$|fu\rangle_{j}$ state created by flipping every alternate elementary
plaquette in the reference state. Flipped plaquettes with respect to the
reference state are shown shaded in blue. The entanglement cut used to
calculate the bipartite entanglement entropy of the system after mapping it to
an open chain of pseudospins divides the system into two halves with
$(N_{p}/2)\pm 1$ pseudospins as shown below.

This can indeed be checked by monitoring the fidelity $F(t)$ $=$ $|\langle
s|\exp(-iHt)|s\rangle|^{2}$ using ED for these representative Fock states
(denoted by $|s\rangle$) in Fig. 25. Most initial states show a rapid drop in
$F(t)$ within $t\sim O(1)$ which is expected for a high-energy initial state
in an interacting system. However, for the wire, the behaviour of $F(t)$ for
$|fu\rangle_{w}$ and $|fuu\rangle_{w}$ are markedly different, with both
showing periodic revivals with an emergent time-scale $T^{*}\sim 5$ for
$|fu\rangle_{w}$ (Fig. 25, top-left panel) and $T^{*}\sim 4$ for
$|fuu\rangle_{w}$ (Fig. 25, top-right panel). The periodic revivals of $F(t)$
starting from $|fu\rangle_{w}$ show a decaying envelope in time that can be
reasonably described by the envelope function $\exp(-t/\tau_{w})$ with
$\tau_{w}\approx 10$ (Fig. 25, top-left panel). This decaying envelope to the
periodic revivals distinguish this phenomenon from the persistent oscillations
starting from initial Fock states discussed in Sec. 2.3. In fact, the fidelity
revivals for the $|fuu\rangle_{w}$ state shows a very interesting finite-size
effect with such revivals being strongest when $N_{p}=3n+1$ where $n$ is an
integer. For example, the peak value of the first fidelity revival in time
equals $0.52$ for $N_{p}=19$ while it is much smaller for $N_{p}=18$ and
$N_{p}=20$ ($0.22$ and $0.18$ respectively). Furthermore, the $N_{p}=19$ data
for $F(t)$ starting from the initial state $|fuu\rangle_{w}$ shows no sign of
a decaying exponential envelope till $t=50$ (see Fig. 25, top-right panel).
For the junction of two wires with $N_{p}=23$, we again see rapid decay of
$F(t)$ starting from $|r\rangle_{j}$ (Fig. 25, bottom-left panel) while $F(t)$
shows non-trivial periodic revivals from $|fu\rangle_{j}$ with the same
$T^{*}\sim 5$ as in the wire case. The periodic revivals are weaker for the
junction of two wires compared to the single wire and again have a decaying
exponential envelope described by $\exp(-t/\tau_{j})$ with a smaller
$\tau_{j}\approx 7$, but these fidelity revivals are nontheless clearly
visible up to $t\sim 20$.

It is useful to point out that the enhanced revivals observed for
$|fuu\rangle_{w}$ for $N_{p}=3n+1$ for the wire, which is equivalent to a
$|\mathbb{Z}_{3}\rangle$ initial state in a 1D PXP chain with $N_{p}$ sites
and OBC, was not pointed out in the literature previously and additional terms
were added to the PXP Hamiltonian to cause enhancement of fidelity revivals
from the $|\mathbb{Z}_{3}\rangle$ state [79]. As is well-known from the 1D PXP
chain [12, 13], these fidelity revivals from certain special initial states is
due to a large overlap with approximate towers of QMBS that are equally spaced
in energy. These towers are most clearly seen by plotting the overlaps of the
initial Fock state $|fuu\rangle_{w}$ with the many-body eigenstates
$|E\rangle$ as a function of energy (see Fig. 26). We see that at $N_{p}=3n+1$
(Fig. 26, middle panel), these towers are much more clearly formed compared to
$N_{p}=3n$ (Fig. 26, left panel) and to $N_{p}=3n+2$ (Fig. 26, right panel).
We also note that at the system sizes, $N_{p}=3n+1$, the $|fuu\rangle_{w}$
Fock state becomes orthogonal to the zero mode subspace of the system (up to
machine precision) even though the initial state has zero average energy. A
deeper understanding of all these striking finite-size effects at $N_{p}=3n+1$
for the wire/open PXP chain would be highly desirable.

Figure 25: The behavior of fidelity $F(t)$ shown for the wire with two
different initial Fock states with the top-left panel for $|fu\rangle_{w}$,
and the the top-right panel for $|fuu\rangle_{w}$. The bottom-left (bottom-
right) panel shows the fidelity as a function of time with the initial state
being $|r\rangle_{j}$ ($|fu\rangle_{j}$) for a junction of two wires.

