# Magnetization of Antiferromagnetic Cactus Graph Model with Exact Dimer
Ground State

Pratyay Ghosh<EMAIL_ADDRESS>Institut für Theoretische Physik
und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat,
Universität Würzburg, Am Hubland Campus Süd, Würzburg 97074, Germany

###### Abstract

We introduce and explore the magnetization behavior in a quantum spin system
on a cactus graph, deemed the Cactus Graph Model (CGM), featuring an exact
dimer singlet ground state. We analyze the singlet-triplet gap, interactions
among excited triplets, and correlated hopping under an external magnetic
field using a strong coupling expansion. Employing an effective hard-core
boson representation, we unveil density wave magnetization plateaus and
supersolid phases, revealing the intricate interplay between interactions and
hopping dynamics. Furthermore, we conjecture that correlated hopping gives
rise to multi-triplet bound states, and potentially engendering to the
stabilization of additional low-lying magnetization plateaus.

##  I Introduction

In the field of condensed matter physics, the Cayley tree (CT) [1] and Bethe
lattice (BL) [2] are fundamental graph structures used to explore spin systems
and their complex statics and dynamics. Despite the shift in contemporary
graph theory towards realistic random graphs [3] and complex networks [4, 5,
6], the enduring significance of BL and CT lies in their precise tree-like
architecture devoid of cycles [7]. This characteristic positions BL and CT as
instructive models for conducting exact calculations [8], providing a simple
yet insightful comprehension of interactions and cooperative behaviors, and
capturing emergent behaviors, critical phenomena, and phase transitions in
spin systems [9, 10].

While exploring spin systems, the investigation of exactly solvable models
presents a captivating avenue [11, 12, 13, 14, 15]. A typical example is the
famous Shastry-Sutherland model [12], a frustrated spin system on a lattice of
orthogonal dimers, which exhibits an exact dimer singlet ground state. The
exact solvability of this model offers profound insights into exotic phases,
quantum entanglement, and novel phase transitions [15, 16, 17, 18, 19, 20, 21,
22]. Thus, the fusion of the BL’s analytically tractable nature with the
intricate details of exact dimer models can create a fertile foundation for
comprehending the diverse behaviors of quantum spin systems.

Figure 1: The cactus graph model (CGM): a graph structure constructed from a
Bethe lattice of degree $z=4$. The construction involves generating a graph of
quadrangles with shared corners. The depicted diagonals represent the dimer
bonds of the model. The singlets residing on the dimer bonds represent an
eigenstate of (1). Figure 2: (a) A speculative quantum phase diagram of (1).
The schematics illustrate three states: the exact dimer singlet state, the
plaquette singlet state, and the Néel phase. The purple ellipses represent
exact dimer singlets, and the orange quadrangles represents plaquette
singlets. Like the other exact dimer models [12, 13, 14], the transition out
of the dimer singlet phase is likely to be first order. The phase transition
from the plaquette singlet phase to the Néel phase may occur at a deconfined
quantum critical point [21] or they could be separated by an intermediate spin
liquid phase [22]. (b) For the CGM with classical spins, i.e. $S\to\infty$,
there exists an incommensurate spiral state for $J^{\prime}<J$, which is
different from the Néel state. This state is highly degenerate and
incommensurate except for some particular ratios of $J^{\prime}/J$ where
$\arccos(J^{\prime}/J)$ is a rational fraction of $\pi$. One such
configuration for $J^{\prime}=J/2$ is shown.

In this article, we start by constructing a quantum spin model on a cactus
graph (CG) that admits an exact dimer singlet eigenstate. We commence with a
BL of degree $z=4$ and follow the protocol in [14] to generate a CG comprising
quadrangles with shared corners. This involves decorating all edges of the BL
with new vertices, which are the physical sites hosting spin-$1/2$’s, and then
employing a Y-$\Delta$-type transformation (Fig. 1) to eliminate the original
vertices and establish pairwise connections between the spins. These newly
formed edges act as inter-dimer bonds. At this stage, the engineered graph
structure (the medial graph of the original BL) is a CG. Next, we connect one
diagonal of each quadrangle (indicated by the thick green lines), avoiding
site sharing between any two dimers. These diagonals constitute our dimer
bonds (depicted in Fig. 1). The model Hamiltonian is given by

