Strong Hilbert space fragmentation via emergent quantum drums in two
dimensions

Anwesha Chattopadhyay1,2, Bhaskar Mukherjee3, K. Sengupta1, Arnab Sen1*

1 School of Physical Sciences, Indian Association for the Cultivation of
Science, Jadavpur, Kolkata 700032, India.

2 Department of Physics, School of Mathematical Sciences, Ramakrishna Mission
Vivekananda Educational and Research Institute, Belur, Howrah 711202, India.

3 Department of Physics and Astronomy, University College London, Gower
Street, London WC1E 6BT, United Kingdom

⋆<EMAIL_ADDRESS>
## Abstract

We introduce a disorder-free model of $S=1/2$ spins on the square lattice in a
constrained Hilbert space where two up-spins are not allowed simultaneously on
any two neighboring sites of the lattice. The interactions are given by ring-
exchange terms on elementary plaquettes that conserve both the total
magnetization as well as dipole moment. We show that this model provides a
tractable example of strong Hilbert space fragmentation in two dimensions with
typical product states in the computational basis evading thermalization.
Given any product state, the system can be decomposed into disjoint spatial
regions made of edge and/or vertex sharing plaquettes that we dub as “quantum
drums”. These quantum drums come in many shapes and sizes and specifying the
plaquettes that belong to a drum fixes its spectrum. The spectra of some small
drums is calculated analytically. We study two bigger quasi-one-dimensional
drums numerically, dubbed “wire” and a “junction of two wires” respectively.
We find that these possess a chaotic spectrum but also support distinct
families of quantum many-body scars that cause periodic revivals from
different initial states. The wire is shown to be equivalent to the one-
dimensional PXP chain with open boundaries, a paradigmatic model for quantum
many-body scarring, while the junction of two wires represents a distinct
constrained model.

###### Contents

  1. 1 Introduction
  2. 2 Model and its properties
    1. 2.1 Quantum drums
      1. 2.1.1 Constructing drums from a product state
      2. 2.1.2 Shielding region and closest approach of drums
      3. 2.1.3 Recursive construction of drums
      4. 2.1.4 Wires, junctions of wires, other quasi-1D and 2D drums
    2. 2.2 Exact diagonalization of small lattices and deciphering spectrum using drums
    3. 2.3 Eigenstates with integer energies from packing of one-plaquette drums
  3. 3 Strong Hilbert space fragmentation
    1. 3.1 Numerical evidence from exact diagonalization
    2. 3.2 Wire decomposition of quantum drums
    3. 3.3 Absence of thermalization from typical product states
  4. 4 Analytical study of small quantum drums
    1. 4.1 Tree structure
    2. 4.2 Wire
    3. 4.3 Junction units
  5. 5 Numerical study of large quantum drums
    1. 5.1 Tree generating algorithm and equivalence of wire to 1D PXP chain
    2. 5.2 Hilbert space dimension and level statistics
    3. 5.3 Zero modes and index theorem
    4. 5.4 QMBS and related diagonastics
  6. 6 Discussion

##  1 Introduction

A generic isolated quantum system with many degrees of freedom is expected to
“self-thermalize” as it evolves unitarily under the dynamics of its own
Hamiltonian [1]. This implies that pure states obtained from the time
evolution of different initial states that share the same energy density
cannot be distinguished from each other at late times using only local probes.
A microscopic justification for this self-thermalization is provided by the
eigenstate thermalization hypothesis (ETH) [2, 3, 4, 5] that posits that high-
energy eigenstates of such systems appear locally thermal with the temperature
being set by the energy density of the eigenstate.

Rapid progress in producing and manipulating well-isolated quantum simulators
such as ultracold gases [6, 7], trapped ions [8], Rydberg atom arrays [9] and
superconducting qubits [10] has made it possible to study thermalization and
its violations in such platforms. In particular, the experimental observation
of late-time coherent oscillations from certain simple high-energy initial
states in a kinetically-constrained chain of $51$ Rydberg atoms [11] generated
great interest in understanding thermalization in interacting theories with
constrained Hilbert spaces. The revivals reported in Ref. [11] were shown to
arise due to the large overlap of some simple initial states with a small set
of nonthermal high-energy eigenstates, dubbed quantum many-body scars (QMBS)
in Refs. [12, 13], in an otherwise non-integrable PXP model [14, 15] that
served as the minimal model for the experiment.

Subsequent theoretical studies have shown a plethora of interesting non-
ergodic behavior in various models with constrained Hilbert spaces, including
Hamiltonian formulations of lattice gauge theories [16, 17, 18, 19, 20] that
may be realizable on quantum simulators [21, 22, 23]. These include different
varieties of QMBS [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45], disorder-free localization [46, 47, 48, 49] as
well as a richer ergodicity-breaking paradigm dubbed Hilbert space
fragmentation [50, 51]. Such forms of ETH-violation are distinct from the
breakdown of ETH due to many-body localization [52, 53, 54] where strong
disorder plays a crucial role.

Systems with Hilbert space fragmentation [55, 56, 57, 58, 59, 60, 61, 62, 63,
64, 65, 66, 67, 68] often feature multiple conservation laws [50, 51] which
severely restrict the mobility of excitations. In such cases, the Hilbert
space can split into exponentially many dynamically disconnected fragments.
These fragments cannot be distinguished by any obvious global symmetries of
the Hamiltonian [50, 51]. Such fragments can either be finite or infinite-
dimensional matrices in the thermodynamic limit and can show vastly different
dynamical properties, such as integrability [57, 65], disorder-free
localization [46, 47, 48, 49, 56, 60] or QMBS [66, 64] though large fragments
are expected to typically satisfy a Krylov-restricted version of ETH [55].
Both weak and strong fragmentation is known to exist in one-dimensional (1D)
models [50, 51], with the two cases distinguished by whether the fraction of
eigenstates violating the ETH are a set of measure zero or not in the
thermodynamic limit. Weakly fragmented systems are similar to systems with
QMBS since both situations lead to weak ergodicity breaking where typical
initial states still thermalize [69]. However, strongly fragmented systems
present a distinct form of ergodicity breaking that is different from systems
with QMBS.

In Ref. [50], 1D spin models with both global charge and dipole conservation
laws were considered and it was argued that such dipole-conserving models
should exhibit Hilbert space fragmentation in any dimension [50, 51] (for
examples of fragmentation without global dipole conservation, see Refs. [64,
57, 60, 67, 58, 62, 66, 31]). One of the tell-tale signs of fragmentation in
such models is an exponential number of completely inert states that form
$1\times 1$ fragments on their own. While examples of both weak and strong
fragmentation are known in one dimension, it is not clear whether global
dipole conservation alone is sufficient to lead to strong fragmentation in
higher dimensions. This extra conservation ensures that Hilbert space
fragments of different sizes can be constructed by embedding suitable “active”
regions into “inert” backgrounds and surrounding the “active” regions by
“shielding” regions; the shielding region, however, turns out to be of the
same size or bigger than the active region it isolates [51]. This makes it
difficult to construct explicit examples of strong fragmentation in two or
higher dimensions.

In this paper, we will construct a model that shows strong Hilbert space
fragmentation in two dimensions by considering $S=1/2$ spins (equivalently,
hard-core bosons) on the square lattice with ring-exchange terms on elementary
plaquettes that are consistent with total magnetization (equivalently, boson
number) conservation as well as global dipole moment conservation. The
important additional ingredient in the model is the presence of a kinematic
constraint that no two nearest neighbor sites can have two up-spins (bosons)
simultaneously. Similar models with ring-exchange and other competing terms,
but without the additional hard-core constraints, are known to have
interesting low-energy phases and transitions [70, 71, 72]. High-energy
properties of the unconstrained model with only the ring-exchange terms were
studied recently in Ref. [68] where it was realized that such terms imply
subsystem symmetries associated with the conservation of magnetization along
each column and row of the square lattice. This leads to global dipole
conservation and consequently Hilbert space fragmentation. However, as we will
show here, the enforcement of the kinematic constraints leads to several
additional features, including strong ergodicity breaking and the emergence of
“quantum drums”, that were absent in the model considered in Ref. [68]. The
quantum drums here can be viewed as the “active” regions which can then be
surrounded by “shielding” regions of $O(1)$ thickness (in lattice units).
Crucially, the thickness of the shielding regions does not grow with the size
of the quantum drums. Each quantum drum is made of edge and/or vertex sharing
elementary plaquettes and specifying the plaquettes that make a drum uniquely
fixes its spectrum, thus justifying this particular nomenclature. We refer the
reader to Fig. 1 for an example of quantum drums and their corresponding
shielding regions that emerge from a particular initial state.

Figure 1: An initial state on the square lattice where the up-spins (bosons)
are indicated in red while the other sites have down-spins (no bosons). The
five quantum drums that correspond to this initial state are shown with their
elementary plaquettes shaded. Plaquettes that are colored as green can have
two up-spins (bosons) along both its diagonals during quantum evolution as
explained in the text. The pink plaquettes, also labeled by “u”, can have two
up-spins (bosons) only along one of its two diagonals. Sites that do not
belong to any of these five drums have inert up/down spins fixed by the
initial condition. Each drum generates a separate fragment in Hilbert space
with the corresponding fragment dimension being $3\times 3$ for the top-left
drum, $7\times 7$ for the bottom-left drum, $24\times 24$ for the middle drum
and $2\times 2$ for both the top-right and bottom-right drums. The shielding
regions of the drums are shown using thick blue lines.

All the Hilbert space fragments of this model that are not $1\times 1$, i.e.,
that do not correspond to inert Fock states, can be generated from a
combination of appropriate quantum drums embedded in an otherwise inert
background (which may itself shrink to zero for certain drums) (see Fig. 1).
These quantum drums come in a variety of shapes and sizes and can be made of a
finite number or an arbitrarily large number of plaquettes in the
thermodynamic limit. Crucially, the scaling of the size of the Hilbert space
fragments generated from large drums allows for a proof of lack of
thermalization for any typical Fock state by identifying either (a) an
extensive number of single spin correlators or (b) an extensive number of
next-nearest neighbor two-spin correlators whose expectation values stay
pinned to their initial (non-thermal) values. To the best of our knowledge,
this interacting theory provides the first example of strong Hilbert space
fragmentation in two dimensions.

The rest of the paper is arranged as follows. In Sec. 2, we introduce the
model and summarize some of its important properties. In Sec. 2.1, we discuss
the quantum drums that emerge in this model in more details. The classical
construction of the drums, given an initial state, is explained in Sec. 2.1.1.
The construction of the shielding regions of drums and closest approach of two
drums such that these can still be considered independent of each other is
explained in Sec. 2.1.2. In Sec. 2.1.3, a recursive procedure to generate
bigger quantum drums starting from the most elementary one-plaquette drum is
detailed. We introduce some particular drums, dubbed wires and different
junctions of wires and some other quasi-one dimensional (1D) and two-
dimensional (2D) drums in Sec. 2.1.4. In Sec. 2.2, we give numerical evidence
that the energy eigenvalues and their associated degeneracies from exact
diagonalization (ED) on small systems can be completely understood in terms of
the spectra of the quantum drums. We construct a large class of eigenstates
with integer eigenvalues (including zero modes) from the packing of the
simplest one-plaquette quantum drums in Sec. 2.3. Evidence for strong Hilbert
space fragmentation for this model is presented in Sec. 3. The numerical
evidence from ED is presented in Sec. 3.1. A wire decomposition of drums is
introduced in Sec. 3.2 which allows us to derive the scaling of the dimension
of the Hilbert space fragments associated with large 2D drums and also
determine which kinds of drums dominate statistically given a certain density
of up-spins (bosons) in Sec. 3.3. This insight leads to the proof that typical
initial product states do not thermalize in Sec. 3.3 by identifying either an
extensive number of single-spin correlators or two-spin correlators that stay
pinned to their initial non-thermal values. The analytical study for the
spectra of certain small quantum drums is given in Sec. 4. A tree structure to
represent the action of $H$ in the Fock space of a drum is explained in Sec.
4.1. The spectra of small wires is calculated in Sec. 4.2 while the spectra of
other small quantum drums that can be viewed as building blocks of more
complicated wire junctions is calculated in Sec. 4.3. The spectra of two
different classes of bigger quasi-1D quantum drums, a wire and a particular
junction of two wires, are addressed numerically using exact diagonalization
(ED) in Sec. 5. Both these large quantum drums can be interpreted as effective
quasi-1D models with a spectrum that is symmetric around zero energy. A tree
generating algorithm is described and the equivalence of the wire to the 1D
PXP model on an open chain is shown in Sec. 5.1. The Hilbert space dimensions
for both the drums are calculated analytically and level statistics are
computed numerically in Sec. 5.2. The Hilbert space structure of the junction
of two wires turns out to be completely different from that of the wire as
discussed in both Sec. 5.1 and Sec. 5.2. One of these fragments is shown to
have a macroscopically large number of exact zero modes while the other
fragment has no zero modes in Sec. 5.3. Both fragments satisfy Krylov-
restricted ETH but also support distinct families of QMBS that result in
periodic revivals from different simple initial states as discussed in Sec.
5.4. Our numerical results for the wire show that open PXP chains of length
$3n+1$, where $n$ is an integer, lead to enhanced fidelity revivals for the
period-$3$ ordered initial $|\mathbb{Z}_{3}\rangle$ state without adding any
optimal perturbations to the bare Hamiltonian; a feature which may have
experimental consequence for Rydberg chains. The junction of two wires also
shows QMBS and simple initial states from which clear revivals in fidelity are
observed. Finally, we summarize our main results and conclude in Sec. 6.

##  2 Model and its properties

The Hamiltonian of the model is given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle
J\sum_{j_{x},j_{y}}\left(\sigma^{+}_{j_{x},j_{y}}\sigma^{+}_{j_{x}+1,j_{y}+1}\sigma^{-}_{j_{x}+1,j_{y}}\sigma^{-}_{j_{x},j_{y}+1}+{\rm
h.c.}\right)$ (1)

where $\sigma^{\alpha}_{j_{x},j_{y}}$ for $\alpha=x,y,z$ represent spin-half
Pauli matrices at sites $(j_{x},j_{y})$ of a 2D square lattice,
$\sigma_{j_{x},j_{y}}^{\pm}=(\sigma^{x}_{j_{x},j_{y}}\pm
i\sigma_{j_{x},j_{y}}^{y})/2$, and the lattice spacing has been set to unity.
The Hamiltonian is supplemented by the constraint that two up-spins can not
occupy neighboring sites of the lattice; this is implemented by the operator
relation

$\displaystyle\left(1+\sigma^{z}_{j_{x},j_{y}}\right)\left(1+\sigma^{z}_{j_{x}\pm
1,j_{y}}\right)$ $\displaystyle=$
$\displaystyle\left(1+\sigma^{z}_{j_{x},j_{y}}\right)\left(1+\sigma^{z}_{j_{x},j_{y}\pm
1}\right)=0$ (2)

For finite $L_{x}\times L_{y}$ rectangular lattices, we will consider open
boundary conditions and the constraint (Eq. 2) is then applied to the
three/two nearest neighbors of $(j_{x},j_{y})$ for the edge/corner sites.