Even though the junction of two wires cannot be reduced to the 1D PXP chain,
this model also admit approximate towers of QMBS that are equidistant in
energy. In Fig. 27 (two panels), the overlap behavior of the $|fu\rangle_{j}$
Fock state (Fig. 24, right panel) with the eigenstates of the junction of two
wires with $N_{p}=23$ is shown for the $\mathcal{R}_{j}=+1$ and the
$\mathcal{R}_{j}=-1$ sectors respectively. In this case, the towers of states
with higher overlap to the Fock state are somewhat less clearly separated from
the bulk of the spectrum as compared to the 1D PXP model, explaining the
weaker fidelity revivals in the junction of two wires as compared to the
single wire case (Fig. 25, top left and bottom right panels). Since
$[\mathcal{R}_{j},\mathcal{C}_{j}]\neq 0$, the overlaps are not symmetric with
respect to zero energy; rather, the overlap behavior for $\mathcal{R}_{j}=+1$
sector is a mirror image (with the mirror axis being $E=0$) of the
$\mathcal{R}_{j}=-1$ sector.

Figure 26: Density plots showing the overlap of the $|fuu\rangle_{w}$ state
with energy eigenstates of the wire with $N_{p}=18$ (left panel), $N_{p}=19$
(middle panel) and $N_{p}=20$ (right panel) respectively. In all the panels,
the density of states is indicated by the same color map where warmer color
corresponds to higher density of states.

Figure 27: Density plots that show the overlap of the $|fu\rangle_{j}$ state
for a junction of two wires with energy eigenstates of the junction with two
wires with $N_{p}=23$ for the $\mathcal{R}_{j}=+1$ (left panel) and the
$\mathcal{R}_{j}=-1$ (right panel) sectors respectively. In both panels, the
density of states is indicated by the same color map where warmer color
corresponds to higher density of states.

Another tell-tale signature for the presence of QMBS is that such states have
anomalously low bipartite entanglement entropy compared to neighboring
eigenstates with similar energies. The bipartite entanglement entropy is given
by

$\displaystyle S(A)=-\mathrm{Tr}[\rho_{A}\ln\rho_{A}]$ (45)

for each eigenstate $|\Psi\rangle$ where
$\rho_{A}=\mathrm{Tr}_{\overline{A}}|\Psi\rangle\langle\Psi|$ where $\rho_{A}$
represents the reduced density matrix obtained by partitioning the system in
to two spatial regions, $A$ and its complement $\overline{A}$. We find it
convenient to compute the bipartite entanglement entropy by adopting the one-
to-one mapping of Fock states in a wire or a junction of two wires to
pseudospins with values $0,\pm 1$ in a 1D open chain with $N_{p}$ sites, with
the mapping explained in Sec. 5.1. We then take $\overline{A}$ to be the first
$N_{p}/2$ sites of the 1D chain for the wire (as shown in Fig. 23) and the
first $(N_{p}/2)-1$ sites of the 1D chain for the junction of two wires (as
shown in Fig. 24). The results of such a computation from ED are shown in Fig.
28 for the wire (top-left panel) and the junction of two wires (top-right
panel) respectively. While both the panels show a presence of several
anomalous eigenstates with lower bipartite entanglement entropy than the bulk
of the spectrum, the wire shows a broader distribution of values especially in
the neighborhood of $E=0$ compared to the junction of two wires.

Figure 28: The behavior of the bipartite entanglement entropy for each
eigenstate shown for (top left panel) a wire with $N_{p}=18$ in the
$\mathcal{R}_{w}=+1$ sector and for (top right panel) a junction of two wires
with $N_{p}=19$ for both $\mathcal{R}_{j}=\pm 1$ sectors together. The
expectation value of a local diagonal operator defined on an elementary
plaquette for each energy eigenstate shown for the wire with $N_{p}=22$
(bottom left panel) and the junction of two wires with $N_{p}=23$ (bottom
right panel) respectively, both in the symmetry sector $\mathcal{R}_{w/j}=+1$.
The dotted lines in both the lower panels indicate the thermal values as a
function of the energy $E$. In all the panels, the density of states is
indicated by the same color map where warmer color corresponds to higher
density of states.

