range $\eta$ pairing in the Hubbard model,” Phys. Rev. Lett. 123, 030603 (2019).
  * Pai  _et al._ (2019) Shriya Pai, Michael Pretko, and Rahul M. Nandkishore, “Localization in fractonic random circuits,” Phys. Rev. X 9, 021003 (2019).
  * Moudgalya  _et al._ (2019) Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B Andrei Bernevig, “Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian,” arXiv e-prints (2019), arXiv:1910.14048 [cond-mat.str-el] .
  * Moudgalya  _et al._ (2020c) Sanjay Moudgalya, B. Andrei Bernevig, and Nicolas Regnault, “Quantum many-body scars in a Landau level on a thin torus,” Phys. Rev. B 102, 195150 (2020c).
  * Pretko  _et al._ (2020) Michael Pretko, Xie Chen, and Yizhi You, “Fracton phases of matter,” International Journal of Modern Physics A 35, 2030003 (2020).
  * Wannier (1962) Gregory H. Wannier, “Dynamics of band electrons in electric and magnetic fields,” Rev. Mod. Phys. 34, 645–655 (1962).
  * Schulz  _et al._ (2019) M. Schulz, C. A. Hooley, R. Moessner, and F. Pollmann, “Stark many-body localization,” Phys. Rev. Lett. 122, 040606 (2019).
  * van Nieuwenburg  _et al._ (2019) Evert van Nieuwenburg, Yuval Baum, and Gil Refael, “From Bloch oscillations to many-body localization in clean interacting systems,” Proceedings of the National Academy of Sciences 116, 9269–9274 (2019).
  * Sikora  _et al._ (2011) Olga Sikora, Nic Shannon, Frank Pollmann, Karlo Penc, and Peter Fulde, “Extended quantum u(1)-liquid phase in a three-dimensional quantum dimer model,” Phys. Rev. B 84, 115129 (2011).
  * Lan  _et al._ (2018) Zhihao Lan, Merlijn van Horssen, Stephen Powell, and Juan P. Garrahan, “Quantum slow relaxation and metastability due to dynamical constraints,” Phys. Rev. Lett. 121, 040603 (2018).
  * Gopalakrishnan and Zakirov (2018) Sarang Gopalakrishnan and Bahti Zakirov, “Facilitated quantum cellular automata as simple models with non-thermal eigenstates and dynamics,” Quantum Science and Technology 3, 044004 (2018).
  * Hudomal  _et al._ (2020) Ana Hudomal, Ivana Vasić, Nicolas Regnault, and Zlatko Papić, “Quantum scars of bosons with correlated hopping,” Communications Physics 3, 99 (2020).
  * Zhao  _et al._ (2020) Hongzheng Zhao, Joseph Vovrosh, Florian Mintert, and Johannes Knolle, “Quantum many-body scars in optical lattices,” Phys. Rev. Lett. 124, 160604 (2020).
  * Ok  _et al._ (2019) Seulgi Ok, Kenny Choo, Christopher Mudry, Claudio Castelnovo, Claudio Chamon, and Titus Neupert, “Topological many-body scar states in dimensions one, two, and three,” Phys. Rev. Research 1, 033144 (2019).
  * Wildeboer  _et al._ (2020) Julia Wildeboer, Alexander Seidel, N. S. Srivatsa, Anne E. 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] .
  * Surace  _et al._ (2020a) Federica Maria Surace, Giuliano Giudici, and Marcello Dalmonte, “Weak-ergodicity-breaking via lattice supersymmetry,” Quantum 4, 339 (2020a).
  * Shiraishi (2019) Naoto Shiraishi, “Connection between quantum-many-body scars and the Affleck–Kennedy–Lieb–Tasaki model from the viewpoint of embedded Hamiltonians,” Journal of Statistical Mechanics: Theory and Experiment 2019, 083103 (2019).
  * Langlett  _et al._ (2021) Christopher M. Langlett, Zhi-Cheng Yang, Julia Wildeboer, Alexey V. Gorshkov, Thomas Iadecola, and Shenglong Xu, “Rainbow scars: From area to volume law,” (2021), arXiv:2107.03416 [cond-mat.str-el] .
  * Kitaev (2003) A.Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics 303, 2–30 (2003).