This system maps exactly to hard-core bosons with the following
transformations:

$\displaystyle 2b^{\dagger}_{j_{x},j_{y}}b_{j_{x},j_{y}}-1$ $\displaystyle=$
$\displaystyle\sigma^{z}_{j_{x},j_{y}}$ $\displaystyle
b^{\dagger}_{j_{x},j_{y}}$ $\displaystyle=$
$\displaystyle\sigma^{+}_{j_{x},j_{y}}$ (3)

where $b^{\dagger}_{j_{x},j_{y}}$ is the boson creation operator and
$n_{j_{x},j_{y}}$ $=$ $b^{\dagger}_{j_{x},j_{y}}b_{j_{x},j_{y}}$ is the boson
number operator at site $(j_{x},j_{y})$. For the rest of this work, we shall
set $J=1$. The terms in Eq. 1 can be viewed as ring-exchange terms on
elementary plaquettes which convert a clockwise arrangement of $\sigma^{z}$
from being $(+1,-1,+1,-1)$ to $(-1,+1,-1,+1)$ (equivalently, an arrangement of
bosons from $(1,0,1,0)$ to $(0,1,0,1)$) and vice-versa and annihilate other
arrangements on a plaquatte. It is convenient to define a vacuum state where
all sites of the lattice have down-spins, i.e., no bosons for future
reference. This model has the following properties:

  * •

The many-body spectrum of $H$ is symmetric around the energy $E=0$ for any
finite $L_{x}\times L_{y}$ lattice with open boundary conditions (OBC). This
is because the operator defined by

$\displaystyle\mathcal{C}=\prod_{{(j_{x},j_{y})\in(\mathrm{even},\mathrm{even})}}\sigma^{z}_{j_{x},j_{y}}$
(4)

satisfies $\\{H,\mathcal{C}\\}=0$ where
$\prod_{{(j_{x},j_{y})\in(\mathrm{even},\mathrm{even})}}$ denotes a product
over all the sites $(j_{x},j_{y})$ of the lattice such that both $j_{x},j_{y}$
are even. This implies that any many-body eigenstate of $H$ with an energy $E$
and denoted by $|E\rangle$ has a partner $\mathcal{C}|E\rangle$ that has the
energy $-E$.

  * •

Apart from discrete symmetries like rotations by $\pi/2$ (for $L_{x}=L_{y}$
lattices) and $\pi$ (for $L_{x}\neq L_{y}$ lattices), the model has a discrete
reflection symmetry $\mathcal{R}$ where the axis of reflection can be taken to
be the diagonal through $(0,0)$ for $L_{x}=L_{y}$ or the perpendicular
bisector of the longer side when $L_{x}\neq L_{y}$. $\mathcal{R}$ commutes
both with the Hamiltonian $H$ and the “chirality” operator $\mathcal{C}$. This
has the important consequence that the spectrum has exact zero modes whose
number scales exponentially with the system size due to an index theorem shown
in Ref. [73]. These zero modes are the only eigenstates of $H$ that also
possess a definite “chiral charge” of $\pm 1$ under the action of
$\mathcal{C}$.

  * •

The model conserves the total magnetization (boson number) defined by
$S^{z}_{\rm tot}=\sum_{j_{x},j_{y}}\sigma^{z}_{j_{x},j_{y}}$. More
interestingly, it conserves the following dipole moments in the $x$ and $y$
directions:

$\displaystyle
D_{x}=\sum_{j_{x},j_{y}}j_{x}\sigma^{z}_{j_{x},j_{y}},\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}D_{y}=\sum_{j_{x},j_{y}}j_{y}\sigma^{z}_{j_{x},j_{y}}.$
(5)

This property follows from the fact that the total magnetization on each
column and each row of the square lattice is separately conserved under the
dynamics induced by $H$ (Eq. 1) as pointed out earlier in Ref. [68] in a
similar model, but without the Hilbert space constraints. Models with the
simultaneous conservation of total charge and dipole moment have been shown to
have the property of Hilbert space fragmentation [50, 51]. This model is also
fragmented due to the same reasons.

  * •

The simultaneous conservation of magnetization on each column and each row of
the $L_{x}\times L_{y}$ lattice also implies disorder-free localization for a
large class of initial states. To see this, let us consider the vacuum state
on a $L_{x}\times L_{y}$ lattice with OBC and then create an excitation by
flipping a subset of spins to $\sigma^{z}_{j}=+1$ such that the sites labelled
by $j$ are contained inside or on the boundaries of a rectangle of finite
extent smaller than the entire lattice. The aforementioned conservation
property then ensures that these $\sigma^{z}=+1$ spins cannot be transported
outside this bounding rectangle since all the rows/columns outside this region
have their magnetizations pinned to their lowest possible value.

  * •

This model has an exponentially large number (in system size) of zero modes
that are simply inert states, i.e., Fock states in the computational basis
that are annihilated by all the local terms in $H$, a property shared by other
models that simultaneously conserve total charge and dipole moment. However,
the constrained nature of the Hilbert space also leads to an an exponentially
large number of non-trivial zero modes that emerge from Hilbert space
fragments of various dimensions larger than $1\times 1$, ranging from $3\times
3$ to $c^{L_{x}L_{y}}\times c^{L_{x}L_{y}}$ fragments, with $c>1$, for
$L_{x},L_{y}\gg 1$.

  * •

This model possesses exact non-zero integer eigenstates whose number also
scales exponentially in $L_{x}L_{y}$ for integer eigenvalues ranging from $\pm
1$ to $\pm O(\sqrt{L_{x}L_{y}})$ for $L_{x},L_{y}\gg 1$.

###  2.1 Quantum drums

Due to the structure of $H$ (Eq. 1) and the nature of the constrained Hilbert
space (Eq. 2), elementary plaquettes can have a maximum of two up-spins
(bosons), along any one of the two diagonals, and these are the only local
configurations that can have any dynamics. Furthermore, a plaquette with two
up-spins (bosons) can influence the number of possible local configurations in
neighboring two-spin plaquettes even if it can have the two up-spins (bosons)
only along one of the diagonals but not the other due to kinematic constraints
(Eq. 2). These two facts lead to the emergence of dynamically disconnected
spatial structures called quantum drums on a $L_{x}\times L_{y}$ lattice with
OBC.

To understand these drums, let us imagine a classical Markov process in which
the transition from one Fock state to another is caused by a ring-exchange on
some elementary plaquette with two up-spins (bosons). In the presence of the
hard-core constraints specified in Eq. 2, the configuration space splits into
mutually inaccessible fragments, with all configurations within a fragment
being mutually accessible via some finite sequence of the allowed transitions.
Crucially, each such fragment can be associated to a unique real-space
structure composed of a collection of connected elementary plaquettes that
share edges and/or vertices. The Hamiltonian $H$ (Eq. 1) acts in the space of
mutually accessible configurations of each such fragment to generate the
spectra of these quantum drums. From a dynamical point of view, the precise
nature of the quantum drums is imprinted in the particular initial state that
the system starts from.

We will specify two complementary construction procedures for quantum drums
below, one which starts from a given product state in the computational basis
(Sec. 2.1.1) and the other where the drums are constructed recursively
starting from the most elementary one-plaquette drum (Sec. 2.1.3). Some of the
important properties of quantum drums, which will be detailed out in the rest
of the paper, are summarized below:

  * •

Quantum drums are constructed of connected elementary plaquettes that share
edges/vertices. A drum has no site that contains an inert up-spin (boson).

  * •

All many-body eigenstates of $H$ (Eq. 1) can be expressed in terms of the
tensor product of eigenstates of appropriate quantum drums and of the
remaining inert (up/down) spins (if any) on sites that do not belong to any
quantum drum. This point is illustrated in detail using ED results in Sec.
2.2.

  * •

The spectrum of a drum is uniquely fixed once the plaquettes that belongs to
it are specified. The spectrum of any quantum drum is symmetric around $E=0$.
This follows from the above mentioned point and implies the existence of a
corresponding chiral operator $\mathcal{C}_{\mathrm{drum}}$ for each such
drum.

  * •

A class of quasi-1D and 2D quantum drums have an internal reflection symmetry
$\mathcal{R}_{\mathrm{drum}}$ that commutes with $\mathcal{C}_{\mathrm{drum}}$
resulting in an exponential number of exact zero modes as the size of the drum
is increased.

  * •

Any quantum drum conserves the total magnetization when only the spins
(bosons) on the sites that belong to the drum are considered.

  * •

Any quantum drum satisfies an internal subsystem symmetry of simultaneous
conservation of magnetizations along each column and each row (where the
column and row is defined with respect to the background $L_{x}\times L_{y}$
lattice) of the drum.

####  2.1.1 Constructing drums from a product state

Figure 2: Illustration of the recursive construction of a quantum drum given
two different initial states (marked by A and B in the figure) on a $7\times
7$ lattice with OBC where the filled red dots indicate up-spins. The shaded
plaquettes form part of a drum with the top-right and bottom-right panels
indicating the drums for the initial states marked A and B respectively. A
cross at the center of a plaquette indicates that a ring-exchange move is
carried out for that plaquette. The green (pink) plaquettes in the top-right
and bottom-right panels follow the same convention as used in Fig. 1.

We first give a construction procedure that fixes all the quantum drums given
a classical Fock state on a $L_{x}\times L_{y}$ lattice with OBC. An initial
Fock state and its associated drums are shown in Fig. 1. The construction
procedure is schematically shown in Fig. 2 for two drums starting from
different Fock states. Given the Fock state, firstly all plaquettes with two
up-spins (bosons) are shaded. Ring-exchange moves are attempted on such
plaquettes to see whether any additional plaquettes with two up-spins (bosons)
are generated which are again shaded. This process is repeated with the newly
shaded plaquettes until no additional shaded plaquettes are generated. The
shaded plaquettes are then subdivided into connected regions that comprise of
elementary plaquettes that share edges and/or vertices. A final check has to
be performed on each of these connected regions separately to construct the
quantum drums. If the mutually accessible Fock states from a connected region
have certain sites where any up-spin (boson) remains the same in each of the
configurations, these up-spins (bosons) are then labelled as inert and the
shaded plaquettes containing any inert up-spins (bosons) are unshaded. The
remaining shaded plaquettes that are still connected to each other via an edge
or a vertex forms a quantum drum. This last check is necessary to rule out
inert structures made entirely of plaquettes with two up-spins (bosons) [see
Fig. 1 for an example composed of three up-spins (bosons) on two edge-sharing
plaquettes] and to find spatial structures that can be decomposed into an
inert region of up-spins (bosons) and a smaller quantum drum [see Fig. 1,
bottom right for an example of such a decomposition].

Two simple examples of this construction are given for initial classical Fock
states on a $7\times 7$ lattice in Fig. 2 (top-left and middle-left panels),
where the filled circles indicate up-spins (bosons) while the other sites have
down-spins (no bosons). Let us first consider the top three panels. The
initial state is given in the top-left panel marked as A and three plaquettes
are shaded at this stage. Implementing ring-exchange moves on two of the
shaded plaquettes indicated by crosses in the top-left panel generates two
more shaded plaquettes as shown in top-middle panel. Implementing ring-
exchange moves on the shaded plaquettes indicated by crosses in that panel
generates another shaded plaquette in the top-right panel and further ring-
exchanges do not generate any additional shaded plaquettes. The quantum drum
generated by this initial state only contains elementary plaquettes that share
vertices.

The initial state in the middle-left panel marked by B gives four shaded
plaquettes. Implementing ring-exchange to this state for the plaquette
indicated by a cross generates two more shaded plaquettes as shown in the
following panel to the right. Carrying out ring-exchange moves on two more
plaquettes as indicated by crosses generates two additional shaded plaquettes.
To generate the other two shaded plaquettes that form the entire quantum drum,
we go back to the initial Fock state shown in the bottom-left panel and
perform two ring exchange moves on the plaquettes indicated by a cross one
after the other as indicated in the bottom panels. The resulting quantum drum
consists of only edge-sharing plaquettes in this case.

Both the quantum drums shown in Fig. 2 generate $11\times 11$ dimensional
Hilbert space fragments respectively, diagonalizing which results in the
following eigenvalues:

$\displaystyle\left(\pm\sqrt{\frac{1}{2}\left(9\pm\sqrt{57}\right)},\pm\sqrt{3},\pm\sqrt{2},0,0,0\right)$
$\displaystyle\left(\pm 2\sqrt{2},\pm\sqrt{3},-1,-1,+1,+1,0,0,0\right)$ (6)

where the top (bottom) line in Eq. 6 refers to the eigenvalues for the quantum
drum shown in the top-right (bottom-right) panel of Fig. 2. These two examples
already illustrate that drums can have non-trivial zero modes, nonzero
integer-valued eigenstates as well as eigenstates with irrational eigenvalues.
We refer the reader to Sec. 4 for the explicit construction of the Hilbert
space fragments associated to some small quantum drums.

####  2.1.2 Shielding region and closest approach of drums

Each quantum drum is associated with a shielding region of its own such that
two quantum drums can fluctuate independently as long as the boundaries of
their corresponding shielding regions do not cross. Given any Fock state
consistent with a single quantum drum composed of a finite number of
elementary plaquettes with the rest of the sites that do not belong to the
drum being $\sigma^{z}=-1$ (no bosons), the corresponding shielding region can
again be fixed by a classical construction. The shielding region only consists
of a subset of the elementary plaquettes that directly share edges/vertices
with the plaquettes on the perimeter of a quantum drum, irrespective of the
size of the drum. Thus, the thickness of the shielding region does not scale
with the size of the quantum drum and remains $O(1)$ in lattice units (e.g.,
see Fig. 1 and Fig. 3).

Figure 3: Two quantum drums are shown in the left and the right panels. The
integers shown inside each plaquette refers to $n_{\Box_{j}}$ given by Eq. 7
on each plaquette of the lattice using all the Fock states generated when the
quantum drum is embedded in the vacuum state. The green (pink) plaquettes in
the two quantum drums follow the same convention as used in Fig. 1. The
shielding regions of both drums are shown using thick blue lines and mark the
boundary between the $n_{\Box_{j}}=1$ and the $n_{\Box_{j}}=0$ plaquettes.

Let us first consider an elementary one-plaquette quantum drum starting from
the vacuum state and then placing two $\sigma^{z}=+1$ spins (bosons) along any
one of the diagonals of an elementary plaquette. Given this Fock state, ring-
exchange is possible on only this elementary plaquette which then generates
another Fock state where the $\sigma_{z}=+1$ spins (bosons) get transported to
the other diagonal of this plaquette. Considering both these Fock states to
compute

$\displaystyle
n_{\Box_{j}}=\mathrm{max}(2+(\sigma^{z}_{j_{x},j_{y}}+\sigma^{z}_{j_{x}+1,j_{y}}+\sigma^{z}_{j_{x},j_{y}+1}+\sigma^{z}_{j_{x}+1,j_{y}+1})/2)$
(7)

on each plaquette of the lattice, we see that $n_{\Box_{j}}=2$ for the
flippable plaquette which is surrounded by $n_{\Box_{j}}=1$ and
$n_{\Box_{j}}=0$ plaquettes, respectively (Fig. 3, left panel). The
$n_{\Box_{j}}=2$ plaquette defines the quantum drum while the $n_{\Box_{j}}=1$
plaquettes along the perimeter of the quantum drum define the shielding region
associated with this drum. The shielding region terminates at the boundary of
these $n_{\Box_{j}}=1$ and the $n_{\Box_{j}}=0$ plaquettes (Fig. 3, left
panel). By construction, the sites at the boundary of the shielding region
cannot have $\sigma^{z}=+1$ spins (bosons). A more complicated quantum drum is
shown in Fig. 3, right panel which can be generated from the vacuum state by,
e.g., placing two $\sigma^{z}=+1$ spins (bosons) each along parallel diagonals
of the left-most and the right-most plaquette contained in the quantum drum
such that the hard-core constraints are not violated. Performing all possible
ring-exchanges for this quantum drum generates two more Fock states.
Considering these three Fock states to compute $n_{\Box_{j}}$ on each
plaquette of the lattice, the four $n_{\Box_{j}}=2$ plaquettes, which are all
connected to each other by edges for this particular drum, now define this
bigger quantum drum (Fig. 3, right panel) while the $n_{\Box_{j}}=1$
plaquettes along the perimeter of the quantum drum define the shielding region
as before (Fig. 3, right panel). The shielding region is more complicated
compared to the one-plaquette drum and its boundary is again defined by the
boundary of the $n_{\Box_{j}}=1$ and the $n_{\Box_{j}}=0$ plaquettes (Fig. 3,
right panel). This classical construction procedure for the shielding region
can be carried out for any arbitrary quantum drum composed of a finite number
of elementary plaquettes.