$\hat{H}=\hat{H}_{0}+\hat{H^{\prime}}$ (1)

with

$\displaystyle\hat{H}_{0}$ $\displaystyle=$ $\displaystyle J\sum_{\langle
ij\rangle}\mathbf{\hat{S}}_{i}\cdot\mathbf{\hat{S}}_{j}$ (2)
$\displaystyle\hat{H^{\prime}}$ $\displaystyle=$ $\displaystyle
J^{\prime}\sum_{\langle
kl\rangle}\mathbf{\hat{S}}_{k}\cdot\mathbf{\hat{S}}_{l}.$ (3)

Here, $\mathbf{\hat{S}}_{i}$ denotes a spin operator acting on a spin-$1/2$
degree of freedom at site $i$. There are two separate summations $ij$, and
$kl$ over the thick green and black bonds, respectively (Fig. 1). We deem this
model as the cactus graph model (CGM). It can be analytically shown that for
antiferromagnetic spin exchange interactions ($J,J^{\prime}>0$), the model has
an exact ground state, i.e. a product state of singlets on the dimer-bonds,
for $J^{\prime}\leq J/2$ [see left most panel of Fig. 2 (a)]. The
demonstration is straightforward. The product state of singlets, i.e.

$|\psi\rangle=\otimes\prod_{\langle ij\rangle}|s\rangle_{ij},$ (4)

is the ground state of the first term in (1). Here, $|s\rangle_{ij}$ denotes a
singlet constituted of the spins at site $i$ and $j$. Now, the matrix element
of the second term in (1) nullifies for any adjacent pair of $J^{\prime}$
bonds due to the singlet’s odd parity concerning the reflection that
interchanges the two spins, leading to $\hat{H^{\prime}}|\psi\rangle=0$. This
makes the dimer state an exact eigenstate of (1) with energy
$e_{0}=-\frac{3}{8}JN$ ($N$ is the total number of spins). This state serves
as the ground state for when $J^{\prime}$ is sufficiently smaller than $J$ –
an analysis based on variational principle finds it to be $J^{\prime}\leq J/2$
(see Refs. [12, 13, 14] for further details). This singlet state can also
remain as the ground state for $J^{\prime}>J/2$, only it cannot be shown
analytically [16, 23, 24, 21, 25, 26, 13, 14].

As the system is bipartite for $J=0$, it accommodates a Néel-type state near
that limit [see Fig. 2 (a)]. Note that the classical phase diagram
($S\to\infty$) of CGM also features a spiral phase for $J^{\prime}<J$ akin to
SSM. The spiral state where the angle between the spin across the $J^{\prime}$
bonds are $\pi-\phi$ and the same between the spins across the $J$ bonds is
$2\phi$, where

$\phi=\arccos(J^{\prime}/J).$ (5)

For $S=1/2$ SSM, this classical highly degenerate incommensurate spiral phase
translates into a plaquette singlet phase separating the exact dimer singlet
phase and the Néel phase [12]. Such a plaquette singlet phase can also be
predicted for CGM due to the same reason [see Fig. 2 (b)], however, the
details of the state are slightly different from the one found in SSM [16, 24,
21, 25]. As the plaquette singlet and the Néel state each break distinct
lattice and spin-rotation symmetries, a second-order transition between them
is beyond the conventional Landau-Ginzburg paradigm, hinting at a deconfined
quantum critical point [27, 21]. Alternatively, there might exist a quantum
spin-liquid sandwiched between the two phases [28, 22].

In this article, we do not study the phase diagram of the model but focus on
another interesting avenue of a model with an exact dimer ground state with a
Zeeman term, $-B\sum_{i}S_{i}^{z}$, with the magnetic field, $B$, of the order
of the dimer singlet-triplet gap, i.e., $B\sim J$. After a perturbative
expansion in $J^{\prime}$, we perform a hard-core boson (HCB) projection onto
a two-dimensional basis, comprising the dimer singlet and the field-aligned
triplet. The effective HCB model exhibits an appealing system devoid of
single-particle dynamics and dictated by two-body interactions and correlated
hoppings [29, 30, 31]. As a result, the system can break translation symmetry
to form density waves, correlated hopping may lead to bound states [17, 32,
18, 33, 20, 31], and supersolid non-plateau states [32, 34, 22] where density
wave and superfluidity coexist.

