# Tensor network study of the spin-$1/2$ Heisenberg anti-ferromagnet on the
Shuriken lattice

Philipp Schmoll  Dahlem Center for Complex Quantum Systems and Institut für
Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin,
Germany  Augustine Kshetrimayum  Dahlem Center for Complex Quantum Systems and
Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14,
14195 Berlin, Germany  Helmholtz-Zentrum Berlin für Materialien und Energie,
Hahn-Meitner-Platz 1, 14109 Berlin, Germany  Theory Division, Saha Institute
of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064, India  Jan Naumann
Dahlem Center for Complex Quantum Systems and Institut für Theoretische
Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany  Jens
Eisert  Dahlem Center for Complex Quantum Systems and Institut für
Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin,
Germany  Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-
Platz 1, 14109 Berlin, Germany  Yasir Iqbal<EMAIL_ADDRESS>Department of Physics and Quantum Centers in Diamond and Emerging Materials
(QuCenDiEM) group, Indian Institute of Technology Madras, Chennai 600036,
India

###### Abstract

We investigate the ground state of the spin $S=1/2$ Heisenberg anti-
ferromagnet on the Shuriken lattice, also in the presence of an external
magnetic field. To this end, we employ two-dimensional tensor network
techniques based on infinite projected entangled pair and simplex states
considering states with different sizes of the unit cells. We show that a
valence bond crystal with resonances over length six loops emerges as the
ground state (at any given finite bond dimension) yielding the lowest reported
estimate of the ground state energy $E_{0}/J=-0.4410\pm 0.0001$ for this
model, estimated in the thermodynamic limit. We also study the model in the
presence of an external magnetic field and find the emergence of $0$, $1/3$
and $2/3$ magnetization plateaus with states respecting translation and point
group symmetries that feature loop-four plaquette resonances instead.

##  I Introduction

Systems of anti-ferromagnetically interacting quantum spins decorated on
corner sharing arrangements of triangles continue to attract much interest as
promising platforms for realizing novel quantum phases [1]. Indeed, the
arrival of candidate quantum spin liquid materials based on the iconic Kagome
lattice such as the celebrated Herbertsmithite [2, 3, 4] and Kapellasite [5]
have provided an impetus to the field of frustrated magnetism. Their
intriguing properties have triggered a flurry of experimental and theoretical
studies which established the Kagome lattice as a fertile host for a myriad of
exotic states. The parameter space of its Heisenberg Hamiltonian in the
presence of long-range interactions is known to be host to quantum spin
liquids including chiral states, spin and lattice nematics, and valence bond
crystals. Recently, a class of materials based on a different corner sharing
arrangement of triangles — the so called _Shuriken lattice_ (also called
square-Kagome, Squagome, and squa-Kagome lattice.) — have come into limelight
as promising candidate _quantum spin liquid materials_ [6, 7]. No sign of
magnetic ordering down to 50 mK has been observed in the spin $S=1/2$ Cu2+
based materials KCu6AlBiO4(SO4)5Cl [8] and Na6Cu7BiO4(PO4)4[Cl,(OH)]3 [9]
despite them having large negative Curie-Weiss temperatures of
$-237\text{\,}\mathrm{K}$ and $-212\text{\,}\mathrm{K}$, respectively. This
reveals a scenario similar to Herbertsmithite for which dominant anti-
ferromagnetic interactions on this highly frustrated lattice prevent the onset
of magnetic order. Such studies can be traced back to early work hinting at
quantum materials featuring that lattice structure [10]. On the theoretical
front, previous investigations into the nature of the ground state of the
$S=1/2$ Heisenberg anti-ferromagnet have provided compelling evidence for a
magnetically disordered ground state while revealing a subtle competition
between different types of nonmagnetic ground states which remains debated
[11, 12, 13, 14, 15, 16, 17, 18, 19].

In this work, we employ instances of two-dimensional _tensor network_ (TN)
algorithms formulated directly in the thermodynamic limit towards resolving
the nature of the ground state. Tensor networks are quantum-information
inspired tools that use entanglement as a resource for studying strongly
correlated quantum many-body systems [20, 21, 22, 23]. They naturally build in
quantum correlations and are suited to capture non-local entanglement by
construction. They do not suffer from the sign problem plaguing quantum Monte
Carlo simulations on frustrated systems. Moreover, these techniques can be
used to study large system sizes including the thermodynamic limit, thus
mitigating finite size effects. In two spatial dimensions, they are known as
projected entangled pair states (PEPS) or iPEPS [24, 25] in their infinite
instance and have become a state-of-the-art numerical tool for studying
strongly interacting systems. These techniques have recently proven to be
quite successful in studying frustrated model Hamiltonians [26, 27, 28, 29,
30], real materials [31, 32, 33], open systems [34, 35, 36, 37, 38, 39] and
non-equilibrium phenomena [40, 41, 42, 43, 44, 45].