We can now ask for the closest approach of any two quantum drums such that
both the drums can be viewed to be independent of each other. The answer is
that the boundaries of their corresponding shielding regions, which do not
carry $\sigma^{z}=+1$ spins (bosons), cannot penetrate each other but can at
most touch each other. For example, this is the case in the example considered
in Fig. 1 which explains why the different quantum drums can be considered to
be independent of each other. If the boundaries cross each other, the
shielding regions have to necessarily change which also changes the
corresponding quantum drums in their interior according to one of the
following three possibilities: (i) the two quantum drums fuse to produce a
bigger quantum drum, (ii) a spatial structure is produced such that it can be
decomposed into an inert region of up-spins (bosons) and a smaller quantum
drum, and (iii) a fully inert region of up-spins (bosons) is formed.

####  2.1.3 Recursive construction of drums

We now present a complementary drum construction procedure to the one
explained in Sec. 2.1.1 which does not need the specification of a product
state on the entire $L_{x}\times L_{y}$ lattice. Instead, this recursive
construction creates larger drums starting from smaller ones. We start with a
Fock state consistent with a particular quantum drum composed of certain
connected plaquettes, where the sites that do not belong to the drum are
assigned $\sigma^{z}=-1$ spins (no bosons). This is equivalent to embedding
the quantum drum in the vacuum state. By definition, the boundary of the
shielding region of such a drum has $\sigma^{z}=-1$ (no bosons). A natural way
to construct bigger drums is to choose a subset of the sites that belong to
this boundary of the shielding region and then replace $\sigma^{z}=-1$ by
$\sigma^{z}=+1$ at these selected sites. This generates a new Fock state from
which, using the procedure of identifying a drum from a Fock state explained
in Sec. 2.1.1, one gets one of the following three possibilities: (i) a bigger
quantum drum with no inert up-spins (bosons), (ii) a partially active
structure that can be decomposed into a smaller quantum drum and a non-zero
number of inert up-spins (bosons), and (iii) a completely frozen structure
with all up-spins (bosons) being inert.

Figure 4: The left panel shows how adding up-spins (bosons) to the sites at
the boundary of the shielding region (indicated by thick blue lines and also
labelled from p1 to p12) of an elementary one-plaquette drum (marked as A)
leads to a variety of larger quantum drums with are grouped according to the
addition of $n$⃝ up-spins (bosons) where $n=1,2,3,4$. The right panel
illustrates the same concept for a more complicated quantum drum (marked as B)
where three up-spins (bosons) (indicated by black dots) are added to three
sites of the boundary of the shielding region (indicated by thick blue lines).
Filled red dots indicate up-spins (bosons). The green (pink) plaquettes in all
the quantum drums follow the same convention as used in Fig. 1.

Two examples of this recursive construction to generate bigger drums starting
from a smaller drum are shown in Fig. 4. We first start with a Fock state
consistent with an elementary one-plaquette drum in the left panel of Fig. 4,
marked as A. In this case, the boundary of the shielding region is a square
that consists of twelve sites, labelled as p1, $\cdots$, p12 in Fig. 4. New
Fock states, consistent with larger drums, can be created by adding
one/two/three or four up-spins (bosons) in this boundary region as indicated
by the groups labelled by $1$⃝, $2$⃝, $3$⃝ and $4$⃝ in Fig. 4, left panel.
Adding a single up-spin (boson) at p1 or p4 generates Fock states consistent
with a drum composed of two elementary plaquettes that share a vertex as shown
in Fig. 4, left panel, group labelled by $1$⃝. Adding two up-spins (bosons) on
the boundary of the shielding region in different ways leads leads to Fock
states consistent with three different drums as shown in Fig. 4, left panel,
group labelled by $2$⃝. E.g., adding up-spins (bosons) at p1 and p7 leads to a
Fock state consistent with a drum with three plaquettes that share vertices
along a single diagonal (leftmost drum shown in the group labelled by $2$⃝ in
left panel of Fig. 4), at p1 and p5 leads to a Fock state consistent with a
drum with four plaquettes that are connected by edges (middle drum shown in
the group labelled by $2$⃝ in left panel of Fig. 4), and at p4 and p7 leads to
a Fock state consistent with a drum with three plaquettes that again share
vertices, but not along a single diagonal (rightmost drum shown in the group
labelled by $2$⃝ in left panel of Fig. 4)). Adding three up-spins (bosons) at,
e.g., p1, p4 and p7, leads to a Fock state consistent with a quantum drum with
four plaquettes that are connected by vertices as shown in the group labelled
by $3$⃝ in left panel of Fig. 4. Finally, adding four up-spins (bosons) at p1,
p4, p7, and p10 leads to a Fock state consistent with a quantum drum with five
plaquettes connected by vertices (Fig. 4, left panel, group labelled by $4$⃝).
To illustrate possibility (ii), we can add four up-spins (bosons) at p1, p3,
p7 and p11 (Fig. 4, left panel) which leads to a Fock state consistent with a
single-plaquette quantum drum containing sites p6, p7 and p8 while the other
up-spins (bosons) become inert. To illustrate possibility (iii), we can add a
single up-spin (boson) at p5 (Fig. 4, left panel) to generate a Fock state
that has only inert up-spins (bosons).

This recursive procedure can be carried forth for the bigger quantum drums to
produce more complicated quantum drums. An example is shown in panel marked as
B (Fig. 4, right panel) where three up-spins (bosons), indicated by filled
black dots, are placed on the boundary of the shielding region of a quantum
drum, previously produced by adding two up-spins (bosons) at the boundary of
the shielding region of the elementary single-plaquette drum, which leads to a
bigger quantum drum with ten elementary plaquettes that are connected by
edges. In principle, this recursive procedure can be used to generate and
enumerate all possible quantum drums until a given stage of the recursion
starting from the most elementary one-plaquette drum, but we leave this for a
possible future investigation.

####  2.1.4 Wires, junctions of wires, other quasi-1D and 2D drums

Figure 5: The quantum drums marked from A-D can be viewed as examples of
different kinds of junctions of wires. A and B show examples of junctions of
two wires while C (D) shows an example of a junction of three (four) wires.
The green (pink) plaquettes in all the quantum drums follow the same
convention as used in Fig. 1. The filled red dots indicate up-spins in all the
panels and represent just one of the many possible Fock states of the
corresponding drum.

As is already evident from the examples that we have constructed so far,
quantum drums come in several shapes and sizes, from being composed of a
single elementary plaquette (Fig. 1) to a finite number of plaquettes (Fig. 1,
Fig. 2, Fig. 4). One can even construct quantum drums with an arbitrarily
large number of plaquettes in the thermodynamic limit. These varieties of
drums can be quasi-1D or 2D in nature. We dub the simplest quasi-1D drum as a
wire. A wire is composed of $N_{p}$ plaquettes that share vertices along a
single diagonal and resemble straight wires (see Fig. 4, left panel for three
such drums with $N_{p}=1,2,3$). Such a wire can be constructed with any
$N_{p}\geq 1$ that leads to a quasi-1D structure for $N_{p}\gg 1$.
Interestingly, one can create other quantum drums that resemble different
kinds of junctions of such wires. Examples of such quantum drums are shown in
Fig. 5. In the top panel, the quantum drums marked by A and B can be viewed as
two different junctions of two wires, while in the bottom panel, the quantum
drum marked by C (D) can be viewed as a junction of three (four) quantum
wires. Wires can be used to build still more intricate quasi-1D as well as 2D
drums (see Sec. 3.2 for details). The fragment sizes for large quasi-1D (2D)
drums scale as $\alpha^{l}\times\alpha^{l}$
($\beta^{l^{2}}\times\beta^{l^{2}}$) where $\alpha>1$ ($\beta>1$) as $l\gg 1$
where $l$ represents the linear dimension of the drum and $\alpha$ ($\beta$)
depend on the nature of the quantum drum under consideration. Each such
quantum drum can be viewed as an interesting example of an interacting
quasi-1D/2D model with a constrained Hilbert space that also satisfies an
internal subsystem symmetry of simultaneous conservation of magnetizations
along each column and each row of the drum, where the columns/rows are defined
with respect to the $L_{x}\times L_{y}$ lattice in which the drum is embedded,
when only the sites that belong to the drum are considered.

###  2.2 Exact diagonalization of small lattices and deciphering spectrum
using drums

The constrained nature of the Hilbert space reduces the number of allowed Fock
states from $2^{L^{2}}$ to $\kappa^{L^{2}}$ where $\kappa\approx 1.503\cdots$
is the hard square entropy constant [74] for a square lattice with $L\gg 1$.
This growth of the Hilbert space dimension with $L$ is, nonetheless, still too
large to perform ED for the full spectrum for even moderately large values of
$L$. However, analysing the numerical results for small $L\times L$ lattices
is already instructive.

Figure 6: (Left panel) Histogram of the energy eigenvalues for a $5\times 5$
lattice with $5$ up-spins and OBC. The vertical dotted blue lines indicate the
allowed eigenvalues from Eq. 8. (Right panel) The allowed quantum drums in
this system where the up-spins (bosons) are indicated by filled red dots and
the plaquettes that belong to quantum drums are shaded. The green (pink)
plaquettes in all the quantum drums follow the same convention as used in Fig.
1. The inert up-spins (bosons) are indicated by a blue circle around the
filled red dot. Each quantum drum is consistent with more than one Fock state
with only one representative Fock state shown here. The different eigenstates
can be viewed as modes of these quantum drums.

Let us first consider a $5\times 5$ lattice and focus on the total
magnetization sector with $5$ up-spins (bosons). This gives a Hilbert space
dimension of $10741$ from direct enumeration taking the hard-core constraints
in account. Plotting the histogram of the energy eigenvalues obtained from
full ED reveals that the eigenvalues are clustered around only a few special
values (up to machine precision) (Fig. 6, left panel) unlike what is expected
of a generic interacting system with a similar Hilbert space dimension.
Furthermore, while an explicit construction shows that there are $4559$ inert
Fock states that are trivially annihilated by $H$ (Eq. 1) in this
magnetization sector, ED reveals that there are a total of $5525$ zero modes
(with zero eigenvalue within machine precision) implying the presence of $966$
non-trivial zero modes. ED also shows the presence of $1580$ eigenmodes with
eigenvalue $+1$ ($-1$) and $196$ eigenmodes with eigenvalue $+2$ ($-2$). Such
non-zero integer eigenvalues are unexpected in generic interacting models
which have highly irrational eigenvalues that cannot be expressed in any
simple closed form.

These and other features of the full ED data can be completely understood in
terms of quantum drums (Fig. 6, right panel). The $10741\times 10741$ matrix
for $H$ in the computational basis gets fragmented into $4559$ $(1\times 1)$,
$1552$ $(2\times 2)$, $434$ $(3\times 3)$, $324$ $(4\times 4)$, $32$ $(5\times
5)$, $32$ $(6\times 6)$, $16$ $(7\times 7)$ and $2$ $(8\times 8)$ Hilbert
space fragments. The $1\times 1$ fragments simply correspond to the inert Fock
states that are annihilated by all local terms of $H$ (Eq. 1) (and are denoted
collectively by panel marked E in Fig. 6, right panel). All the other
fragments can be viewed as being generated from a collection of appropriate
quantum drums and any remaining inert up-spins (bosons) that are not part of
any drum. Given the lattice dimensions and the number of up-spins (bosons),
only certain drums are allowed with specific degeneracies set by the lattice.
All $2\times 2$ fragments are generated by a quantum drum with two up-spins
(bosons) on an elementary plaquette and the rest of the $3$ up-spins (bosons)
being inert (these are represented collectively by panel marked D in Fig. 6,
right panel). Given the lattice dimension, there are $1552$ distinct ways of
choosing the drum plaquette and the $3$ inert up-spins (bosons) which explains
the notation D[1552] used in Eq 8. The $3\times 3$ fragments are generated by
$128$ B3-type quantum drums and $306$ C-type quantum drums, where the former
drum is composed of $4$ up-spins (bosons) while the latter drum is composed of
$3$ up-spins (bosons) while the rest of the up-spins are inert ((these two
cases are represented collectively by panels marked B3 and C in Fig. 6, right
panel). It is useful to stress here that while the B3-type and C-type drums
are evidently different from each other, they have identical spectra (Eq. 8).
The $4\times 4$ fragments are generated by $8$ A5-type, $16$ A6-type, $8$
A7-type, $96$ B1-type and $196$ B4-type quantum drums. While A5, A6, and
A7-type drums contain $5$ up-spins (bosons), B1 and B4-type drums contain $4$
up-spins (bosons) (these are represented collectively by panels marked A5, A6,
A7, B1, and B4 in Fig. 6, right panel). A5, A7 and B1-type quantum drums have
identical spectra even though these three types of drums are distinct from
each other. The $5\times 5$ fragments are generated by $12$ A8-type and $20$
B2-type quantum drums (these are represented collectively by panels marked A8
and B2 in Fig. 6, right panel). The $6\times 6$ fragments are generated by
$16$ A3-type, $12$ A4-type and $4$ A9-type quantum drums (these are
represented collectively by panels marked A3, A4, and A9 in Fig. 6, right
panel). Lastly, the $7\times 7$ ($8\times 8$) fragments are generated by $16$
($2$) A2 (A1)-type quantum drums (these are represented collectively by panels
marked A1 and A2 in Fig. 6, right panel). The eigenspectra of all the
fragments that arise from these quantum drums, barring A2-type drums, can be
expressed in closed form and show a variety of eigenvalues including zero
modes, non-zero integer modes and irrational modes (Eq. 8). The extra non-
trivial zero modes and their degeneracies can also be understood as zero modes
of quantum drums like A1, A2, A3, A4, A6, A8, B2, B3, B4 and C (Eq. 8).

$\displaystyle\mathrm{A1}[2]$ $\displaystyle\rightarrow$
$\displaystyle(\pm\sqrt{4+\sqrt{10}},\pm\sqrt{2},\pm\sqrt{4-\sqrt{10}},0,0)$
$\displaystyle\mathrm{A2}[16]$ $\displaystyle\rightarrow$ $\displaystyle(\pm
2.47367\cdots,\pm 1.25235\cdots,\pm 0.559107\cdots,0)$
$\displaystyle\mathrm{A3}[16]$ $\displaystyle\rightarrow$
$\displaystyle(\pm\sqrt{3+\sqrt{3}},\pm\sqrt{3-\sqrt{3}},0,0)$
$\displaystyle\mathrm{A4}[12]$ $\displaystyle\rightarrow$
$\displaystyle(\pm\sqrt{5},\pm 1,0,0)$
$\displaystyle\mathrm{A5}[8],\mathrm{A7}[8],\mathrm{B1}[96]$
$\displaystyle\rightarrow$
$\displaystyle\left(\pm\frac{1}{2}(1\pm\sqrt{5})\right)$
$\displaystyle\mathrm{A6}[16]$ $\displaystyle\rightarrow$
$\displaystyle(\pm\sqrt{3},0,0)$ $\displaystyle\mathrm{A8}[12]$
$\displaystyle\rightarrow$ $\displaystyle(\pm\sqrt{3},\pm 1,0)$
$\displaystyle\mathrm{A9}[4]$ $\displaystyle\rightarrow$
$\displaystyle(\pm(1\pm\sqrt{2}),\pm 1)$ $\displaystyle\mathrm{B2}[20]$
$\displaystyle\rightarrow$
$\displaystyle\left(\pm\sqrt{\frac{1}{2}(5\pm\sqrt{17})},0\right)$
$\displaystyle\mathrm{B3}[128],\mathrm{C}[306]$ $\displaystyle\rightarrow$
$\displaystyle(\pm\sqrt{2},0)$ $\displaystyle\mathrm{B4}[196]$
$\displaystyle\rightarrow$ $\displaystyle(\pm 2,0,0)$
$\displaystyle\mathrm{D}[1552]$ $\displaystyle\rightarrow$ $\displaystyle\pm
1,\mbox{~{}~{}~{}~{}~{}}\mathrm{E}[4559]\rightarrow 0$ (8)

The degeneracies associated with the different quantum drums indicated inside
$[]$ for each case in Eq. 8 arise from the number of distinct ways in which
such drums can be placed on the $5\times 5$ lattice with OBC. For example, A1
has a degeneracy of two because there are two diagonals along which such a
drum may be placed. Similarly, A2 has a degeneracy of sixteen since there are
sixteen distinct ways to place a “L” composed of four connected plaquettes on
this lattice. The other degeneracies given in Eq. 8 can be computed similarly.