##  II Singlet-triplet gap

In the exact dimer phase, i.e. $J^{\prime}\lesssim J/2$, the ground state of
(1) is a product state of spin singlets on all dimer bonds. Isolated dimer
eigenstates comprise a singlet state $|s\rangle$ and three triplet states
$|t_{0,\pm 1}\rangle$ (the subscript denotes the total $S^{z}$ quantum
number). The second term in (1), i.e. the inter-dimer interactions, is
incorporated perturbatively on the basis of dimer eigenstates [35]. The
singlet-triplet gap then reads:

$\Delta=J-B-\frac{J^{\prime 2}}{J}-\frac{J^{\prime 3}}{2J^{2}}-\frac{J^{\prime
4}}{8J^{3}}+\mathcal{O}\left(J^{\prime 5}\right).$ (6)

For exact dimer models, such as the SSM [12], Maple leaf model (MLM) [13], or
the current Cactus Graph model (CGM), the singlets’ parity and the matrix
elements of $J^{\prime}$ terms establish limitations on the hopping dynamics
of the triplets. Specifically, a triplet can move solely to two out of its
four adjacent dimers [see Fig. 3 (a)] as

$\hat{h}^{\prime}_{\alpha\beta}\ket{s}_{\alpha}\ket{t_{\pm 1,0}}_{\beta}=0$
(7)

where $\hat{h}^{\prime}_{\alpha,\beta}$ is the $J^{\prime}$ terms acting
between the dimers $\alpha$ and $\beta$. In the process, dictated by the
matrix elements

$\displaystyle\hat{h}^{\prime}_{\beta\gamma}\ket{t_{\pm
1}}_{\beta}\ket{s}_{\gamma}$ $\displaystyle=$
$\displaystyle\pm\frac{J^{\prime}}{2}\left(\ket{t_{\pm
1}}_{\beta}\ket{t_{0}}_{\gamma}-\ket{t_{0}}_{\beta}\ket{t_{\pm
1}}_{\gamma}\right)$ (8)
$\displaystyle\hat{h}^{\prime}_{\beta\gamma}\ket{t_{0}}_{\beta}\ket{s}_{\gamma}$
$\displaystyle=$
$\displaystyle\frac{J^{\prime}}{2}\left(\ket{t_{1}}_{\beta}\ket{t_{-1}}_{\gamma}-\ket{t_{-1}}_{\beta}\ket{t_{1}}_{\gamma}\right),$
(9)

it leaves another triplet behind [29, 31]. For MLM and SSM, the underlying
lattice of dimers contains cycles of girth three and four, respectively, and,
hence, single triplet hoppings start in fourth- and sixth-order in
perturbation, respectively [29, 30, 31]. For CGM, however, the graph of dimers
is a BL, i.e. cycle-free, and thus, the single triplet hopping is entirely
suppressed. The same is also true for a spin chain of linked tetrahedra [36].
As the single particle dynamics lowers the spin-gap, the $\Delta$ for CGM
provides an upper bound to the singlet-triplet gap for all exact dimer models.

##  III Interactions between excited triplets.

We, now, discuss the interactions between the excited $S^{z}=1$ triplets
($|t_{1}\rangle$) determined within the perturbation theory. We estimate the
energy needed to generate two triplet excitations from the ground state. The
computed energy contains two components: (i) the spin-gap energy required for
two triplet excitations ($2\Delta$) including the self-energy corrections in
(6), and (ii) the interaction energy between the excited triplets ($V_{n}$’s).
Fig. 3 (b) provides a visual depiction of the interactions occurring between
two triplets within third-order perturbation theory. The $V_{n}$’s are given
by,