  * Rokhsar and Kivelson (1988) Daniel S. Rokhsar and Steven A. Kivelson, “Superconductivity and the quantum hard-core dimer gas,” Phys. Rev. Lett. 61, 2376–2379 (1988).
  * Lesanovsky and Katsura (2012) Igor Lesanovsky and Hosho Katsura, “Interacting Fibonacci anyons in a Rydberg gas,” Phys. Rev. A 86, 041601 (2012).
  * Feiguin  _et al._ (2007) Adrian Feiguin, Simon Trebst, Andreas W. W. Ludwig, Matthias Troyer, Alexei Kitaev, Zhenghan Wang, and Michael H. Freedman, “Interacting anyons in topological quantum liquids: The golden chain,” Phys. Rev. Lett. 98, 160409 (2007).
  * Sun and Robicheaux (2008) B. Sun and F. Robicheaux, “Numerical study of two-body correlation in a 1d lattice with perfect blockade,” New Journal of Physics 10, 045032 (2008).
  * Olmos  _et al._ (2009) B. Olmos, R. González-Férez, and I. Lesanovsky, “Collective Rydberg excitations of an atomic gas confined in a ring lattice,” Phys. Rev. A 79, 043419 (2009).
  * Turner  _et al._ (2018b) C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, “Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations,” Phys. Rev. B 98, 155134 (2018b).
  * Rigol  _et al._ (2007) Marcos Rigol, Vanja Dunjko, Vladimir Yurovsky, and Maxim Olshanii, “Relaxation in a completely integrable many-body quantum system: An Ab Initio study of the dynamics of the highly excited states of 1d lattice hard-core bosons,” Phys. Rev. Lett. 98, 050405 (2007).
  * Mukherjee  _et al._ (2020) 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).
  * Michailidis  _et al._ (2020a) A. A. Michailidis, C. J. Turner, Z. Papić, D. A. Abanin, and M. Serbyn, “Slow quantum thermalization and many-body revivals from mixed phase space,” Phys. Rev. X 10, 011055 (2020a).
  * Michailidis  _et al._ (2020b) A. A. Michailidis, C. J. Turner, Z. Papić, D. A. Abanin, and M. Serbyn, “Stabilizing two-dimensional quantum scars by deformation and synchronization,” Phys. Rev. Research 2, 022065 (2020b).
  * Lin  _et al._ (2020a) Cheng-Ju Lin, Vladimir Calvera, and Timothy H. Hsieh, “Quantum many-body scar states in two-dimensional Rydberg atom arrays,” Phys. Rev. B 101, 220304 (2020a).
  * Iadecola  _et al._ (2019) Thomas Iadecola, Michael Schecter, and Shenglong Xu, “Quantum many-body scars from magnon condensation,” Phys. Rev. B 100, 184312 (2019).
  * Lin and Motrunich (2019) 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).
  * van Voorden  _et al._ (2020) Bart van Voorden, Jiří Minář, and Kareljan Schoutens, “Quantum many-body scars in transverse field Ising ladders and beyond,” Phys. Rev. B 101, 220305 (2020).
  * Sugiura  _et al._ (2021) Sho Sugiura, Tomotaka Kuwahara, and Keiji Saito, “Many-body scar state intrinsic to periodically driven system,” Phys. Rev. Research 3, L012010 (2021).
  * Mizuta  _et al._ (2020) Kaoru Mizuta, Kazuaki Takasan, and Norio Kawakami, “Exact Floquet quantum many-body scars under Rydberg blockade,” Phys. Rev. Research 2, 033284 (2020).
  * Fendley  _et al._ (2004) Paul Fendley, K. Sengupta, and Subir Sachdev, “Competing density-wave orders in a one-dimensional hard-boson model,” Phys. Rev. B 69, 075106 (2004).
  * Lesanovsky (2012) Igor Lesanovsky, “Liquid ground state, gap, and excited states of a strongly correlated spin chain,” Phys. Rev. Lett. 108, 105301 (2012).
  * Lin  _et al._ (2020b) Cheng-Ju Lin, Anushya Chandran, and Olexei I. Motrunich, “Slow thermalization of exact quantum many-body scar states under perturbations,” Phys. Rev. Research 2, 033044 (2020b).
  * Surace  _et al._ (2021a) Federica Maria Surace, Matteo Votto, Eduardo Gonzalez Lazo, Alessandro Silva, Marcello Dalmonte, and Giuliano Giudici, “Exact many-body scars and their stability in constrained quantum chains,” Phys. Rev. B 103, 104302 (2021a).