###  2.3 Eigenstates with integer energies from packing of one-plaquette drums

Eigenstates composed of only elementary one-plaquette quantum drums and inert
spins already generate non-trivial zero modes and non-zero integer
eigenvalues. These can be viewed as the 2D generalization of bubble
eigenstates discussed in a 1D model of Hilbert space fragmentation [64].
Hilbert space fragments with $n_{0}$ such independent one-plaquette drums have
a dimensionality of $2^{n_{0}}$ since each such elementary quantum drum is
consistent with two configurations on the plaquette. An extensive number of
such elementary quantum drums are needed to form finite energy-density
eigenstates of $H$ with a macroscopic number of up-spins (bosons) (Fig. 7).
The closest packing of these elementary quantum drums such that the boundaries
of their shielding regions do not overlap is shown in Fig. 7 which yields the
maximum possible value of $n_{0}$ $=$ $L^{2}/9$ for a $L\times L$ square
lattice when $L\gg 1$ thus fixing the corresponding fragment’s dimension to be
equal to

$\displaystyle(2^{1/9})^{L^{2}}\approx(1.08006\cdots)^{L^{2}}.$ (9)

The corresponding matrix can be immediately diagonalized by noting that the
form of $H$ projected to any $n_{0}\neq 0$ fragment produced solely by
elementary one-plaquette quantum drums equals

$\displaystyle H_{\mathrm{eff}}=\sum_{i=1}^{n_{0}}\tau_{i}^{x}$ (10)

where $i$ denotes the center of an elementary drum plaquette, and
$\tau^{x}_{i}$ locally flips an arrangement of $(+1,-1,+1,-1)$ to
$(-1,+1,-1,+1)$ and vice-versa on that drum in the computational basis. This
“non-interacting” $H_{\mathrm{eff}}$ only leads to integer eigenvalues for any
$n_{0}$. If $n$ ($n_{0}-n$) of the elementary quantum drums are associated
with an eigenvalue $\tau_{i}^{x}=+1(-1)$, the resulting eigenstate has energy
$E=2n-n_{0}$. Clearly, there are $\binom{n_{0}}{n}$ distinct eigenstates that
have the same energy $E=2n-n_{0}$. Assuming that both $n_{0},n\gg 1$, the
degeneracy $\Omega(n)$ of such eigenstates is bounded below by

$\displaystyle\Omega(n)>2^{n_{0}}\sqrt{\frac{2}{\pi
n_{0}}}\exp\left(2n_{0}\left(x-\frac{1}{2}\right)^{2}\right)$ (11)

where $n_{0}=L^{2}/9$ for the largest such fragment (Fig. 7) and $x=n/n_{0}$.
This bound immediately shows that the number of such integer eigenstates is
exponentially large in the system size for integer energies that range from
$E=0$ to $|E|\sim O(L)$ (while the maximum value of the integer energy
$|E|=L^{2}/9$ when $L\gg 1$ for a $L\times L$ square lattice with OBC). These
high-energy eigenstates satisfy a strictly area law scaling of entanglement
entropy with the entanglement entropy of an arbitrary bipartition,
$S_{\mathrm{bp}}=bL$, where $b$ can range from $0$ to $\ln(2)/3$ (examples of
two such bipartition cuts which give the extreme values of $b$ are shown as
thick lines in Fig. 7), depending on the nature of the bipartition.

Figure 7: Close packing of elementary one-plaquette quantum drums shown, where
the drum plaquettes are shaded, alongwith the boundaries of the shielding
regions for two such drums (shown as thick blue lines). An initial Fock state
which is consistent with this arrangement of quantum drums and where up-spins
(bosons) are indicated by filled red dots is also shown.

Any Fock state consistent with $n_{0}$ independent one-plaquette drums (e.g.,
one such Fock state is shown in Fig. 7 where the red filled dots represent up-
spins (bosons)) shows persistent oscillations with a time-period $T=\pi$ under
unitary evolution under $H$ for a class of local operators. This can be
directly related to the non-interacting nature of $H_{\mathrm{eff}}$ in Eq. 10
which leads to the following emergent dynamical symmetry [75]:

$\displaystyle\left[P_{\mathrm{eff}}HP_{\mathrm{eff}},\frac{\tau_{i}^{y}+i\tau_{i}^{z}}{2}\right]=\omega\left(\frac{\tau_{i}^{y}+i\tau_{i}^{z}}{2}\right)$
(12)

where $P_{\mathrm{eff}}$ is a projection operator to the Fock space with
$n_{0}$ one-plaquette drums and $\omega=2$ given the form of
$H_{\mathrm{eff}}$ in Eq. 10. Thus, for any such initial Fock state, any local
operator with a finite overlap with any of the
$\left(\frac{\tau_{i}^{y}+i\tau_{i}^{z}}{2}\right)$ operators will show
persistent oscillations with a time period $T=2\pi/\omega=\pi$.

##  3 Strong Hilbert space fragmentation

In this section, we show strong Hilbert space fragmentation for this
kinematically constrained 2D model (Eq. 2) defined on a $L_{x}\times L_{y}$
rectangular lattice with OBC as $L_{x},L_{y}\gg 1$. We first discuss numerical
evidence from ED in Sec. 3.1. In Sec. 3.2, we introduce a decomposition of
quantum drums in terms of parallel wires which reveals how to construct
quantum drums with the largest fragment dimension, given a fixed number of up-
spins (bosons). The construction of some 2D quantum drums with large fragment
dimensions and the lack of thermalization for typical initial product states
with an extensive number of up-spins (bosons) will be discussed in Sec. 3.3.

###  3.1 Numerical evidence from exact diagonalization

Figure 8: (Left panel) The total Hilbert space dimension $\mathcal{D}$ as a
function of the number of up-spins, $N_{b}$, for rectangular lattices
$L_{x}\times L_{y}$ of various dimensions. The thin dotted vertical line is at
$n=N_{b}/(L_{x}L_{y})=1/4$. (Right panel) The behavior of the ratio of the
dimension of the largest Hilbert space fragment and the total Hilbert space
dimension for the magnetization sector for densities $n=1/4$ (blue), $n=3/10$
(green) and $n=1/3$ (red) shown for rectangular lattices with dimension
$L_{x}\times L_{y}$ and OBC as a function of the system size $N=L_{x}L_{y}$.

One procedure to distinguish between weak and strong fragmentation [50, 51]
involves monitering the ratio of the largest Hilbert space fragment (denoted
by $\mathrm{max}[\mathcal{D}_{f,n}]$) to the total Hilbert space dimension
(denoted by $\mathcal{D}_{n}$) in a sector with a fixed density (denoted by
$n$) of up-spins (bosons) for different system sizes. We stress here that only
the global symmetry of total magnetization conservation and its associated
density is relevant for this analysis since internal symmetries like
reflections etc can always be removed by adding suitable diagonal terms to $H$
in the computational basis that do not connect the different Hilbert space
fragments. If the ratio $\mathrm{max}[\mathcal{D}_{f,n}]$/$\mathcal{D}_{n}$
behaves as $\exp(-\gamma N)$ with $\gamma>0$ as the number of sites in the
system, $N$, diverges, it implies strong fragmentation; in contrast, if it
approaches $1$ as $N\gg 1$, it implies weak fragmentation.

Using exact enumeration techniques, we calculate the Hilbert space dimension,
$\mathcal{D}$, for a fixed number of up-spins (bosons), $N_{b}$, for a variety
of rectangular lattices of dimension $L_{x}\times L_{y}$ with OBC (see Fig. 8,
left panel) which shows that $\mathcal{D}$ is maximized when
$n=N_{b}/(L_{x}L_{y})=1/4$. We then focus on this particular density of up-
spins (bosons) $n=1/4$ as well as two other densities $n=3/10$ and $n=1/3$ to
show the scaling of $\mathrm{max}[\mathcal{D}_{f,n}]/\mathcal{D}_{n}$ for
fixed $n$ as a function of $N=L_{x}L_{y}$ in Fig. 8, right panel using data
from exact enumeration. The data for these limited system sizes already
clearly indicate that $\mathrm{max}[\mathcal{D}_{f,n}]$/$\mathcal{D}_{n}$
$\sim$ $\exp(-\gamma N)$ with $\gamma$ depending on the density of up-spins
(bosons), $n$, and thus points towards strong Hilbert space fragmentation in
this 2D model.

###  3.2 Wire decomposition of quantum drums

Figure 9: Examples of wire decomposition of quantum drums shown here. The
drums labeled A1 and A2 can be constructed from two parallel wires both with
$N_{p}=1$, the drums labeled B1 and B2 can be constructed from two parallel
wires with $N_{p}=3$ and $N_{p}=2$ while the drums labeled C1, C2, and C3 can
be constructed from three parallel wires, where two of them have $N_{p}=1$ and
one has $N_{p}=4$. The wires in all the panels are indicated by double dashed
lines and also labeled by w1, w2, w3. The red filled dots in all panels
represent up-spins (bosons). The perpendicular wires that can be generated
from the Fock state shown for each drum are indicated by bold lines in blue
and orange. Additionally, the plaquettes marked by open blue (orange) circles
at their centers in B1 and C1 represent the locations of one-plaquette
excitations that generate additional Fock states that cannot be represented by
excitations of alternate parallel wires in these two cases. In the drum C2, a
smaller drum made of $10$ edge-sharing plaquettes (indicated by thick blue
region enclosing it) fluctuates simultaneously, with the wire w3. The green
(pink) plaquettes in the drums shown in all panels follow the same convention
as used in Fig. 1.

As already introduced earlier, wires represent the basic quantum drums that
can be generated for any given number of plaquettes, $N_{p}$, by arranging the
plaquettes in a vertex-sharing pattern along any one of the two diagonal
directions of the parent $L_{x}\times L_{y}$ lattice. A reference Fock state
of the wire can be taken to be all the $N_{p}+1$ up-spins (bosons) to be
arranged along the length of the drum. The shielding region around a wire
consists of all plaquettes that share either an edge or a vertex with any of
the $N_{p}$ plaquettes that belong to the drum. As we will show here, a
reference Fock state for more complicated drums can be constructed from two or
more parallel wires with differing lengths in general. Each of these wires can
be initialized in its reference state where the wires are placed close enough
such that they cannot fluctuate independently. This, in turn, indicates that
the boundaries of their shielding regions cross. We refer the reader to Sec.
4.2 (Sec. 5) where the spectrum for wires with small (large) $N_{p}$ shall be
discussed. For now, it is sufficient to note that the number of Fock states
generated by a wire with $N_{p}$ plaquettes equals $F_{N_{p}+2}$ (a Fibonacci
number) where $F_{0}=0$, $F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for $n>1$ (see
Sec. 5.2 for the derivation).

In this section, we will show that

  1. 1.

Drums composed of only vertex-sharing plaquettes can be generated by parallel
wires that can all fluctuate simultaneously being at a distance of $2\sqrt{2}$
in lattice units from each other. All the Fock states of such drums can be
generated from the fluctuations of these parallel wires and possibly, other
sets of parallel wires in the same direction or perpendicular to the direction
of the original set of wires.

  2. 2.

Drums composed of only edge-sharing plaquettes can be generated by parallel
wires that cannot fluctuate simultaneously being at a distance of
$(3/2)\sqrt{2}$ in lattice units from each other, but only do so if alternate
wires are kept in their reference states. The Fock states of such drums can be
generated from the fluctuations of the alternate parallel wires and possibly,
other sets of alternate parallel wires in the same direction or perpendicular
to the original set of wires. However, not all simultaneous fluctuations of
such consecutive wires are disallowed and these can be represented as
additional excitations of elementary plaquettes that are separated by $3$
lattice units along $x$ or $y$ such that these plaquettes can fluctuate
simultaneously.

  3. 3.

Drums with both edge-sharing as well as vertex-sharing plaquettes can be built
from parallel wires such that while all the wires cannot fluctuate
simultaneously, some consecutive wires can do so if the other wires are kept
in their reference state.

We start with the simplest case of two $N_{p}=1$ wires in their reference
state that are placed parallel to each other. If such wires fluctuated
independently, then the two independent quantum drums would have produced a
total of $F_{3}\times F_{3}=4$ Fock states. There are two distinct ways of
placing these wires with respect to each other such that they do not fluctuate
independently and no inert up-spins (bosons) are created. These are shown as
the quantum drums marked by A1 and A2 in Fig. 9. While the drum A1 generates a
Hilbert space fragment with $3$ Fock states, which is less than the fragment
size from two independently fluctuating $N_{p}=1$ wires, the drum A2 generates
one with $5$ Fock states, which is more than the fragment size from two
independently fluctuating $N_{p}=1$ wires. In the drum indicated by A1 (Fig.
9), the wire w1 (w2) (indicated by double dotted lines in Fig. 9) can
fluctuate to generate both its Fock states only if w2 (w1) is held fixed in
its reference state. Thus, the two wires w1 and w2 cannot fluctuate
simultaneously in A1 and produce $2F_{3}-1=3$ states. On the other hand, in
the drum A2 (Fig. 9), both the wires w1 and w2 (indicated by double dotted
lines in Fig. 9) can fluctuate simultaneously without producing a Fock state
that violates the hard-core constraints. Additionally, performing a ring-
exchange from the reference state on both the plaquettes that represent w1 and
w2 generates the reference state for another wire with $N_{p}=3$ that is
perpendicular to w1 and w2 (shown as a blue line in the drum marked A2 in Fig.
9). The Fock state obtained from a ring-exchange on the middle plaquette from
the reference state of this $N_{p}=3$ wire cannot be represented by combining
any of the Fock states generated from the w1 and w2 wires and accounts for the
total $F_{3}^{2}+1=5$ Fock states for the drum A2. In the case of A1 [A2], the
minimum distance between the parallel wires w1 and w2 equals $(3/2)\sqrt{2}$
[$2\sqrt{2}$] in lattice units (Fig. 9).