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\frac{J^{\prime}}{2}+\frac{J^{\prime 2}}{2J}-\frac{J^{\prime
3}}{8J^{2}}+\mathcal{O}\left(J^{\prime 4}\right)$ (10) $\displaystyle V_{2}$
$\displaystyle=$ $\displaystyle\frac{J^{\prime
3}}{8J^{2}}+\mathcal{O}\left(J^{\prime 4}\right)$ (11) $\displaystyle V_{3}$
$\displaystyle=$ $\displaystyle\frac{J^{\prime 2}}{2J}+\frac{3J^{\prime
3}}{4J^{2}}+\mathcal{O}\left(J^{\prime 4}\right).$ (12)

Note that the $V_{n}$’s are closely related to the two triplet interactions
found in SSM. The interaction energies display strong directional dependence
and lack monotonicity. These characteristics are rooted in the geometry of the
model. The triplet-triplet interactions extend beyond third-order, yet higher-
order calculations are analytically intricate. Here advanced numerical
techniques [37, 20, 38] will be advantageous and are to be pursued in the
future.

Figure 3: (a) The singlet’s parity and the matrix elements of the $J^{\prime}$
terms [31] restrict the hopping of the triplets only to two of its four
adjacent dimers. The possible hopping directions are indicated using arrows on
the edges. (b) The interactions between pairs of triplets (reference triplet
in black) in the CGM. The interactions are determined within third-order
perturbation theory. (c) Correlated hopping in CGM, which is the movement of
one triplet (filled in purple or orange) assisted by the other (filled in
black).

##  IV Correlated Hopping

A notable feature inherent to the dimer model lies in its ability to
facilitate two-triplet hopping, termed correlated hopping, which involves the
motion of two triplets in a concerted manner [see Fig. 3 (c)]. Such a process
defies categorization as either single-triplet hopping or the creation-
annihilation of a triplet pair. This correlated hopping involving two triplets
starts from second-order perturbations. The hopping amplitudes (which are the
same for SSM) are given by

$\displaystyle\tau_{1}$ $\displaystyle=$ $\displaystyle\frac{J^{\prime
2}}{4J}+\frac{3J^{\prime 3}}{8J^{2}}+\mathcal{O}\left(J^{\prime 4}\right)$
(13) $\displaystyle\tau_{2}$ $\displaystyle=$ $\displaystyle\frac{J^{\prime
2}}{4J}+\frac{5J^{\prime 3}}{16J^{2}}+\mathcal{O}\left(J^{\prime 4}\right).$
(14)

Given the suppressed single triplet dynamics, correlated hopping assumes a
pivotal role. That has likewise been predicted in the SSM and MLM, where
correlated hopping leads to the formation of multi-triplet bound states [17,
32, 18, 33, 20, 31].

Figure 4: Magnetization phase diagram of the spin Hamiltonian achieved via
Matsubara-Matsuda transformation of the effective hard-core boson Hamiltonian
(15) (a) without and (b) with correlated hopping for a $1457$-site Cayley
tree. The diagrams highlight magnetization plateaus at specific fractional
values. Correlated hopping introduces additional supersolid (SS) phases that
influence the stability of density wave magnetization plateaus. The wavy-
patterned regions are not accessible within our current calculations. (c) The
density waves and a representative supersolid state (central $485$ sites are
shown). The color represents the local densities of the hard-core bosons. The
density wave phase exhibits a regular spatial pattern, indicating a stable
arrangement of particles. The supersolid phase demonstrates the coexistence of
density wave behavior and superfluidity.