Figure 1: Nearest-neighbour spin-spin correlations for the ground state
configuration of the pinwheel VBC (a) and the loop-six VBC (b) states at
$\chi_{B}=12$. The expectation values of the spin-$z$ component are typically
$<10^{-3}$ and only shown for completeness. (c) Shuriken lattice with
spin-$1/2$ degrees of freedom on the sites. The elementary unit cell (gray
rectangles) consists of six sites, which are coarse-grained to map the
Shuriken lattice to a regular square lattice. Straight lines denote virtual
bond indices, curly lines denote physical indices in the TN structure.

Recent theoretical works investigating the ground state of the isotropic
$S=1/2$ Heisenberg anti-ferromagnet on the Shuriken lattice have identified
two competing _valence bond crystals_ (VBCs) involving resonating loops of
different lengths: (i) a pinwheel VBC which maximizes the number of smallest
possible loops of length four [see Fig. 1(a)] and (ii) a VBC pattern
comprising only of loops of length six [see Fig. 1(b)]. Surprisingly, it has
been shown within an effective _resonating valence bond_ (RVB) theory that the
tunneling processes can be renormalized in such a way that the smallest loops
are not always the most relevant in capturing the correct ground state
correlations [15]. Indeed, based on a quantum dimer model approach it has been
shown in Ref. [15] that the loop-six VBC is more stable energetically compared
to the pinwheel VBC when non-local processes outside the nearest-neighbor
valence bond basis were invoked. Complementing this, based on energetic
considerations alone, the pinwheel VBC should conventionally be the expected
ground state as found in a recent variational Monte Carlo study [11]. This
opens a delicate question on how to properly account for such non-local
quantum correlations and patterns of long-range entanglement in highly
degenerate frustrated systems.

Here, we use a _tensor network_ approach to simulate the model directly in the
thermodynamic limit. TNs represent the state vector of a many-body system,
e.g., reflecting the ground state, as a contraction of a network of local
tensors, that are connected by auxiliary indices (bond indices). This enables
efficient numerical simulations with only a polynomial scaling in the number
of constituents [23, 46, 20, 21]. In this work, we employ the _infinite
projected entangled pair state_ (iPEPS) [24] and _infinite projected entangled
simplex state_ (iPESS) (a variant of iPEPS) [47] techniques with an ansatz
based on different and specifically tailored unit cell sizes for optimizing
the ground state of our model. In this context, the TN is used as an ansatz
for the full many-body state vector, consisting of a unit cell of different
tensors that generates a translationally invariant state. The accuracy of the
ansatz can be systematically improved by increasing the bond dimension of the
TN, which is the dimension of the virtual indices connecting the local
tensors, see Fig. 1(c) [see Appendix]. It controls the number of variational
parameters in the ansatz and is a measure for the amount of quantum
entanglement that can be captured. We mainly employ the so-called simple
update [48] to optimize the ground state tensors, which is expected to work
well for the gapped model at hand [49, 50, 48]. In order to verify its
accurate functioning and ability to resolve the close competition between the
two candidate ground states, we additionally employ a variational update [51]
for this task. The corner transfer matrix renormalization group (CTMRG) [52,
53, 54, 55] is then used to compute the expectation values of the ground state
energy in a variational manner, the spin-$z$ operator as well as the two-point
correlations to decipher the nature of the ground state. We also employ
additional $SU(2)$-symmetric simulations [56, 30] for the model. Given the
flexibility of the framework, we apply an external magnetic field to study the
magnetization process of the model and provide a compelling picture of the
nature of phases corresponding to different magnetization plateaus.

##  II Model and methods

The model we are considering is the $S=1/2$ Heisenberg anti-ferromagnet on the
Shuriken lattice

$\hat{H}=\sum_{\langle
i,j\rangle}\mathbf{\hat{S}}_{i}\cdot\mathbf{\hat{S}}_{j}-h\sum_{i}{\hat{S}}^{z}_{i}\
$ (1)

in the presence of an external magnetic field, where $\mathbf{\hat{S}}_{i}$
are the $S=1/2$ operators on site $i$ and $\langle i,j\rangle$ denotes
nearest-neighbours. The Shuriken lattice [see Fig. 1(c)] features corner-
sharing triangles, and thus leads to only a marginal alleviation of geometric
frustration in the presence of anti-ferromagnetic couplings. Being composed of
corner-sharing triangles, it is locally similar to the Kagome lattice. At the
same time, the Shuriken lattice shares two inequivalent sublattices, rendering
this lattice ideal to study effects of lattice anisotropy, for which our
methods are ideally suited.