The drums labeled B1 and B2 in Fig. 9 represent more complicated cases that
arise when two parallel wires in their reference states, one with $N_{p}=3$
and another with $N_{p}=2$, are brought close to each other such that the
minimum distance between the wires equal $(3/2)\sqrt{2}$ and $2\sqrt{2}$
respectively. If these two wires fluctuated independently, these would have
generated $F_{5}\times F_{4}=15$ Fock states. However, the drum B1 generates a
fragment with $11$ Fock states while the drum B2 generates a fragment with
$18$ Fock states. In the drum B1 (Fig. 9), the wire w1 (w2), shown by double
dotted lines in Fig. 9, can fluctuate to generate all its Fock states only if
the other wire w2 (w1) is held fixed in its reference state. Such wire
fluctuations lead to $F_{5}+F_{4}-1=7$ states. Two cases where fluctuations of
a perpendicular wire (indicated by the top blue line and the bottom orange
line respectively in the drum marked B1 in Fig. 9) when the other wire
parallel to it at a distance $(3/2)\sqrt{2}$ (indicated by the bottom blue
line and the top orange line respectively in Fig. 9) is kept fixed in its
reference states generates an additional $2$ Fock states. The remaining $2$
Fock states in the fragment are generated by two separate cases of ring-
exchanges on two plaquettes together [indicated by the plaquettes with an open
circle of the same color (blue and orange) at their centers] that are
separated by $3$ lattice units along $x$/$y$ as shown in Fig. 9 (panel marked
B1). On the other hand, in the drum B2 (Fig. 9), both the wires w1 and w2
(indicated by double dotted lines in Fig. 9) can fluctuate simultaneously to
generate all their Fock states without violating the hard-core constraints.
Fluctuations of w1 and w2 in the drum B2 cannot, however, generate any Fock
state with two up-spins (bosons) along the diagonal parallel to w1, w2 on any
of the two plaquettes that are not part of w1 and w2. The extra
$18-F_{5}\times F_{4}=3$ Fock states are generated from ring-exchange moves in
any one of these two plaquettes starting with Fock states obtained from the
fluctuations of w1 and w2 that can be represented as the reference state of a
$N_{p}=3$ wire perpendicular to both w1 and w2 and containing one of these two
plaquettes (shown by a blue and an orange line perpendicular to w1, w2 in Fig.
9).

Finally, we consider a case where three parallel wires in their reference
states, w1 with $N_{p}=1$, w2 with $N_{p}=4$, and w3 with $N_{p}=1$, are
brought close to each other to generate three different drums labeled C1, C2,
and C3 in Fig. 9. While independent fluctuations of these three wires generate
$F_{3}\times F_{6}\times F_{3}=32$ Fock states, the fragment generated by the
drum C1 contains $24$ Fock states, by the drum C2 contains $28$ Fock states,
and by the drum C3 contains $42$ Fock states, respectively. In the drum C1
(Fig. 9), the wire w1 (w3) can only access all its Fock states if w2 is held
fixed in its reference state (with these wires indicated by double dotted
lines in the drum C1 in Fig. 9). Similarly, the wire w2 can only access all
its Fock states if both w1 and w3 are fixed to their reference states in C1.
This generates a total $F_{3}^{2}+F_{6}-1=11$ Fock states. An additional $5$
Fock states of drum C1 are generated by similar wire fluctuations of parallel
wires separated by $(3/2)\sqrt{2}$ but perpendicular to w1, w2, w3 (indicated
by blue lines in drum C1 in Fig. 9). Finally, the remaining $8$ Fock states in
C1 are generated by simultaneous ring-exchanges on two/three of the four
corner plaquettes (marked by blue circles at the centres of the corresponding
plaquettes in C3 in Fig. 9) that are separated from each other/from a corner
plaquette by $3$ lattice units in the x/y direction. On the other hand, in the
drum C3 (Fig. 9), all the wires, w1, w2 and w3 (indicated by double dotted
lines in Fig. 9), can fluctuate simultaneously without violating the hard-core
constraints. Furthermore, fluctuations in w1, w2 and w3 generates a new open
channel of fluctuations in the form of a wire with $N_{p}=5$ plaquettes in the
direction perpendicular to these wires (indicated by a blue line in C3 in Fig.
9) which generates an additional $10$ Fock states besides the $F_{3}\times
F_{6}\times F_{3}=32$ Fock states generated from w1, w2, w3. The drum marked
as C2 in Fig. 9 represents an interesting intermediate case between C1 and C3
where the parallel wires w1 and w2 (w2 and w3) are at a distance
$(3/2)\sqrt{2}$ ($2\sqrt{2}$) from each other. The smaller drum containing
wires w1, w2 and composed of $10$ edge-sharing plaquettes (marked by the blue
region in the drum C2 in Fig. 9) can fluctuate simultaneously with the wire
w3. This leads to a total of $13\times 2=26$ Fock states (see Fig. 16 for the
$13$ Fock states of the smaller drum made by the wires w1 and w2). The
additional states are generated from an extra open channel for fluctuations
along a $N_{p}=3$ wire perpendicular to w1, w2, w3 containing the w3 plaquette
as its right-most plaquette as indicated by the blue line perpendicular to w1,
w2, w3 in the drum C2 in Fig. 9. This results in $2$ Fock states where a ring-
move is performed on the reference state of this $N_{p}=3$ wire using the
plaquette excluded from both the smaller drum composed of $10$ edge-sharing
plaquettes and w3.

Figure 10:  The drums A1 to C3 shown in Fig. 9 can be constructed using the
overlap of the shielding regions of parallel wires as shown. The wires are
indicated by doble dashed lines in each panel. The red filled dots in all
panels represent up-spins (bosons). The boundaries of the shielding region of
each wire is shown using thick lines of different colors and the plaquettes
formed by the overlap of the shielding regions are indicated by “o”. The green
(pink) plaquettes in the drums shown in all panels follow the same convention
as used in Fig. 1.

These examples demonstrate the general principle that given $n$ parallel
wires, with unequal lengths in general, it is optimal to place the wires such
that all the parallel wires can fluctuate simultaneously and that these
fluctuations additionally generate the maximum number of longest-possible
wires perpendicular to the original wires as extra open channels of
fluctuations to maximize the fragment size generated by the resulting drum.
Both these conditions are satisfied by appropriate drums composed of only
vertex-sharing plaquettes as shown in Fig. 9 (panels marked A2, B2 and C3).

The fact that drums with only vertex-sharing plaquettes are generated when the
parallel wires can simultaneously fluctuate can be most clearly seen by using
the overlap of the shielding regions of different parallel wires (see Fig. 10)
to construct the drums shown in Fig. 9. For the drums A1, A2, B1, B2 and C3,
it is sufficient to consider the overlap of the shielding regions of the
parallel wires shown in Fig. 9 as can be seen from Fig. 10, where the
overlapping plaquettes of the shielding regions have been denoted by “o” in
all panels of Fig. 10. The drums C1 and C2 present more interesting cases
where this construction only identifies a subset of plaquettes that belong to
the corresponding drum (Fig. 10). However, starting from the reference state
of the original parallel wires, it is easy to perform ring-exchange moves on a
subset of the plaquettes that belong to these wires to create a Fock state
that can be viewed as parallel wires in their reference state, but in the
perpendicular direction to the original wires (Fig. 10). The overlap of the
shielding regions of these new wires gives the remaining plaquettes that are
part of the quantum drum for both C1 and C2 (Fig. 10). Given that all the
parallel wires can fluctuate simultaneously but not independently, the overlap
of the shielding regions only generate vertex-sharing plaquettes with the
parallel wires being separated by a distance of $2\sqrt{2}$ in terms of
lattice units. Finally, stacking two parallel wires even closer such that
these are at a distance of $\sqrt{2}$ in terms of lattice units freezes both
the wires in their reference states. Hence, the last category of wire
arrangements, where any two parallel wires are only separated by $\sqrt{2}$,
are not relevant for constructing quantum drums.

Figure 11: Two parallel wires w1 and w2 (see left panel) and two other
parallel wires w${}_{1}^{{}^{\prime}}$ and w${}_{2}^{{}^{\prime}}$ (see right
panel) indicated by double dotted lines, both in their reference state and
both with $N_{p}=4$, seperated by a distance of $3\sqrt{2}$ in lattice units.
The wires w1, w2 are perpendicular to the wires w${}_{1}^{{}^{\prime}}$,
w${}_{2}^{{}^{\prime}}$. In both panels, two up-spins (bosons) are located on
two common sites between the boundaries of the shielding regions, shown using
thick lines, of the wires. The red filled circles in both panels indicate up-
spins. The green plaquettes in the drums shown in both panels follow the same
convention as used in Fig. 1, with the right panel showing all the plaquettes
that belong to this drum.

For completeness, we note that if two parallel wires are separated by a
distance of $3\sqrt{2}$, but no more, these wires can still be coupled to each
other to make a larger drum by placing up-spins (bosons) in a subset of the
common sites between the boundaries of the shielding regions of both the
wires. An example of this is illustrated in Fig. 11 where two parallel wires
w1 and w2, both with $N_{p}=4$ and in their reference state, are placed at a
distance of $3\sqrt{2}$ from each other (Fig. 11, left panel). Two up-spins
(bosons) are placed on two of the common sites of the boundaries of the
shielding regions of both the wires (the boundaries of the shielding regions
are shown using thick green (blue) lines for w1 (w2) in Fig. 11, left panel).
The wires w1 and w2 can fluctuate simultaneously to generate all their Fock
states in spite of the up-spins (bosons) on the boundaries of the shielding
regions. Performing appropriate ring-exchanges on this reference state (Fig.
11, left panel), it is easy to get a Fock state that can be viewed as two
parallel wires, w${}_{1}^{{}^{\prime}}$ and w${}_{2}^{{}^{\prime}}$, that are
both perpendicular to w1, w2 and again separated by $3\sqrt{2}$ with two up-
spins (bosons) on two of the common sites shared by the boundaries of the
shielding regions of w${}_{1}^{{}^{\prime}}$ and w${}_{2}^{{}^{\prime}}$. This
shows that four other plaquettes (apart from the ones shaded in Fig. 11, left
panel) are part of the bigger drum and that all the Fock states can be
generated from simultaneous fluctuations of either w1, w2 or
w${}_{1}^{{}^{\prime}}$, w${}_{2}^{{}^{\prime}}$, again generating a quantum
drum with only vertex-sharing plaquettes.

###  3.3 Absence of thermalization from typical product states

Figure 12: Three examples of 2D drums shown here. The green (pink) plaquettes
in all the quantum drums follow the same convention as used in Fig. 1. The
filled red dots indicate up-spins in all the panels and represent just one of
the many possible Fock states of the corresponding drum. The wire
decomposition of each drum is indicated by thin dotted lines in each of the
three panels. Panels A and B are composed of parallel wires that can fluctuate
simultaneously and hence contain only vertex-sharing plaquettes. The
checkerboard drum in panel B represents the maximum packing of such wires that
gives the density of up-spins (bosons) to be $1/4$. The close packed drum in
panel C represents the maximum packing of wires such that none of them are
inert that gives the density of up-spins (bosons) to be $1/3$. The plaquettes
where ring-exchanges can be simultaneously carried out are indicated by open
blue circles at their centres in panel C.

We now consider the fate of typical unentangled product states in the
computational basis for a large system, say a $L\times L$ lattice with OBC
where $L\gg 1$. Since $H$ (Eq. 1) is purely off-diagonal in the computational
basis, all such product states have an average energy of $\langle E\rangle=0$.
Given the $E$ to $-E$ symmetry of the many-body spectrum, ETH will imply that
such a product state with a macroscopic number of up-spins (bosons) should
thermalize to the infinite temperature ensemble (ITE) with the density of up-
spins (bosons) fixed by the initial condition as far as local operators are
concerned.

Given a product state with an extensive number of up-spins (bosons), it can be
categorized in one of the following five classes:

  1. 1.

The initial state is an inert Fock state which forms a $1\times 1$ fragment on
its own.

  2. 2.

The initial state is consistent with a finite number of finite-sized drums,
i.e., drums composed of a finite number of plaquettes, when $L\gg 1$.

  3. 3.

The initial state is consistent with an extensive number of finite-sized drums
when $L\gg 1$.

  4. 4.

The initial state is consistent with the presence of one or more quasi-1D
drums with a typical linear dimension of $O(L)$ as $L\gg 1$.

  5. 5.

The initial state is consistent with the presence of one or more 2D drums with
a typical linear dimension of $O(L)$ as $L\gg 1$.

Initial states in class $1$ do not thermalize since all spins are inert. In
class $2$, an extensive number of sites do not belong to any drum and the
corresponding spins are, therefore, inert and retain memory of their initial
condition. Initial states in class $3$ again contain an extensive number of
inert down-spins when all the boundary sites of the shielding regions of the
drums are considered together.

For initial states in class $4$, one simply needs to consider local operators
that have support from sites on opposite sides of a quasi-1D drum of linear
dimension $O(L)$. Such local operators evade thermalization since all the
sites that compose such a local operator cannot be part of the same quantum
drum and are, therefore, dynamically disconnected and retain memory of the
initial state.

Initial states in class $5$ are more subtle since almost all local operators
contain sites in the interior of a single 2D quantum drum. We take such a 2D
quantum drum to cover (almost) the entire $L\times L$ lattice without any loss
of generality. Following Sec. 3.2, a reference Fock state for such a 2D drum
can be composed by bringing $O(L)$ parallel wires together, typically of
length $O(L)$, in their reference states such that these wires cannot
fluctuate independently. Examples of some 2D drums are shown in Fig. 12 (three
panels) and in Fig. 13 (right panel).

Given that the number of Fock states accessible to a single wire of length $l$
equals $F_{l+2}$ (Eq. 41) where $F_{n}$ are the Fibonacci numbers, the
fragment size scales exponentially with increasing $l$ as $\varphi^{l}$ for
$l\gg 1$, where $\varphi$ $=$ $(1+\sqrt{5})/2$ $\approx$ $1.618\cdots$ is the
golden ratio. Thus, if a 2D drum can be decomposed into parallel wires that
can fluctuate simultaneously, then the corresponding number of Fock states
generated equals

$\displaystyle\varphi^{(L_{1}+L_{2}+L_{3}+L_{4}\cdots)}$ (13)

where $L_{1},L_{2},L_{3},L_{4}$ etc denote the lengths of these wires. Since a
2D quantum drum contains an extensive number of sites,
$L_{1}+L_{2}+L_{3}+\cdots=O(L^{2})$, Eq. 13 already shows that the
corresponding Hilbert space fragment grows exponentially with the system size.
Given a fixed density of up-spins (bosons), 2D quantum drums with the largest
fragment dimensions dominate statistically, provided that the other fragments
at the same density are exponentially smaller, and control the behavior of
typical initial states in class $5$.

We will now consider two important cases of such 2D drums. We first consider a
“checkerboard” drum (see panel B in Fig. 12) which represents the closest
packing of parallel wires (indicated by dotted lines in panel B of Fig. 12) in
their reference state such that all the wires can fluctuate simultaneously.
Comparing the representative Fock state of this drum shown in Fig. 12 (panel
B) to the inert state with the maximum density of up-spins (bosons) that
equals $n=1/2$ (Fig. 13, left panel), we see that the former may be obtained
from the latter by removing the up-spins (bosons) on alternate parallel wires
from the inert state. This fixes the density of up-spins (bosons) to be
$n=(1/2)\times(1/2)=1/4$ for the checkerboard drum. Since this drum only has
vertex-sharing plaquettes, following Sec. 3.2 all the Fock states of this drum
can be generated (in fact, overcounted) by considering simultaneous
fluctuations of wires along either of the diagonal directions of the square
lattice with mutual separation of $2\sqrt{2}$ and also their shifted
counterparts with a shift of $\sqrt{2}$ perpendicular to the direction of the
wires. This immediately establishes that

$\displaystyle(\varphi^{1/4})^{L^{2}}<\mathcal{N}_{\mathrm{HSD,ch}}<4(\varphi^{1/4})^{L^{2}}$
(14)

where $\mathcal{N}_{\mathrm{HSD,ch}}$ equals the number of Fock states for
this drum when $L\gg 1$. Thus, we get that

$\displaystyle\mathcal{N}_{\mathrm{HSD,ch}}\sim(\varphi^{1/4})^{L^{2}}\approx(1.12784\cdots)^{L^{2}}$
(15)

for the 2D checkerboard drum that accomodates the maximum density of
simultaneously fluctuating parallel wires, resulting in a density of up-spins
(bosons) that we denote as $n_{\mathrm{ch}}=1/4$ henceforth.