  * Mondragon-Shem  _et al._ (2020) Ian Mondragon-Shem, Maxim G. Vavilov, and Ivar Martin, “The fate of quantum many-body scars in the presence of disorder,” (2020), arXiv:2010.10535 [cond-mat.quant-gas] .
  * Schecter and Iadecola (2018) Michael Schecter and Thomas Iadecola, “Many-body spectral reflection symmetry and protected infinite-temperature degeneracy,” Phys. Rev. B 98, 035139 (2018).
  * Sutherland (1986) Bill Sutherland, “Localization of electronic wave functions due to local topology,” Phys. Rev. B 34, 5208 (1986).
  * Inui  _et al._ (1994) M. Inui, S. A. Trugman, and Elihu Abrahams, “Unusual properties of midband states in systems with off-diagonal disorder,” Phys. Rev. B 49, 3190 (1994).
  * Surace  _et al._ (2021b) Federica Maria Surace, Marcello Dalmonte, and Alessandro Silva, “Quantum local random networks and the statistical robustness of quantum scars,” (2021b), arXiv:2107.00884 [cond-mat.stat-mech] .
  * Karle  _et al._ (2021) Volker Karle, Maksym Serbyn, and Alexios A. Michailidis, “Area-law entangled eigenstates from nullspaces of local hamiltonians,” (2021), arXiv:2102.13633 [quant-ph] .
  * Choi  _et al._ (2019) Soonwon Choi, Christopher J. Turner, Hannes Pichler, Wen Wei Ho, Alexios A. Michailidis, Zlatko Papić, Maksym Serbyn, Mikhail D. Lukin, and Dmitry A. Abanin, “Emergent su(2) dynamics and perfect quantum many-body scars,” Phys. Rev. Lett. 122, 220603 (2019).
  * Bull  _et al._ (2020) Kieran Bull, Jean-Yves Desaules, and Zlatko Papić, “Quantum scars as embeddings of weakly broken Lie algebra representations,” Phys. Rev. B 101, 165139 (2020).
  * Khemani  _et al._ (2019) Vedika Khemani, Chris R. Laumann, and Anushya Chandran, “Signatures of integrability in the dynamics of Rydberg-blockaded chains,” Phys. Rev. B 99, 161101 (2019).
  * Bull  _et al._ (2019) Kieran Bull, Ivar Martin, and Z. Papić, “Systematic construction of scarred many-body dynamics in 1d lattice models,” Phys. Rev. Lett. 123, 030601 (2019).
  * Mehta (2004) Madan Lal Mehta,  _Random matrices_ , Vol. 142 (Elsevier, 2004).
  * Arnol’d (2013) Vladimir Igorevich Arnol’d,  _Mathematical methods of classical mechanics_ , Vol. 60 (Springer Science & Business Media, 2013).
  * Leviatan  _et al._ (2017) Eyal Leviatan, Frank Pollmann, Jens H. Bardarson, David A. Huse, and Ehud Altman, “Quantum thermalization dynamics with matrix-product states,” (2017), arXiv:1702.08894 [cond-mat.stat-mech] .
  * Hallam  _et al._ (2019) A. Hallam, J. G. Morley, and A. G. Green, “The Lyapunov spectra of quantum thermalisation,” Nature Communications 10, 2708 (2019).
  * Lieb and Robinson (1972) E. H. Lieb and D. Robinson, “The finite group velocity of quantum spin systems,” Commun. Math. Phys. 28, 251–257 (1972).
  * Heller (1991) E. J. Heller, “Wavepacket dynamics and quantum chaology,” in  _Chaos and quantum physics_ , Vol. 52 (North-Holland: Amsterdam, 1991).
  * Schnirelman (1974) A. I. Schnirelman, “Ergodic properties of eigenfunctions,” Uspekhi Mat. Nauk 29, 181 (1974).
  * De Verdiere (1985) Y Colin De Verdiere, “Ergodicité et fonctions propres du laplacien,” Communications in Mathematical Physics 102, 497–502 (1985).
  * Zelditch  _et al._ (1987) Steven Zelditch _et al._ , “Uniform distribution of eigenfunctions on compact hyperbolic surfaces,” Duke mathematical journal 55, 919–941 (1987).