##  V Effective hard-core boson model

We, now, proceed to establish a mapping of the initial Heisenberg model onto a
HCB model on the lattice of the dimers, i.e. a $z=4$ BL. For $B>0$, the
system’s lowest energy states with magnetization $m\geq 0$ consist of
$|s\rangle$ and $|t_{1}\rangle$. These states are projected as low-energy
physical degrees of freedom. By treating $|t_{1}\rangle$ as a HCB and
$|s\rangle$ as a vacancy, we execute a projection, yielding an effective HCB
model containing two-particle interactions and correlated hopping up to third
order in perturbation theory. In the perturbative treatment, $|t_{1}\rangle$
triplets are treated as physical magnetic entities, and other dimer states —
$|t_{0}\rangle$ and $|t_{-1}\rangle$ — serve as intermediate virtual states
within the perturbation theory. The resultant effective Hamiltonian can be
expressed in the general form:

$\displaystyle\hat{H}_{\text{eff}}=$
$\displaystyle\Delta\sum_{r}\hat{n}_{r}+\sum_{n}V_{n}\sum_{(r,r^{\prime})_{n}}\hat{n}_{r}\hat{n}_{r^{\prime}}$
(15)
$\displaystyle+\sum_{m}\tau_{m}\sum_{\left[r,r^{\prime},r^{\prime\prime}\right]_{m}}\hat{n}_{r}\left[\hat{b}_{r^{\prime}}^{\dagger}\hat{b}_{r^{\prime\prime}}+h.c.\right].$

The resulting Hamiltonian contains onsite potential, mutual repulsion, and
correlated hopping. The index $r$ runs over the sites of the effective BL of
dimer bonds, $(r,r^{\prime})_{n}$ sums span the pair of sites that interact
via $V_{n}$ [see Fig. 3 (b)], and
$\left[r,r^{\prime},r^{\prime\prime}\right]_{m}$ sum represents the correlated
hoppings with amplitude $\tau_{m}$, as shown in Fig. 3 (c). The operator
$\hat{b}_{r}^{\dagger}$ ($\hat{b}_{r}$) creates (annihilates) a
$|t_{1}\rangle$ particle at bond $r$, and
$\hat{n}_{r}=\hat{b}_{r}^{\dagger}\hat{b}_{r}$ represents the particle number
operator.

To study the Hamiltonian of HCBs (15), we take a classical approach, whereby
HCB operators are mapped to spin-$1/2$ operators using the Matsubara-Matsuda
transformation [39, 40, 41]:

$\hat{n}_{r}=\frac{1}{2}-\hat{T}_{r}^{z}\text{,\ \
}\hat{b}_{r}=\hat{T}_{r}^{+}\text{,\ \
}\hat{b}_{r}^{\dagger}=\hat{T}_{r}^{-}.$ (16)

Note that, this transformation is exact. Following this transformation, (15)’s
onsite potential and repulsion terms manifest as Ising terms and a
longitudinal magnetic field, while correlated hopping introduces $XX$-type
terms, along with additional three-spin interactions.

The diverse phases of the bosonic Hamiltonian, $\hat{H}_{\text{eff}}$, can be
read off from the spin configuration – an empty site is represented as a spin-
up state, while an occupied site corresponds to a spin-down state. Therefore,
the density $\hat{n}$ is replaced by the magnetization per site
$m=\frac{1}{2}-\hat{n}$. Note that, the occupation can only happen when the
spin gap $\Delta$ is closed by the physical magnetic field, $B$. Nonetheless,
the robust repulsive interactions prevent all singlets from transitioning to
triplets upon gap closure. Due to the interplay between the energy gains from
triplet excitations and the repulsive energy, for a given $B$, triplets would
adopt a distinct superstructure (density wave) to optimize energy. This
density wave, with constant magnetization, remains a stable configuration
within a finite range of $B$. It demonstrates long-range correlations and
displays a spin gap.

For other plausible phases of the bosonic Hamiltonian, such as the superfluids
and supersolids, $\langle\hat{b}_{r}\rangle=\langle\hat{T}_{r}^{+}\rangle$
serves as the order parameter. A superfluid state is a ferromagnetic state
with all the spins having a non-zero projection onto the $xy$-plane. A
supersolid phase, which is a superfluid state on the background of a density
wave, becomes a long-range order with a fraction of spins pointing down and
the rest having a non-zero projection onto the $xy$-plane.