As shown in Sec. 3.2, the closest distance of approach between two parallel
wires in their reference state equals $(3/2)\sqrt{2}$ in lattice units such
that these do not become inert. Extending this to 2D, one gets a “close packed
drum” as shown in Fig. 12 (panel C) where the parallel wires are indicated by
dotted lines. Unlike the checkerboard drum, this 2D drum is composed of only
edge-sharing plaquettes and its interior has no unshaded plaquettes that do
not belong to the drum. The density of up-spins (bosons) for this close packed
drum equals $n=1/3$ which can be seen by comparing the reference Fock state
shown in panel C of Fig. 12 to the inert state with the maximum density of up-
spins (bosons), $n=1/2$, (Fig. 13, left panel). We see that the former may be
obtained from the latter by deleting the up-spins (bosons) on every two
parallel wires and keeping every third wire intact from the inert state in a
$1-0-0$ pattern. This gives one set of simultaneously flippable wires of the
close packed drum, implying that $n=(1/6+1/6)=1/3$. As discussed in Sec. 3.2,
all Fock states of such structures where the consecutive parallel wires are at
a distance of $(3/2)\sqrt{2}$ in lattice units can be generated from two types
of excitations: (a) simultaneous fluctuations of every alternate parallel wire
and (b) excitations of elementary plaquettes that are separated by $3$ lattice
units along $x$ or $y$ and thus simultaneously flippable. One set of such
parallel wires (indicated by parallel dotted lines) and elementary plaquettes
(indicated by open blue circles in the centers of the corresponding
plaquettes) are shown in Fig. 12 (panel C). The scaling of the number of Fock
states associated with the wire fluctuations can be simply calculated using
Eq. 13 and gives $(\varphi^{1/6})^{L^{2}}$. The number of states generated
from the simultaneously flippable elementary plaquettes can be calculated by
simply noting that the result is identical to the one discussed in Sec. 2.3
since the elementary plaquettes have the same spatial arrangement in Fig. 7 as
the marked plaquettes in Fig. 12 (panel C) thus giving the number of such
excitations as $(2^{1/9})^{L^{2}}$ (Eq. 9). Furthermore, all the Fock states
can be generated (in fact, overcounted) by considering all combinations of
such parallel wires as well as their perpendicular counterparts and the
simultaneously flippable elementary plaquettes and their lattice translations.
This gives that

$\displaystyle(\varphi^{1/6})^{L^{2}}<\mathcal{N}_{\mathrm{HSD,cp}}<4[(\varphi^{1/6})^{L^{2}}+(2^{1/9})^{L^{2}}]$
(16)

where $\mathcal{N}_{\mathrm{HSD,cp}}$ equals the fragment dimension for this
drum when $L\gg 1$. Importantly, since $2^{1/9}/\varphi^{1/6}$ $\approx$
$0.996819\cdots$, the above equation can be simplified to give

$\displaystyle\mathcal{N}_{\mathrm{HSD,cp}}\sim(\varphi^{1/6})^{L^{2}}\approx(1.08351\cdots)^{L^{2}}$
(17)

for the close packed drum that accomodates the maximum density of non-inert
parallel wires, resulting in a density of up-spins (bosons) that we denote as
$n_{\mathrm{cp}}=1/3$ henceforth.

A straightforward approach to maximize the number of states produced by wire
fluctuations, given a certain density of up-spins (bosons), is to use Eq. 13
and consider 2D drums where all the parallel wires that compose the drums can
fluctuate simultaneously which automatically lead to structures made of only
vertex-sharing plaquettes. Since such wires can only have a minimum separation
of $2\sqrt{2}$ in lattice units, this sets an upper bound on the density of
up-spins (bosons) to be $n_{\mathrm{ch}}=1/4$ (the density for the
checkerboard drum). For initial states in class $5$ where the density of up-
spins (bosons), $n\in(0,1/4]$, vertex-sharing drums dominate statistically. An
example of such a 2D drum at a lower density compared to $n=1/4$ is shown in
panel A of Fig. 12. Comparing the reference Fock state shown in panel A of
Fig. 12 to the inert state with the maximum density of up-spins (bosons),
$n_{b}=1/2$, (Fig. 13, left panel), we see that $n=(1/6)\times(1+1/3)=2/9$ for
this quantum drum. The parallel wires in panel A of Fig. 12 can be obtained by
deleting every two wires in the inert state and keeping every third wire
intact in a $1-0-0$ pattern while the up-spins that do not belong to any wire
in panel A of Fig. 12 can again be obtained by taking the same $1-0-0$ pattern
of wires and deleting every two up-spins (bosons) and keeping every third up-
spin (boson) in the surviving wires. Crucially, any 2D drum composed of
vertex-sharing plaquettes alone contain an extensive number of unshaded
plaquettes in its interior (Fig. 12, panels A and B) that do not belong to the
drum. By definition, the two-spin local correlators
$\langle(\sigma^{z}_{j_{x},j_{y}}+1)(\sigma^{z}_{j_{x}+1,j_{y}+1}+1)\rangle$
and
$\langle(\sigma^{z}_{j_{x},j_{y}+1}+1)(\sigma^{z}_{j_{x}+1,j_{y}}+1)\rangle$
stay pinned to zero for any such unshaded plaquette during the time evolution
induced by $H$. On the other hand, these correlators should relax to non-zero
positive values at late times for the plaquettes that belong to the 2D drum
assuming Krylov-restricted ETH simply because two up-spins (bosons) are
allowed on both diagonals of these plaquettes. However, since $H$ is local in
space and respects translational symmetry on the lattice, the ITE at any
finite density of up-spins (bosons) cannot give such an effect. Hence, typical
initial states in class $5$ with $n\leq 1/4$ evade ETH-predicted
thermalization due to the presence of such two-spin correlators. Furthermore,
while the checkerboard drum (Fig. 12, panel B) does not contain any inert spin
in its interior, vertex-sharing drums with a lower density of up-spins
(bosons) (e.g., Fig. 12, panel A) also contain an extensive number of sites in
their interior that do not belong to the drum and thus harbor inert spins.

The thermalization properties of typical initial states in class $5$ where the
density of up-spins (bosons), $n\in(1/4,1/2]$, cannot be determined by just
considering 2D drums composed of vertex-sharing plaquettes. We will first show
that such states always contain an extensive number of inert up-spins (bosons)
for $n\in(1/3,1/2]$. As already discussed, the close packing of parallel wires
such that neighboring wires can still fluctuate generates the close packed
drum (Fig. 12, panel C) and a corresponding density of $n_{\mathrm{c}p}=1/3$.
Adding an excess amount of up-spins (bosons) in the system such that
$n=(1/3)+\delta$, where $\delta\in(0,1/6)$, necessarily leads to an extensive
number of inert wires locked in their reference state, with the density of
such wires scaling as $\delta/6$. Thus, the number of inert up-spins (bosons)
in typical initial states in class $5$ should scale as $\delta L^{2}/6$ for
$n=(1/3)+\delta$ with $\delta\in(0,1/6)$. Thus, these typical states in class
$5$ also evade ETH-predicted thermalization.

We now come to the case of typical initial states in class $5$ where the
density of up-spins (bosons) equals $n\in(1/4,1/3)$ such that we can write
$n=(1/4)+\gamma$ with $\gamma\in(0,1/12)$. One can start with the reference
state of the checkerboard drum in Fig. 12 (panel B) and insert extra parallel
wires in their reference state such that the distance between parallel wires
equals $\sqrt{2}$ for a linear extent of $4\gamma L$ of the system while the
rest of the system still has parallel wires that can fluctuate simultaneously.
However, this immediately produces $O(4\gamma L^{2})$ inert up-spins (bosons)
in the system and thus such initial states evade thermalization. Since the
size of the spatial region that harbors wires that can simultaneously
fluctuate reduces from $L^{2}$ to $(1-4\gamma)L^{2}$, the fragment size scales
as $\left(\varphi^{\frac{1}{4}-\gamma}\right)^{L^{2}}$ using Eq. 15 in this
case.

Figure 13:  (Left panel) A section of the inert state with the maximum density
of up-spins (bosons) equal to $1/2$ shown with the filled red dots indicating
up-spins. This state can be viewed in terms of parallel wires (indicated by
dotted lines and shaded plaquettes of different colors) that are placed so
close that they cannot fluctuate out of their reference states. (Right panel)
Example of a 2D quantum drum with a density of up-spins (bosons) between $1/4$
and $1/3$. In this particular example, the density equals $5/18$. The green
(pink) plaquettes in this quantum drum follows the same convention as used in
Fig. 1. The filled red dots indicate up-spins and represents just one of the
many possible Fock states of the corresponding drum. The wire decomposition of
the drum is indicated by thin dotted lines. The blue circles indicate inert
down-spins that are not part of this quantum drum.

There are two distinct ways of producing reference states for 2D drums with
large fragment dimensions when $n\in(1/4,1/3)$ such that an extensive number
of inert up-spins (bosons) are not produced.
(i) One can have “phase-separated” drums, with the phase separation in one
direction, such that different macroscopic regions are composed of close
packed parallel wires in their reference state. One of these sets have
parallel wires at a distance $(3/2)\sqrt{2}$ (with a local density $n=1/3$)
for a total linear extent of $(12\gamma)L$. The other regions comprise of
parallel wires in their reference state at a distance $2\sqrt{2}$ (with a
lower local density of $n=1/4$) for the rest of the system to get the correct
up-spin (boson) density. We refer the reader to the drum marked as C2 in Fig.
9 for an illustration of this principle for a small drum where the parallel
wires w1 and w2 (w2 and w3) are at a distance $(3/2)\sqrt{2}$ ($2\sqrt{2}$)
with respect to each other. The leading scaling for the Hilbert space size of
such drums can simply be obtained by considering independent fluctuations of
these “checkerboard drum” and “close packed drum” regions. Using Eq. 15 and
Eq. 17, this immediately gives

$\displaystyle\mathcal{N}_{\mathrm{HSD,phase-
sep}}\sim[\varphi^{1/6}]^{12\gamma
L^{2}}[\varphi^{1/4}]^{(1-12\gamma)L^{2}}\sim\left(\varphi^{\frac{1}{4}-\gamma}\right)^{L^{2}}.$
(18)

(ii) One can start with the close packed drum reference state and delete up-
spins (bosons) in such a manner that another 2D drum with a lower density of
up-spins (bosons) is created. We refer the reader to the 2D drum shown in the
right panel of Fig. 13 where the reference Fock state of this particular drum
can be produced from the reference Fock state of the close packed drum by
deleting every third up-spin (boson) from alternate wires in their reference
state. This gives a reduced density of up-spins (bosons) to be $(1/6)(1+2/3)$
$=$ $5/18$ $\approx$ $0.2777\cdots$ as well as an extensive density of inert
down-spins from sites that are not part of the drum. However, the scaling of
the fragment size for such drums is upper-bounded by
$\left(\varphi^{\frac{1}{6}}\right)^{L^{2}}$.

Thus, the “phase-separated” drums in (i) dominate statistically for
$n\in(1/4,1/3)$ and typical initial states in class $5$, that do not contain
an extensive number of inert spins, nonetheless contain $O((1-12\gamma)L^{2})$
unshaded plaquettes in the interior of such phase-separated drums where two-
spin local correlators
$\langle(\sigma^{z}_{j_{x},j_{y}}+1)(\sigma^{z}_{j_{x}+1,j_{y}+1}+1)\rangle$
and
$\langle(\sigma^{z}_{j_{x},j_{y}+1}+1)(\sigma^{z}_{j_{x}+1,j_{y}}+1)\rangle$
stay pinned to zero for any such plaquette, thus evading ETH-predicted
thermalization. This completes our analysis for the lack of ETH-predicted
thermalization in typical initial states in class $5$ for all $n\neq 1/3$ by
considering 2D quantum drums that are statistically dominant from the wire
construction. We see that such initial states have either an extensive number
of inert spins or an extensive number of two-spin next-nearest neighbor
correlators that are pinned to zero or both.

The case of the close packed drum with density of up-spins (bosons) $n=1/3$
with a Hilbert space fragment whose dimension scales as
$\left(\varphi^{\frac{1}{6}}\right)^{L^{2}}$ seems more subtle since it
contains neither inert spins nor unshaded plaquettes in its bulk. In fact,
initial states that arise from a wire pattern composed of $(2/3)L^{2}$ of the
system in a checkerboard drum pattern, with a local density $n=1/4$, and the
rest of the system being fully inert with a local $n=1/2$ also yields the same
leading scaling of its fragment size as
$\left(\varphi^{\frac{1}{6}}\right)^{L^{2}}$. We leave the issue of
thermalization, or lack of it, or of an even more exotic feature like a
behavior intermediate to both weak and strong Hilbert space fragmentation for
the particular density of up-spins (bosons), $n=1/3$, as an interesting open
problem. For completeness, we note that ED results on small systems points
towards a strong Hilbert space fragmentation scenario even at $n=1/3$ (Fig. 8,
right panel).

##  4 Analytical study of small quantum drums

In this section, we shall study the spectrum of the simplest quantum drums of
the model analytically. We first discuss how the connection diagrams between
different Fock states of a drum, where the connections are generated by $H$,
can be represented by unidirectional trees in Sec. 4.1. Such tree structures
turn out to be particularly useful in finding the non-zero matrix elements of
$H$ for large drums (e.g., see Sec. 5). We will subsequently study the case of
a wire with $N_{p}$ plaquettes for small $N_{p}$ (Sec. 4.2), and then consider
some other examples of small quantum drums that can be viewed as building
blocks of the different kinds of wire junctions shown in Fig. 5 (Sec. 4.3).

###  4.1 Tree structure

It is convenient to represent the connection diagram between different Fock
states in the Hilbert space of a drum as nodes of a tree with the non-zero
matrix elements of $H$ (which equals $1$ due to the form of $H$ in Eq. 1)
between two such states denoted as a link between the corresponding nodes.
Such a tree can be build in the forward direction with the different levels
being denoted by integers starting from level-$0$ for the top level and being
incremented by $1$ for each of the levels below. The top level consists of a
single node that can be represented by any “reference state”. It is optimal to
choose a reference state such that the Fock state maximizes the number of
flippable plaquettes for the corresponding drum, but this is not a necessary
condition. A single application of $H$ on this reference state at level-$0$
generates all the nodes at level-$1$, where the corresponding Fock states have
exactly $1$ flipped plaquette with respect to the reference state with the
location of the flipped plaquette uniquely identifying the corresponding Fock
state. Links are then formed between nodes at level-$0$ and level-$1$.
Applying $H$ on each of the level-$1$ nodes generates level-$2$ nodes where
the corresponding Fock state has another flipped plaquette with respect to the
level-$1$ state with the locations of the two flipped plaquettes
characterizing the generated Fock state uniquely. The possibility that
different level-$i$ nodes may generate the same level-$(i+1)$ node first
arises at $i=1$. New links are then drawn between the appropriate nodes at
level-$1$ and level-$2$. This process is continued recursively at each
subsequent level-$i$ to go forward to level-$(i+1)$ during which the links
between appropriate level-$i$ and level-$(i+1)$ nodes are also generated.
Carrying out this forward construction of the tree, one also encounters “dead
nodes” which are Fock states at level-$i$ from which no other Fock states with
an extra flipped plaquette can be generated to go to the next level ($i+1$).
The forward construction of the tree terminates when the last level is reached
which is characterized by all its nodes being dead nodes. A plaquette, once
flipped, cannot be unflipped in the tree construction which makes the
construction unidirectional.

###  4.2 Wire

Figure 14: Left Panel:(a) Schematic representation of the basis states of the
simplest fragment with $N_{p}=1$ and (b) the corresponding tree between states
in the Hilbert space. Center Panel: Same as the left panel but corresponding
to $N_{p}=2$. Right Panel: Same as the left panel but corresponding to
$N_{p}=3$. In all plots, the red filled dots indicates sites with up-spins
(bosons). The green plaquettes in the drums follow the same convention as used
in Fig. 1.