  * Berry (1983) M. V. Berry, in  _Chaotic Behaviour of Deterministic Systems_ , Vol. 36, edited by Alain Chenciner, Anthony Patera, Daniel D Joseph, Robert McCredie May, Sheldon E Newhouse, Joseph Albert Libchaber, Jean-Pierre Eckmann, Jerry P Gollub, Michel Hénon, Michał Misiurewicz, _et al._ (North-Holland, 1983).
  * Sridhar (1991) S. Sridhar, “Experimental observation of scarred eigenfunctions of chaotic microwave cavities,” Phys. Rev. Lett. 67, 785–788 (1991).
  * Wilkinson  _et al._ (1996) P. B. Wilkinson, T. M. Fromhold, L. Eaves, F. W. Sheard, N. Miura, and T. Takamasu, “Observation of ’scarred’ wavefunctions in a quantum well with chaotic electron dynamics,” Nature 380, 608–610 (1996).
  * Wintgen and Hönig (1989) D. Wintgen and A. Hönig, “Irregular wave functions of a hydrogen atom in a uniform magnetic field,” Phys. Rev. Lett. 63, 1467–1470 (1989).
  * Turner  _et al._ (2021) C. J. Turner, J.-Y. Desaules, K. Bull, and Z. Papić, “Correspondence principle for many-body scars in ultracold Rydberg atoms,” Phys. Rev. X 11, 021021 (2021).
  * Gazeau (2009) Jean-Pierre Gazeau,  _Coherent States in Quantum Physics_ (Wiley-VCH, 2009).
  * Green  _et al._ (2016) A. G. Green, C. A. Hooley, J. Keeling, and S. H. Simon, “Feynman path integrals over entangled states,” (2016), arXiv:1607.01778 [cond-mat.str-el] .
  * Sciolla and Biroli (2011) Bruno Sciolla and Giulio Biroli, “Dynamical transitions and quantum quenches in mean-field models,” J. Stat. Mech.: Theory Exp 2011, P11003 (2011).
  * Mori (2017) Takashi Mori, “Classical ergodicity and quantum eigenstate thermalization: Analysis in fully connected Ising ferromagnets,” Phys. Rev. E 96, 012134 (2017).
  * Christandl  _et al._ (2004) Matthias Christandl, Nilanjana Datta, Artur Ekert, and Andrew J. Landahl, “Perfect state transfer in quantum spin networks,” Phys. Rev. Lett. 92, 187902 (2004).
  * Labuhn  _et al._ (2016) Henning Labuhn, Daniel Barredo, Sylvain Ravets, Sylvain de Léséleuc, Tommaso Macrì, Thierry Lahaye, and Antoine Browaeys, “Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models,” Nature 534, 667–670 (2016).
  * Browaeys and Lahaye (2020) Antoine Browaeys and Thierry Lahaye, “Many-body physics with individually controlled Rydberg atoms,” Nature Physics 16, 132–142 (2020).
  * Bluvstein  _et al._ (2021) D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Semeghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, S. Choi, M. Serbyn, M. Greiner, V. Vuletić, and M. D. Lukin, “Controlling quantum many-body dynamics in driven rydberg atom arrays,” Science 371, 1355–1359 (2021).
  * Else  _et al._ (2020) Dominic V. Else, Christopher Monroe, Chetan Nayak, and Norman Y. Yao, “Discrete time crystals,” Annual Review of Condensed Matter Physics 11, 467–499 (2020).
  * Maskara  _et al._ (2021) Nishad Maskara, Alexios A Michailidis, Wen Wei Ho, Dolev Bluvstein, Soonwon Choi, Mikhail D Lukin, and Maksym Serbyn, “Discrete time-crystalline order enabled by quantum many-body scars: entanglement steering via periodic driving,” (2021), arXiv:2102.13160 [quant-ph] .
  * Keesling  _et al._ (2019) Alexander Keesling, Ahmed Omran, Harry Levine, Hannes Bernien, Hannes Pichler, Soonwon Choi, Rhine Samajdar, Sylvain Schwartz, Pietro Silvi, Subir Sachdev, Peter Zoller, Manuel Endres, Markus Greiner, Vladan Vuletic, and Mikhail D. Lukin, “Quantum Kibble-Zurek mechanism and critical dynamics on a programmable Rydberg simulator,” Nature 568, 207–211 (2019).