In our approach, we adopt a classical treatment of the effective spin
Hamiltonian. This involves assuming that the spin operators,
$\mathbf{\hat{T}}_{r}\equiv\\{\hat{T}^{z},\hat{T}^{+},\hat{T}^{-}\\}$, can be
approximated as classical vectors of length $1/2$. The ground state is then
determined by arranging the spins in a configuration that minimizes the
classical Hamiltonian. This minimization process is conducted numerically
using the iterative minimization technique [42] on a CT comprising $1457$
sites. The approach involves initialization of the system with a random spin
configuration, then iteratively selecting one vector-spin, $\vec{T}_{r}$, at a
time and aligning it antiparallel to its local field, i.e. a directive given
by $\partial\hat{H}_{\text{eff}}/\partial\vec{T}_{r}$. This minimizes the
overall energy of the system. In the course of each iteration, we ensure that,
on average, each spin experiences a single update. We conduct extensive
$10^{7}$ sweeps for each set of parameters, and convergence is recognized when
the total energy difference per iteration falls below the threshold of
$10^{-4}$ in the unit $J$.

The resulting phase diagrams, both in the absence and incorporation of
correlated hopping, are depicted in Fig. 4 (a) and (b), respectively. Notably,
both phase diagrams exhibit magnetization plateaus at $m/m_{\text{sat}}\approx
0.34$, $0.46$, and $0.69$. Note that the finiteness of CT can induce severe
boundary effects on the plateau states. Therefore, in a BL, the magnetization
might differ from the ones presented by the CT. We, however, do not embark on
that study in detail – our purpose here is to demonstrate the essential
physics that the CGM has to offer.

When considering the phase diagram in the context of correlated hopping,
additional supersolid phases emerge in the vicinity of criticality. The
supersolid phases can be understood in the following manner: as an external
field destabilizes a plateau state, the system retains the density wave
characteristic of the preceding phase, while also accommodating extra bosons.
These bosons, when assisted by the bosonic density wave, achieve superfluid-
like mobility in the lattice through correlated hopping. As a consequence, a
supersolid phase, a combination of density wave and the superfluid [32, 34,
22]. Within this phase, the magnetization increases monotonically with
increasing field until the next plateau state is reached. We do not explore
the magnetization beyond the $0.69$ plateau, as from there one encounters a
dense population of HCBs, necessitating the consideration of three-particle
interactions to thoroughly explore higher magnetization plateaus. The various
bosonic density wave patterns and a representative supersolid structure are
depicted in Fig. 4 (c).

Figure 5: (a) Correlated hopping facilitates the motion of two excited
triplets (filled in red), leading to two-triplet bound states indicated in
light-yellow. (b) The dot-dashed line represents $\Delta$, while the dashed
line represents the energy associated with two noninteracting dimer triplets
($2\Delta$), and the solid line corresponds to the energy of a bound state of
two triplets, $\Delta E_{2P}$, (in the unit of $J$) as a function of
$J^{\prime}/J$. Example of (c) an extended multi-triplet bound state (d) a
crystal of two-triplet bound states.

##  VI Triplet bound states

While two excited triplets are distantly separated, each triplet remains
localized. However, when they are adjacent to each other, the phenomenon of
correlated hopping facilitates the coherent movement of the two triplets and
the system gains kinetic energy. When two triplets are placed on nearest-
neighbor dimers, their interaction, linear in $J^{\prime}$, is typically the
dominant factor. They can, however, utilize correlated hopping ($\propto
J^{\prime 2}$) to separate from each other. This separation reduces their
repulsive energy cost to $V_{2}$, which is $\propto J^{\prime 3}$. As a
result, they effectively form a two-particle bound state. See Fig. 5 for a
visual representation of the relative motion of two triplets. By diagonalizing
the two-particle hopping matrix

$2\Delta\times\mathbf{I}_{6}+\begin{bmatrix}V_{2}&0&\tau_{2}&0&0&0\\\
0&V_{2}&-\tau_{2}&0&0&0\\\ \tau_{2}&-\tau_{2}&V_{1}&\tau_{1}&0&0\\\
$0$&0&\tau_{1}&V_{1}&-\tau_{2}&\tau_{2}\\\ $0$&0&0&-\tau_{2}&V_{2}&0\\\
$0$&0&0&\tau_{2}&0&V_{2}\end{bmatrix},$ (17)

we evaluate the energies of a two-triplet excitation to identify a preferred
two-triplet bound state with energy