The simplest quantum drum of the Hamiltonian given by Eq. 1 constitutes a
single plaquette ($N_{p}=1$) with two up-spins (bosons) as shown in the left
panel of Fig. 14. The Hilbert space of this drum constitutes two states; the
Hamiltonian in the space of these two states,
$|\psi_{a}\rangle\equiv|a\rangle$ for $a=1,2$, can be written as (Fig. 14)

$\displaystyle H_{1\ell}$ $\displaystyle=$ $\displaystyle\tau^{x},\quad
H_{1\ell}|\psi_{1(2)}\rangle=|\psi_{2(1)}\rangle$ (19)

where $\tau^{x}$ denotes Pauli matrix in the space of the states in the
Hilbert space. This yields integer eigenvalues $E=\pm 1$.

The next set of quantum drums that we discuss constitutes two elementary
square plaquettes ($N_{p}=2$) with three up-spins (bosons) in total as shown
in the central panel of Fig. 14. The Hilbert space consists of three states,
$|\phi_{a}\rangle\equiv|a\rangle$ for $a=1,2,3$, as shown in the panel. The
action of the Hamiltonian is summarized by the tree given in the central panel
of Fig. 14. In the space of these states, the Hamiltonian can be represented
as

$\displaystyle H_{2\ell}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&1&1\\\ 1&0&0\\\
1&0&0\end{array}\right)$ (23)

This yields eigenvalues $E=0,\pm\sqrt{2}$. Thus these fragments leads to
eigenenergies which can be represented by simple irrational numbers as well as
a zero mode.

Finally, we consider a wire with $N_{p}=3$ where the states have 4 up-spins
(bosons). The basis states spanning the $5$-dimensional Hilbert space of such
a fragment is charted in the right panel of Fig. 14 and the corresponding tree
is shown in the bottom panel of the figure. As can be read off from the tree,
in the space of these states, the $5\times 5$ Hamiltonian matrix can be
written as

$\displaystyle H_{3\ell}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}0&1&1&1&0\\\ 1&0&0&0&1\\\
1&0&0&0&1\\\ 1&0&0&0&0\\\ 0&1&1&0&0\end{array}\right)$ (29)

The corresponding eigenvalues are given by $E=0,\pm\sqrt{5\pm\sqrt{17}}$
leading to eigenvalues represented by non-trivial irrational numbers and a
zero mode. The spectrum of these wires for larger $N_{p}$ gets complicated and
these shall be studied in details numerically in Sec. 5.

###  4.3 Junction units

Figure 15: Left Panel:(a) Schematic representation of the basis states of the
quantum drum corresponding to a junction of two wires as shown in Fig. 5 (A)
with $N_{p}=3$ and (b) the corresponding tree between the four states in the
Hilbert space. Right Panel: Same as the left panel but corresponding to the
simplest quantum drum (with $N_{p}=4$) that can be treated as the junction
unit to generate another junction of two wires as shown in Fig. 5 (B). In all
plots, the red filled dots indicates sites with up-spins (bosons). The green
(pink) plaquettes in the drums follow the same convention as used in Fig. 1.

In this section, we shall study small quantum drums corresponding to the
simplest junction units that are building blocks of the different junctions of
wires shown in Fig. 5 (A, B, C, D) and calculate their spectra analytically.
Larger quantum drums that resemble a junction of two wires as shown in Fig. 5
(A) shall be studied numerically in Sec. 5.

We begin with the quantum drum corresponding to a junction of two wires as
shown in Fig. 5 (A) with $N_{p}=3$ elementary plaquettes. The basis states
corresponding to such a junction is shown in the left panel of Fig. 15. The
Hilbert space, as can be seen from this figure, is four dimensional. The tree
for the states in the Hilbert space is shown in the bottom of the left panel
Fig. 15. A straightforward analysis shows that $H$ admits a four-dimensional
matrix representation in the space of these states which can be written in
terms of outer product of two sets of Pauli and identity matrices
($\vec{\tau}_{a}$ and $I_{a}$ for $a=1,2$) as

$\displaystyle H_{1j}$ $\displaystyle=$
$\displaystyle\tau_{1}^{x}\otimes(I_{2}+\tau_{2}^{z})/2+I_{1}\otimes\tau_{1}^{x}$
(30)

The corresponding eigenvalues satisfy the characteristic equation
$E^{4}-3E^{2}+1=0$ and yields a solution $E=\pm(1\pm\sqrt{5})/2$. These
eigenvalues therefore yield the golden ratio for this particular drum.

Next, we consider the simplest quantum drum that can be treated as the
junction unit to generate another junction of two wires as shown in Fig. 5
(B). This unit corresponds to a drum with $N_{p}=4$ as shown in the right
panel of Fig. 15. The basis states spanning the six-dimensional Hilbert space
is shown in the top of the right panel of Fig. 15 while the tree for these
states is shown in the bottom of this figure. We find that the Hamiltonian has
a $6\times 6$ matrix representation given by

$\displaystyle H_{3\ell}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}0&1&1&0&0&0\\\ 1&0&0&1&1&0\\\
1&0&0&1&0&1\\\ 0&1&1&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&1&0&0&0\end{array}\right)$
(37)

The characteristics equation for the eigenvalues simplifies to
$E^{2}(E^{2}-1)(E^{2}-5)=0$ and yields eigenvalues $E=0$ (doubly degenerate)
and $E=\pm 1,\pm\sqrt{5}$.

Figure 16: Left Panel: Schematic representation of the basis states of the
simplest quantum drum that can be viewed as the junction unit of the junction
of three wires shown in Fig. 5 (C) with $N_{p}=10$ plaquettes. The red filled
circles indicates sites with up-spins (bosons). Right Panel: The corresponding
tree between the $13$ states in the Hilbert space. The green (pink) plaquettes
in the drum follows the same convention as used in Fig. 1.

Next, we consider the simplest quantum drum that can be viewed as the junction
unit of the junction of three wires shown in Fig. 5 (C). This drum has
$N_{p}=10$ plaquettes in it. The basis states spanning the Hilbert space is
shown in the left panel of Fig. 16. The Hilbert space is $13$ dimensional; the
tree for these states is shown in the right panel of Fig. 16. This allows a
$13\times 13$ dimensional matrix representation of $H$. We do not write this
matrix explicitly here since it can be easily constructed from the tree shown
in Fig. 16. This matrix needs to be numerically diagonalized and yields
eigenvalues $E=$ $\pm 3.18259\cdots$, $\pm 1.91182\cdots$,
$\pm(1\pm\sqrt{5})/2$, $\pm 1$, $0$, and $\pm 0.464856\cdots$.

Figure 17: Left Panel: Schematic representation of a few basis states of the
drum that can be viewed as the junction unit of a junction of four wires as
shown in Fig. 5 (D). The drum contains $N_{p}=16$ plaquettes. The red filled
circles indicates sites with up-spins (bosons). Right Panel: The corresponding
tree between the $24$ states in the Hilbert space. Each of the states can be
obtained from a state connected to it by application of $H$ on it. The green
(pink) plaquettes in the drum follows the same convention as used in Fig. 1.
See text for details

Finally, we study a quantum drum that is the junction unit of a junction of
four wires as shown in Fig. 5 (D). This junction unit is shown in Fig. 17 and
corresponds to $N_{p}=16$ with nine up-spins (bosons) and $24$ basis states. A
few representative such states are shown in the left panel of Fig. 17. Each of
these states belong to a different level in the tree starting from the state
$|1\rangle$ as shown in the right panel of Fig. 17; they can be obtained from
the state in the preceding level shown by application of $H$. The other states
can be analogously obtained following the tree; we do not show them explicitly
to avoid clutter. The tree shows that $H$ admits a $24\times 24$ matrix
representation. Remarkably, this matrix can be analytically diagonalized; its
eigenvalues satisfies the characteristics equation

$\displaystyle
E^{6}(E^{2}-2E-2)(E^{2}-2)(E^{4}-22E^{2}+80)(E^{4}-6E^{2}+6)^{2}(E^{2}+2E-2)$
$\displaystyle=$ $\displaystyle 0$ (38)

These leads to the $24$ eigenvalues given by $0$ (six fold degenerate),
$\pm\sqrt{3\pm\sqrt{3}}$ (each two fold degenerate),
$\pm\sqrt{11\pm\sqrt{41}}$, $\pm\sqrt{2}$, and $\pm 1\pm\sqrt{3}$.

##  5 Numerical study of large quantum drums

In this section, we will numerically calculate the spectrum of large quantum
drums with $N_{p}$ elementary plaquettes using the examples of a wire (Fig.
18, left and middle panels have $N_{p}=4$ and $5$, respectively) and a
particular junction of two equal length wires (Fig. 18, right panel with
$N_{p}=7$). We refer to this latter case as “junction of two wires”
henceforth. Using exact diagonalization (ED), we could calculate the spectrum
up to $N_{p}=22$ for the wire and $N_{p}=23$ for the junction of two wires. We
will show that the spectrum of a wire with $N_{p}$ plaquettes is identical to
the paradigmatic 1D PXP chain [12, 13] with $N_{p}$ sites on a chain with OBC.
This equivalence allows us to extract several features of the high-energy
spectrum of the wire from known results in the literature [12, 13]. However,
our numerical studies also reveal enhanced fidelity revivals from a period-$3$
initial state for $N_{p}=3n+1$, where $n$ is an integer, without the need of
adding any optimal perturbations to the Hamiltonian which was not pointed out
earlier in the literature. While the junction of two wires differs from the
wire by only a “surface term” when $N_{p}$ is large, the structure of the
Hilbert space is completely different and gives a different constrained model
compared to the 1D PXP chain. Thus, the presence or absence of a single
junction leads to interesting differences in the high-energy spectrum that
persist for large drums.

Figure 18: Two wires with $N_{p}=4$ (left panel) and $N_{p}=5$ (middle panel)
respectively and a junction of two wires with $N_{p}=7$ (right panel) are
illustrated here. These quantum drums have a discrete reflection symmetry with
the thick blue line in all the three panels indicating the corresponding axis
of reflection. Figure 19: Tree structure (bottom panel) for a wire with
$N_{p}=4$ shown here. The $i$th level contains Fock states with $i$ flipped
plaquettes (indicated by shaded plaquettes in the top panel) with respect to
the reference state defined in level-$0$. The nodes between two Fock states
imply that these are connected with a single application of $H$. The Fock
states enclosed by double circles represent dead ends of the tree. The
corresponding value of the pseudospin variable $(\pm 1,0)$ is also shown at
the center of each plaquette for each of the Fock states in the top panel.

###  5.1 Tree generating algorithm and equivalence of wire to 1D PXP chain

The concept of a unidirectional tree starting from a reference state has
already been introduced in Sec. 4.1. The reference state for a wire with
$N_{p}$ plaquettes can be taken to be the Fock state with all $N_{p}+1$ up-
spins to be along the wire length as previously done in Sec. 3.2. For the
junction of two wires with $N_{p}=2x+1$ plaquettes, we take the reference
state to be the one where $x+2$ [$x$] up-spins (bosons) are arranged along the
length of the wire of length $x+1$ [$x$] to the right [left] of the central
junction plaquette, including [excluding] the junction plaquette (e.g., see an
example of the reference state marked as $1$⃝ in the left panel of Fig. 20.).
For both the wire and the junction of two wires, computationally it is
convenient to adopt a one-to-one map from a spin configuration on the wire or
a junction of two wires to another defined on an open chain with $N_{p}$ sites
in 1D where each site of the chain can have a pseudospin variable
$\tau^{z}_{i}=\pm 1$ or $0$, where $i=1$ to $N_{p}$. The pseudospins on the
chain represent the plaquettes of the drum sequentially from left to right in
both the cases. For the wire, these variables take the value $+1$ ($-1$) for
elementary plaquettes that have two up-spins along (perpendicular to) the wire
direction and $0$ otherwise (Fig. 19, top panel). For the junction of two
wires, we follow the same convention and remove the ambiguity at the central
plaquette by associating it to the wire to the right of the junction plaquette
(Fig. 20, left panel). While a pseudospin with $0$ has multiple possibilities
associated with an elementary plaquette involving states with zero or one up-
spin, specifying the locations of the $\pm 1$ pseudospins also fixes the spin
state of the other plaquettes on the drum.

The tree generating algorithm then proceeds as follows. One starts with the
reference state which has $\tau_{i}^{z}$ equal to $111\cdots 1$ for the wire
and $11\cdots 1011\cdots 1$ for the junction of two wires where the $0$ in the
latter case represents the plaquette to the immediate left of the $1$-junction
plaquette. For the wire, the states at subsequent levels are generated by
flipping a $1$ to $-1$ and replacing the pseudospins at neighboring site(s) of
the flipped pseudospin by $0$ (Fig. 19). For the junction of two wires, the
rules are practically the same except at the $1$-junction plaquette denoted by
the site $i_{0}$ on the open chain. When $1$ is flipped to $-1$ at $i_{0}$,
while the pseudospin at $i_{0}+1$ is replaced by $0$ as usual, the pseudospin
at $i_{0}-1$ is replaced by $0$ if the pseudospin at $i_{0}-2$ equals $-1$,
else it is replaced by $+1$ (Fig. 20). Following this algorithm, we generate
the tree and the corresponding $H$ matrix for both the quantum drums being
discussed here.

Figure 20: Tree structure for a junction of two wires with $N_{p}=5$ shown
here. The $i$th level contains Fock states with $i$ flipped plaquettes
(indicated by shaded plaquettes) with respect to the reference state defined
in level-$0$. The nodes between two Fock states (where the links imply the
forward construction of the tree) imply that these are connected with a single
application of $H$. The Fock states enclosed by double blue circles represent
dead ends of the tree. The pseudospins on an open chain with $N_{p}=5$ sites
is also shown for each of the Fock states in the left panel where red filled
circles denote up-spins (bosons).

We now show that the wire with $N_{p}$ plaquettes has the same spectrum as
that of the 1D PXP chain with $N_{p}$ sites and OBC, whose Hamiltonian is
defined as follows:

$\displaystyle
H_{\mathrm{PXP}}=\sum_{i=2}^{N_{p}-1}P_{i-1}\mu_{i}^{x}P_{i+1}+\mu_{1}^{x}P_{2}+P_{N_{p}-1}\mu^{x}_{N_{p}}$
(39)

where $\mu^{\alpha}_{i}$ for $\alpha=x,y,z$ represents spin-$1/2$ Pauli
matrices at site $i$ of the open chain with $N_{p}$ sites, and
$P_{i}=(1-\mu_{i}^{z})/2$ is a local projection operator. The constrained
Hilbert space of the PXP chain is defined by the condition that no two nearest
neighbor sites $i,i+1$ can have $\mu^{z}_{i}=+1$ and $\mu^{z}_{i+1}=+1$
together. We now make the following correspondence between the pseudospins
$\tau^{z}_{i}$ for the wire and the spins $\mu_{i}^{z}$ for the PXP chain:

$\displaystyle\tau_{i}^{z}=-1$ $\displaystyle\Rightarrow$
$\displaystyle\mu_{i}^{z}=+1,$ $\displaystyle\tau_{i}^{z}=+1$
$\displaystyle\Rightarrow$ $\displaystyle\mu_{i}^{z}=-1_{\mathrm{f}},$
$\displaystyle\tau_{i}^{z}=0$ $\displaystyle\Rightarrow$
$\displaystyle\mu_{i}^{z}=-1_{\mathrm{uf}}$ (40)

where $\mu_{i}^{z}=-1_{\mathrm{f}}$ ($-1_{\mathrm{uf}}$) implies that flipping
$\mu_{i}^{z}$ from $-1$ to $+1$ is allowed (disallowed) due to the hard-core
constraints of the PXP model. In this language, the reference state of the
wire with $\tau^{z}_{i}=+1$ for all $i$ corresponds to the “Rydberg vacuum”
state of the PXP chain with no Rydberg excitations, i.e.,
$\mu_{i}^{z}=-1_{\mathrm{f}}$ for all $i$. The tree generating algorithm then
constructs a unidirectional tree starting from the reference state at
level-$0$ by flipping a $\tau^{z}_{i}=1$ to $\tau^{z}_{i}=-1$ and replacing
the pseudospins at neighboring site(s) of the flipped pseudospin by
$\tau_{i+1}^{z}=\tau_{i-1}^{z}=0$ for $i\neq 1,N_{p}$ and $\tau_{i+1}^{z}=0$
($\tau_{i-1}^{z}=0$) for $i=1$ ($i=N_{p}$) at each subsequent level of the
tree. The action of $H_{\mathrm{PXP}}$ in Fock space can also be represented
by the same tree structure as the wire using Eq. 40 since flipping any
$\mu_{i}$ from $-1$ to $+1$ starting from the Rydberg vacuum state
automatically makes the previously flippable nearest neighbor site(s) with
$\mu=-1$ unflippable due to the hard-core constraints of the PXP chain.