  * Semeghini  _et al._ (2021) Giulia Semeghini, Harry Levine, Alexander Keesling, Sepehr Ebadi, Tout T. Wang, Dolev Bluvstein, Ruben Verresen, Hannes Pichler, Marcin Kalinowski, Rhine Samajdar, Ahmed Omran, Subir Sachdev, Ashvin Vishwanath, Markus Greiner, Vladan Vuletic, and Mikhail D. Lukin, “Probing topological spin liquids on a programmable quantum simulator,” (2021), arXiv:2104.04119 [quant-ph] .
  * Satzinger  _et al._ (2021) K. J. Satzinger, Y. Liu, A. Smith, C. Knapp, M. Newman, C. Jones, Z. Chen, C. Quintana, X. Mi, A. Dunsworth, C. Gidney, I. Aleiner, F. Arute, K. Arya, J. Atalaya, R. Babbush, J. C. Bardin, R. Barends, J. Basso, A. Bengtsson, A. Bilmes, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. R. Derk, D. Eppens, C. Erickson, E. Farhi, L. Foaro, A. G. Fowler, B. Foxen, M. Giustina, A. Greene, J. A. Gross, M. P. Harrigan, S. D. Harrington, J. Hilton, S. Hong, T. Huang, W. J. Huggins, L. B. Ioffe, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, T. Khattar, S. Kim, P. V. Klimov, A. N. Korotkov, F. Kostritsa, D. Landhuis, P. Laptev, A. Locharla, E. Lucero, O. Martin, J. R. McClean, M. McEwen, K. C. Miao, M. Mohseni, S. Montazeri, W. Mruczkiewicz, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, T. E. O’Brien, A. Opremcak, B. Pató, A. Petukhov, N. C. Rubin, D. Sank, V. Shvarts, D. Strain, M. Szalay, B. Villalonga, T. C. White, Z. Yao, P. Yeh, J. Yoo, A. Zalcman, H. Neven, S. Boixo, A. Megrant, Y. Chen, J. Kelly, V. Smelyanskiy, A. Kitaev, M. Knap, F. Pollmann, and P. Roushan, “Realizing topologically ordered states on a quantum processor,” (2021), arXiv:2104.01180 [quant-ph] .
  * Morong  _et al._ (2021) W. Morong, F. Liu, P. Becker, K. S. Collins, L. Feng, A. Kyprianidis, G. Pagano, T. You, A. V. Gorshkov, and C. Monroe, “Observation of Stark many-body localization without disorder,” (2021), arXiv:2102.07250 [quant-ph] .
  * Guo  _et al._ (2020) Qiujiang Guo, Chen Cheng, Hekang Li, Shibo Xu, Pengfei Zhang, Zhen Wang, Chao Song, Wuxin Liu, Wenhui Ren, Hang Dong, Rubem Mondaini, and H. Wang, “Stark many-body localization on a superconducting quantum processor,” (2020), arXiv:2011.13895 [quant-ph] .
  * Yang  _et al._ (2020a) 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 in a 71-site Bose–Hubbard quantum simulator,” Nature 587, 392–396 (2020a).
  * Desaules  _et al._ (2021) Jean-Yves Desaules, Ana Hudomal, Christopher J. Turner, and Zlatko Papić, “Proposal for realizing quantum scars in the tilted 1d Fermi-Hubbard model,” Phys. Rev. Lett. 126, 210601 (2021).
  * Nandkishore and Sondhi (2017) Rahul M. Nandkishore and S. L. Sondhi, “Many-body localization with long-range interactions,” Phys. Rev. X 7, 041021 (2017).
  * Kormos  _et al._ (2017) Marton Kormos, Mario Collura, Gabor Takács, and Pasquale Calabrese, “Real-time confinement following a quantum quench to a non-integrable model,” Nature Physics 13, 246–249 (2017).
  * Robinson  _et al._ (2019) Neil J. Robinson, Andrew J. A. James, and Robert M. Konik, “Signatures of rare states and thermalization in a theory with confinement,” Phys. Rev. B 99, 195108 (2019).
  * Yang  _et al._ (2020b) Zhi-Cheng Yang, Fangli Liu, Alexey V. Gorshkov, and Thomas Iadecola, “Hilbert-space fragmentation from strict confinement,” Phys. Rev. Lett. 124, 207602 (2020b).