$\displaystyle\Delta E_{2P}=2\Delta$ $\displaystyle+$
$\displaystyle\frac{1}{2}\left[\left(V_{1}+V_{2}-\tau_{1}\right)\vphantom{\sqrt{\left(V_{1}+V_{2}-\tau_{1}\right)^{2}+8\tau_{2}^{2}}}\right.$
(18) $\displaystyle-$
$\displaystyle\left.\sqrt{\left(V_{1}+V_{2}-\tau_{1}\right)^{2}+8\tau_{2}^{2}}\right]$

that occupies the indicated light-yellow region in Fig. 5 (a). This bound
state primarily comprises a linear combination of two-triplet states that
interact via $V_{2}$ and has a kinetic energy gain of $-\frac{J^{\prime
3}}{8J^{2}}+O(J^{\prime 4})$ from the two-particle threshold. Note that, in
SSM [32, 18], the bound state is formed on a cycle of length four. Therefore,
the two-particle bound state in CGM is different from the one in SSM [32, 18]
due to the cycle-free nature of the BL of the dimers. As a consequence of the
two-particle bound states possessing lower energy than two distinct triplets,
the gap associated with these bound states decreases more rapidly compared to
the singlet-triplet gap when $J^{\prime}$ is increased [see Fig. 5 (b)].
Consequently, this leads to the precedence of bound states undergoing
condensation before isolated individual triplets for $0.75\lesssim
J^{\prime}/J\lesssim 0.79$.

In the case of SSM and MLM, it has been predicted that $S^{z}=2$ triplet bound
states cause the emergence of extra magnetization plateaus close to $m=0$
through the creation of bound state crystals [20, 43]. A similar phenomenon
can be anticipated for the CGM also. It must also be mentioned that the
formation of a bound state is not limited to two triplets, but the system can
produce extended bound states with large $S^{z}$. Fig. 5 (c) shows an example
of such extended bound states. In Fig. 5 (d), we predict a crystal of two-
triplet bound states on the BL, which is a supersolid state giving rise to
low-magnetization plateau states.

We expect that these bound states no longer persist as the magnetization
increases. More precisely, single-triplet excitations can gain mobility via
correlated hopping at high magnetizations, and they can amass higher kinetic
energy compared to bound states. This implies a transition of elementary
particles from bound states at low magnetizations to single-triplet
excitations at high magnetization.

##  VII Conclusions and Outlook

In this study, we have introduced and undertaken a comprehensive investigation
of the magnetization behavior of the Cactus Graph Model (CGM), a quantum spin
system defined on a cactus graph which admits an exact dimer singlet ground
state, akin to the celebrated Shastry-Sutherland model. By investigating the
singlet-triplet gaps, interactions between excited triplets, correlated
hopping, and with the help of an effective hard-core boson representation, we
shed light on the magnetization behaviors of CGM. We discover that owing to
the geometry of the system single-triplet excitations are dispersionless. We
unveil the emergence of density wave magnetization plateaus, and supersolid
phases in the presence of an external magnetic field. Moreover, we highlight
the key steps towards the formation of two- and potentially multi-particle
extended bound states arising from correlated hopping. This could potentially
lead to a further stabilization of density wave magnetization plateaus and the
emergence of additional plateaus at lower magnetizations.

Further exploration is required to fully understand the magnetization behavior
of CGM, especially in the vicinity of $m=0$. At present, our examination of
two-particle repulsions only extends to third-order expansions. Nevertheless,
extending this analysis to higher orders could potentially unveil more
extensive repulsion effects between particles, particularly at longer ranges.
One crucial investigation is centered around the crystallization phenomenon in
scenarios involving two-particle, or even multi-particle, bound states under
weaker magnetic fields. Additionally, the impact of three-body interactions at
elevated magnetization levels remains an unexplored domain, warranting
thorough investigation.

Acknowledgments. The work is supported by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through Project-ID 258499086-SFB 1170 and
the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in
Quantum Matter – ct.qmat Project-ID 390858490-EXC 2147.

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