The expectation value of any local operator in a high-energy eigenstate is
supposed to approach the thermal result, with the inverse temperature being
fixed by the energy density of the eigenstate, for a system that satisfies
ETH. In Fig. 28 (bottom panels), we consider the expectation value
$\langle\Psi|\mathcal{O}|\Psi\rangle$ as a function of energy, where
$|\Psi\rangle$ denotes an eigenstate of the wire (bottom left panel) or the
junction of two wires (bottom right panel) and
$\mathcal{O}=\sigma_{1}^{z}\sigma_{3}^{z}+\sigma_{2}^{z}\sigma_{4}^{z}$ where
sites $1,2,3,4$ represent the four sites (in a clockwise manner) of the
$N_{p}/2$-th elementary plaquette from the top-left for a wire and the central
junction plaquette for the junction of two wires. Since this local operator is
located away from the edges of the system, it represents a bulk operator in
both the cases. The thermal result as a function of energy is represented by
dotted curves on both the lower panels of Fig. 28. While the expectation value
of the local operator for the bulk of the spectrum indeed approaches the
thermal result, several eigenstates do show an expectation value that is quite
far from the corresponding thermal result. The wire again shows a much larger
variation in the range of expectation values compared to the junction of two
wires, especially in the vicinity of $E=0$. Interestingly, the latter case
shows tower-like structures that are equidistant in energy (Fig. 28 (bottom
right panel)) with a similar spacing between them as the tower of scar states
visible in the overlap plots (Fig. 26, two panels).

##  6 Discussion

In conclusion, we have considered a spin-$1/2$ model on the two-dimensional
square lattice in a constrained Hilbert space where no two nearest-neighbor
sites can have up-spins simultaneously. The interaction Hamiltonian is
composed of ring-exchange terms on elementary plaquettes that not only
conserve the total magnetization but also the magnetization along each column
and row of the square lattice. These additional subsystem symmetries imply
conservation of a global dipole moment that leads to the phenomenon of Hilbert
space fragmentation. While microscopic models of both weak and strong
fragmentation are known in one dimension, we show that this particular
interacting model with hard-core constraints presents a rare example of strong
Hilbert space fragmentation in higher dimensions.

All the many-body eigenstates of this model can be expressed in terms of the
tensor product of modes of appropriate quantum drums and any left-over inert
spins. Given an initial unentangled product state in the computational basis,
the associated quantum drums get fixed and come in a variety of shapes and
sizes starting from one-plaquette drums to truly extensive structures made of
plaquettes that share edges and/or vertices with each other. Specifying the
plaquettes that belong to a drum uniquely fixes its spectrum. Crucially, these
drums can be “shielded” from each other by shielding regions that only grow as
the perimeter and not the area of such drums.

Large quantum drums and their associated fragment dimensions can be most
easily estimated by using a “wire” decomposition of such drums and then
counting the number of ways in which such wires can fluctuate simultaneously
without violating the kinematic constraints. This allows us to identify the
appropriate drums that dominate statistically for a given density of up-spins
(bosons). It is shown that typical initial product states evade ETH-predicted
thermalization due to the presence of either an extensive number of inert
spins or an extensive number of next-nearest neighbor spin correlations that
are pinned to their initial athermal values.

We consider the spectrum of some small drums analytically to show the
emergence of interesting zero, non-zero integer and irrational modes. Close
packing an extensive number of the elementary one-plaquette drums already
generate many-body eigenstates with integer energies (including zero) and
strict area-law scaling of entanglement entropy. Large quasi-one-dimensional
and two-dimensional quantum drums can be viewed as interesting interacting
systems with constrained Hilbert spaces. A class of these drums harbor a large
number of exact zero modes. The simplest quasi-one-dimensional drum, which we
dub as a wire, is shown to be exactly equivalent to the well-known PXP chain
with open boundary conditions. However, a particular junction of two wires is
also studied which cannot be mapped in to the PXP chain and represents a
different constrained model. Both these quasi-one-dimensional drums support
distinct families of quantum many-body scars that cause periodic revivals from
certain simple initial states. Our numerics for the wire also shows that the
period-$3$ state with Rydberg excitations on every third site shows strong
revivals for open chains of length $3n+1$ without the necessity of adding
further perturbations to the PXP chain. This result can have possible
implications for experiments with Rydberg atoms.

Several possible open directions emerge from our study. Other junctions of
wires, like junctions of three wires and four wires, as well as some of the
two-dimensional drums introduced here should have interesting high-energy
properties. It is further possible to add diagonal interactions in the
computational basis which preserve the fragmented structure of the model.
Using such additional interactions, one can possibly access different phases
and phase transitions at zero temperature in both quasi-one-dimensional and
two dimensional theories in the presence of subsystem symmetries.

Note added: While preparing this manuscript, we came to know of a related work
by Lehmann et al. [80] which discusses strong Hilbert space fragmentation in
higher dimensions using a different Hamiltonian (correlated hopping model).

Acknowledgements: K.S. thanks DST, India for support through SERB project
JCB/2021/000030. AC thanks Madhumita Sarkar for help with cluster facilities
and DST, India for support through SERB project PDF/2021/001134.

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