  * Castro-Alvaredo  _et al._ (2020) Olalla A. Castro-Alvaredo, Máté Lencsés, István M. Szécsényi, and Jacopo Viti, “Entanglement oscillations near a quantum critical point,” Phys. Rev. Lett. 124, 230601 (2020).
  * Magnifico  _et al._ (2020) Giuseppe Magnifico, Marcello Dalmonte, Paolo Facchi, Saverio Pascazio, Francesco V. Pepe, and Elisa Ercolessi, “Real Time Dynamics and Confinement in the $\mathbb{Z}_{n}$ Schwinger-Weyl lattice model for 1+1 QED,” Quantum 4, 281 (2020).
  * Chanda  _et al._ (2020) Titas Chanda, Jakub Zakrzewski, Maciej Lewenstein, and Luca Tagliacozzo, “Confinement and lack of thermalization after quenches in the bosonic Schwinger model,” Phys. Rev. Lett. 124, 180602 (2020).
  * Borla  _et al._ (2020) Umberto Borla, Ruben Verresen, Fabian Grusdt, and Sergej Moroz, “Confined phases of one-dimensional spinless fermions coupled to ${Z}_{2}$ gauge theory,” Phys. Rev. Lett. 124, 120503 (2020).
  * Surace  _et al._ (2020b) Federica M. Surace, Paolo P. Mazza, Giuliano Giudici, Alessio Lerose, Andrea Gambassi, and Marcello Dalmonte, “Lattice gauge theories and string dynamics in Rydberg atom quantum simulators,” Phys. Rev. X 10, 021041 (2020b).
  * Celi  _et al._ (2020) Alessio Celi, Benoît Vermersch, Oscar Viyuela, Hannes Pichler, Mikhail D. Lukin, and Peter Zoller, “Emerging two-dimensional gauge theories in Rydberg configurable arrays,” Phys. Rev. X 10, 021057 (2020).
  * Pancotti  _et al._ (2020) Nicola Pancotti, Giacomo Giudice, J. Ignacio Cirac, Juan P. Garrahan, and Mari Carmen Bañuls, “Quantum East model: Localization, nonthermal eigenstates, and slow dynamics,” Phys. Rev. X 10, 021051 (2020).
  * Roy and Lazarides (2020) Sthitadhi Roy and Achilleas Lazarides, “Strong ergodicity breaking due to local constraints in a quantum system,” Phys. Rev. Research 2, 023159 (2020).
  * Feldmeier  _et al._ (2019) Johannes Feldmeier, Frank Pollmann, and Michael Knap, “Emergent glassy dynamics in a quantum dimer model,” Phys. Rev. Lett. 123, 040601 (2019).
  * Hart  _et al._ (2020) Oliver Hart, Giuseppe De Tomasi, and Claudio Castelnovo, “From compact localized states to many-body scars in the random quantum comb,” Phys. Rev. Research 2, 043267 (2020).
  * Sachdev and Ye (1993) Subir Sachdev and Jinwu Ye, “Gapless spin-fluid ground state in a random quantum Heisenberg magnet,” Phys. Rev. Lett. 70, 3339–3342 (1993).
  * Kitaev (2015) A Kitaev, “A simple model of quantum holography, talk at KITP,” University of California, Santa Barbara USA 7 (2015).
  * Maldacena  _et al._ (2016) Juan Maldacena, Stephen H. Shenker, and Douglas Stanford, “A bound on chaos,” Journal of High Energy Physics 2016, 106 (2016).
  * Alhambra  _et al._ (2020) Álvaro M. Alhambra, Anurag Anshu, and Henrik Wilming, “Revivals imply quantum many-body scars,” Phys. Rev. B 101, 205107 (2020).
  * Omran  _et al._ (2019) 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. Vuletić, and M. D. Lukin, “Generation and manipulation of Schrödinger cat states in Rydberg atom arrays,” Science 365, 570–574 (2019).
  * Dooley (2021) Shane Dooley, “Robust quantum sensing in strongly interacting systems with many-body scars,” PRX Quantum 2, 020330 (2021).
  * Kao  _et al._ (2021) Wil Kao, Kuan-Yu Li, Kuan-Yu Lin, Sarang Gopalakrishnan, and Benjamin L. Lev, “Topological pumping of a 1d dipolar gas into strongly correlated prethermal states,” Science 371, 296–300 (2021).