We note that this equivalence immediately breaks down for the junction of two
wires since flipping a pseudospin $\tau^{z}_{i_{0}}=+1$ to
$\tau^{z}_{i_{0}}=-1$ on the central junction plaquette, denoted by $i_{0}$,
starting from the reference state produces a flippable $\tau^{z}_{i_{0}-1}=+1$
to its immediate left (see Fock states marked by $1$ and $4$ in the left panel
of Fig. 20 for an example) which implies that the junction of two wires cannot
be represented by the same constrained Hilbert space as the PXP model by this
mapping.

###  5.2 Hilbert space dimension and level statistics

Let us calculate the Hilbert space dimension for both these drums for an
arbitrary $N_{p}$ which will justify their interpretation as effective
quasi-1D models since the dimensionality scales exponentially with $N_{p}$ as
$N_{p}\gg 1$ in both cases. Let us denote the number of possible Fock states
for a wire with $N_{p}$ plaquettes to be $\mathcal{N}_{w}(N_{p})$. All the
Fock states for such a wire can be built in either one of the following two
ways. Consider starting from the reference state (Fig. 19, level-$0$ state)
and building all possible Fock states using the first $N_{p}-1$ plaquettes
starting from the top. The number of generated states then equals
$\mathcal{N}_{w}(N_{p}-1)$ and it is easy to see that the last plaquette will
then either have the pseudospin to be $+1$ or $0$. The remaining states of the
wire with $N_{p}$ plaquettes can be generated by starting from the reference
state and fixing the pseudospin of the last plaquette to be $-1$ (i.e.,
flipping this last plaquette). The first $N_{p}-2$ plaquettes from the top can
then be used to generate the missing Fock states whose number equals
$\mathcal{N}_{w}(N_{p}-2)$. Thus, we get that

$\displaystyle\mathcal{N}_{w}(N_{p})$ $\displaystyle=$
$\displaystyle\mathcal{N}_{w}(N_{p}-1)+\mathcal{N}_{w}(N_{p}-2)$ (41)
$\displaystyle=$ $\displaystyle F_{N_{p}+2}$

By construction, $\mathcal{N}_{w}(1)=2$ and $\mathcal{N}_{w}(2)=3$ which
implies that $\mathcal{N}_{w}(N_{p})=F_{N_{p}+2}$ as written above, where
$F_{n}$ are the Fibonacci numbers defined by the recurrence relation
$F_{0}=0$, $F_{1}=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for $n>1$.

Similarly, for the junction of two wires with $N_{p}=2x+1$ plaquettes, all the
Fock states can again be built in one of the following two ways. Consider
starting from the reference state (Fig. 20, level-$0$ state) and building all
possible Fock states of the left wire with $x-1$ plaquettes starting from the
left-bottom plaquette and the right wire with $x+1$ plaquettes starting from
the right-bottom plaquette. The number of such states equal
$\mathcal{N}_{w}(x-1)\mathcal{N}_{w}(x+1)$ and the plaquette to the immediate
left of the central junction plaquette can have a pseudospin of either be $+1$
or $0$. To generate the remaining configurations, we start from the reference
state again and make the pseudospin of this particular plaquette to be $-1$ by
first flipping the central junction plaquette and then flipping the plaquette
to the immediate left of the junction. The number of Fock states generated
from the rest of the plaquettes then equals
$\mathcal{N}_{w}(x-1)\mathcal{N}_{w}(x-2)$, thus giving the relation

$\displaystyle\mathcal{N}_{j}(N_{p}=2x+1)=\mathcal{N}_{w}(x-1)\left[\mathcal{N}_{w}(x-2)+\mathcal{N}_{w}(x+1)\right]=F_{x+1}\left(F_{x}+F_{x+3}\right)$
(42)

where $\mathcal{N}_{j}(N_{p}=2x+1)$ refers to the number of Fock states in a
junction of two wires composed of $N_{p}=2x+1$ elementary plaquettes. Eq. 41
and Eq. 42 show that the number of allowed Fock states scale exponentially for
large $N_{p}$ for both the drums. Note that while Eq. 41 is identical to the
Hilbert space dimension of a 1D PXP chain with $N_{p}$ sites and OBC, as
should be the case from the equivalence of both models shown in Sec. 5.1, the
Hilbert space dimension of the junction of two wires (Eq. 42) cannot be
expressed as $F_{m}$ with an integer $m$ in general showing that the structure
of the constrained Hilbert space of this drum is different from that of the 1D
PXP chain.

We can then ask whether these large quantum drums satisfy a Krylov-restricted
version of the ETH, i.e., whether these quasi-1D models are non-integrable. We
check this using the method of level statistics that can be obtained directly
using the eigenspectrum from ED (e.g., see Ref. [76]). To calculate the level
statistics for large quantum drums, it is important to first project to a
sector where all the commuting global symmetries have been resolved. Since
both the wire and the junction of two wires (Fig. 18) are quasi-1D structures
with open boundaries, momentum is not a good quantum number. The total
magnetization in the computational basis, $S^{z}_{\mathrm{tot}}$, represents a
conserved quantity for these drums. However, all nodes of a tree (Fig. 19 and
Fig. 20) already have the same $S^{z}_{\mathrm{tot}}$ by construction. The
only remaining non-trivial global symmetry turns out to be a reflection
symmetry, denoted by $\mathcal{R}_{w}$ ($\mathcal{R}_{j}$) for the wire
(junction of two wires), which takes a Fock state $|\alpha\rangle$ to another
Fock state $|\beta\rangle=\mathcal{R}_{w/j}|\alpha\rangle$ with the axis of
reflection shown in Fig. 18 for both the drums. For a wire with even (odd)
number of plaquettes, the axis passes through a site (the diagonal of a
square) (Fig. 18, left and middle panels) whereas for a junction of two wires,
it passes through the central junction plaquette as shown in Fig. 18, right
panel. Since $\mathcal{R}_{w/j}^{2}|\alpha\rangle=|\alpha\rangle$ for any Fock
state, the basis states
$(|\alpha\rangle\pm\mathcal{R}_{w/j}|\alpha\rangle)/\sqrt{2}$ define states
with $\mathcal{R}_{w/j}=\pm 1$ respectively. If
$\mathcal{R}_{w/j}|\alpha\rangle=|\alpha\rangle$ for some Fock state(s), then
such Fock state(s) only contribute to the $\mathcal{R}_{w/j}=+1$ sector. This
happens in the case of the wire, where, the reference state provides one
example of such a Fock state. Thus, the number of basis states in
$\mathcal{R}_{w}=+1$ always exceeds the corresponding number for
$\mathcal{R}_{w}=-1$ for a wire whereas these two numbers are equal to each
other for a junction of two wires.

Figure 21: Level spacing ratio distribution $P(\tilde{r})$ versus $\tilde{r}$
for a wire with $N_{p}=22$ (left panel) and a junction of two wires with
$N_{p}=23$ (right panel), with both the data taken in the symmetry sector with
$\mathcal{R}_{w/j}=+1$. The histograms indicate the non-integrability of both
the quasi-1D models.

Restricting to the larger sector with $\mathcal{R}_{w/j}=+1$, we construct the
distribution of consecutive level spacing ratios $\tilde{r}$ (with support in
$[0,1]$) where $\tilde{r}$ is defined as follows:

$\displaystyle\tilde{r}=\mathrm{min}\left\\{r_{n},\frac{1}{r_{n}}\right\\}\leq
1,\mbox{~{}~{}~{}}r_{n}=\frac{s_{n}}{s_{n-1}},\mbox{~{}~{}~{}}s_{n}=E_{n+1}-E_{n}$
(43)

where $E_{n}$ represent the energies of the eigenvectors obtained from ED. For
a non-integrable model, one expects a Gaussian orthogonal ensemble (GOE)
distribution, while an integrable system leads to a Poisson distribution for
$P(\tilde{r})$ [77], where the two distributions have the following forms:

$\displaystyle
P_{\mathrm{GOE}}(\tilde{r})=\frac{27}{4}\frac{\tilde{r}+\tilde{r}^{2}}{(1+\tilde{r}+\tilde{r}^{2})^{5/2}};\mbox{~{}~{}~{}}P_{\mathrm{P}}(\tilde{r})=\frac{2}{(1+\tilde{r})^{2}}.$
(44)

The numerically generated data for $P(\tilde{r})$ versus $\tilde{r}$ is shown
for a wire with $N_{p}=22$ plaquettes and a junction of two wires with
$N_{p}=23$ plaquettes in Fig. 21. The data clearly indicates that
$P(\tilde{r})$ follows $P_{\mathrm{GOE}}(\tilde{r})$ much more closely than
$P_{\mathrm{P}}(\tilde{r})$ for these system sizes giving strong evidence for
the non-integrable nature of both these quasi-1D models.

###  5.3 Zero modes and index theorem

While both the wire and the junction of two wires have a symmetric
eigenspectrum of $H$ around $E=0$ as is expected for any quantum drum, the ED
data further reveals the presence of an ever-increasing number of exact zero
modes (up to machine precision) with increasing $N_{p}$ for the former case
and the absence of any zero mode for the latter case.

Figure 22: Scaling of the total number of zero modes versus $N_{p}$ for a
wire. For even (odd) values of $N_{p}$, the number of zero modes grow as
$\mu_{e}(\sqrt{\varphi})^{N_{p}}$ ($\mu_{o}(\sqrt{\varphi})^{N_{p}}$) where
$\mu_{e}\approx 0.75$ ($\mu_{o}\approx 0.35$) [indicated by dotted lines].

This striking difference between the two drums can be understood in terms of
the index theorem of Ref. [73]. Firstly, a chiral operator
$\mathcal{C}_{w/j}=\prod_{\Box_{j}}\sigma^{z}_{j_{x},j_{y}}$ (where the
subscript $w(j)$ refers to the wire (junction of two wires)) can be defined in
both cases, which involves one site $(j_{x},j_{y})$ per elementary plaquette
contained in the drum (these sites are indicated by crosses in red in all
panels of Fig. 18). This operator satisfies $\\{H,\mathcal{C}_{w/j}\\}=0$ for
the Hilbert space fragments generated by these drums, thus ensuring the
$E\rightarrow-E$ symmetry of the spectrum. Furthermore, as already discussed,
these two drums have a global reflection symmetry, $\mathcal{R}_{w/j}$, that
commutes with $H$ (Fig. 18). Importantly, while
$[\mathcal{R}_{w},\mathcal{C}_{w}]=0$, it turns out that
$[\mathcal{R}_{j},\mathcal{C}_{j}]\neq 0$ which means that the index theorem
of Ref. [73] applies to the wire but not to junction of two wires. This leads
to a macroscopically large number of protected zero modes in the former case
and also explains our numerical data (Fig. 22). The number of zero modes in
the wire show an interesting even-odd effect as a function of $N_{p}$ (Fig.
22) with the even values of $N_{p}$ showing a higher number of zero modes.
This even-odd effect stems from the fact that the axis that defines the
reflection symmetry, $\mathcal{R}_{w}$, passes through a single site shared by
two elementary plaquettes for even values of $N_{p}$; in contrast, it passes
through two sites along a diagonal of an elementary plaquette for odd values
of $N_{p}$ (Fig. 18, left and middle panels). The number of zero modes scale
as $\mu_{e/o}(\sqrt{\varphi})^{N_{p}}$ (with $\varphi$ $=$ $(1+\sqrt{5})/2$
being the golden ratio as defined before) where $\mu_{e}\approx 0.75$
($\mu_{o}\approx 0.35$) for even (odd) values of $N_{p}$ (see Fig. 22).
Identical scaling behavior was also observed for the number of zero modes in
the 1D PXP model [12, 13, 78].

It is useful to point out here that a different type of junction of two equal-
length wires (Fig. 5, panel B) instead of this junction being studied here
will again have an exponentially large number of exact zero modes. Similarly,
a junction of three equal-length wires (Fig. 5, panel C) as well as a junction
of four equal-length wires (Fig. 5, panel D) will also have a macroscopic
number of zero modes due to the index theorem of Ref. [73]. It is interesting
to see how the number of zero modes scales as a function of $N_{p}$ for these
quasi-1D cases as $N_{p}\gg 1$.

###  5.4 QMBS and related diagonastics

While the level statistics distribution of both the wire and the junction of
two wires (Fig. 21) is consistent with these quasi-1D models being non-
integrable for large $N_{p}$ and thus satisfying Krylov-restricted ETH, both
quantum drums also harbor QMBS that give rise to observable dynamical
signatures like periodic revivals from certain simple initial states.

Figure 23: Three Fock states shown for the wire with $N_{p}=10$. (Left panel)
Reference state denoted by $|r\rangle_{w}$ (Middle panel) A $|fu\rangle_{w}$
state created by flipping every alternate elementary plaquette in the
reference state (Right panel) A $|fuu\rangle_{w}$ state created by flipping
every third elementary plaquette of 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 is shown below; such a cut divides
the system into two equal halves with $N_{p}/2$ pseudospins each.

Let us consider three such Fock states for the wire as shown in Fig. 23. Fig.
23 (left panel) shows the reference state which we denote as $|r\rangle_{w}$,
Fig. 23 (middle panel) shows a Fock state obtained by flipping every alternate
elementary plaquette in the reference state which we denote as
$|fu\rangle_{w}$, and Fig. 23 (right panel) shows a Fock state obtained by
flipping every third elementary plaquette in the reference state which we
denote as $|fuu\rangle_{w}$. Similarly, two representative Fock states are
shown in Fig. 24 for the junction of two wires. Fig. 24 (left panel) shows the
reference state which we denote as $|r\rangle_{j}$ (note that there are two
such reference states possible for the junction of two wires) and Fig. 24
(right panel) shows a Fock state obtained by flipping every alternate
elementary plaquette in the reference state which we denote as
$|fu\rangle_{j}$. From the equivalence of the wire to the 1D PXP chain shown
in Sec. 5.1, it is clear that while local operators starting from the state
$|r\rangle_{w}$ will thermalize quickly, since the initial state maps to the
Rydberg vacuum state of the PXP chain, this will not be the case from the
initial states $|fu\rangle_{w}$ and $|fuu\rangle_{w}$ which map to the
period-$2$ $|\mathbb{Z}_{2}\rangle$ and the period-$3$
$|\mathbb{Z}_{3}\rangle$ Fock states of the PXP chain, respectively.

Figure 24: Two Fock states shown for the junction of two wires with $N_{p}=9$.
(Left panel) Reference state denoted by $|r\rangle_{j}$ (Right panel) A