We have applied two different TN structures for the simulation of the Shuriken
lattice. The first ansatz (iPEPS) uses a partial coarse-graining of the
Shuriken lattice to an irregular square lattice. Inspired by the success for
the $S=1/2$ Kagome Heisenberg anti-ferromagnet [26], the second structure is
based on the iPESS ansatz [47] that generalizes iPEPS to lattices with higher
simplices. For the Shuriken lattice, it is defined on its dual lattice, the
so-called $(4,8^{2})$ Archimedean lattice (also referred to as the square-
octagon, Fisher or CaVO lattice). While the simple update for iPESS
incorporates three lattice sites at each update step, it includes six sites
for iPEPS [see Appendix]. In order to compute expectation values, a unit cell
of six sites on the Shuriken lattice is coarse-grained into a single tensor on
the regular square lattice, as shown in Fig. 1(c). This approach is taken both
for the iPESS and the iPEPS simulations, starting from a $(4,8^{2})$
Archimedean lattice and a partially coarse-grained Shuriken lattice,
respectively. A directional CTMRG routine then computes effective environment
tensors for each coarse-grained iPEPS tensors, such that quantum correlations
are fully incorporated when computing expectation values (details are given in
the Appendix). This is achieved by a well chosen environment bond dimension
$\chi_{E}$ — a refinement parameter controlling the approximations in the
CTMRG routine, which is increased until the expectation values converge.

##  III Results on the ground state energy and dimer orders

The ground state energy of the Shuriken Heisenberg model can be
straightforwardly evaluated in the TN representation. The Hamiltonian consists
of a sum of nearest-neighbour terms,

$\displaystyle E_{0}=\frac{1}{N}\sum_{\langle
i,j\rangle}\langle\psi_{0}|h_{i,j}|\psi_{0}\rangle\,$ (2)

where $N$ is the number of lattice sites and $|\psi_{0}\rangle$ is the
normalized ground state vector. The smallest possible geometrical unit cell of
the Shuriken lattice consists of six sites [the dashed regions in Fig. 1(c)].
This ansatz — which imposes translational invariance while being compatible
with quantum spin liquid and lattice nematic candidate ground states — would,
however, fail to capture translation symmetry broken VBC orders such as the
pinwheel and the loop-six VBCs, which are the prime competing ground state
candidates. Therefore, we use different unit cell sizes to search for
competitive TN ansätze with the lowest ground state energy. The different unit
cell configurations are then labeled by the size of the super-unit-cell on the
square lattice, denoted by $(L_{x},L_{y})$. A configuration $(L_{x},L_{y})$
hence corresponds to an TN state with $6\cdot L_{x}\cdot L_{y}$ spins in
total.

Figure 2: Ground state energy without magnetic field [$h=0$ in Eq. (1)] versus
the inverse of the bond dimension $\chi_{B}$. iPESS simulations are denoted by
$(S)$, iPEPS simulations by $(P)$.

In Fig. 2, we show the ground state energy for the iPEPS and iPESS simulations
and square lattice unit cells of $(1,1)$ and $(2,2)$ as a function of the
inverse of the iPEPS bond dimension $\chi_{B}$ (bulk bond dimension). Our
results are compared to a previous iPEPS study of the model in Ref. [11],
using a $(1,1)$ TN ansatz with a coarse-graining to a honeycomb lattice. Based
on our simulations with different sizes of the unit cells, we find the lowest
energy is obtained with a $(2,2)$ configuration (unit cell with 24 sites),
i.e., corresponding to a valence bond crystal ground state. A similar ground
state can be obtained by a $(1,2)$ configuration in a checkerboard
arrangement, which consists of only twelve spins. However, we use the more
general state vector ansatz with 24 spins to be able to incorporate possible
richer patterns of spin correlations.

Figure 3: Comparison of unconstrained and constrained ground state simulations
up to $\chi_{B}=12$ for the different configurations imprinted. A first-order
polynomial fit is used to extract the infinite bond dimension limit for the
three largest bond dimensions.

In our simulation, the main difference in the iPESS and iPEPS calculations is
in the simple update. It is more local in the iPESS with only three sites that
are updated at once, compared to six sites in the iPEPS ansatz [see Appendix].
This, along with a larger number of variational parameters in the iPEPS ansatz
is responsible for a better ground state approximation with lower energies.
For subsequent investigation of the model we therefore use the iPEPS with a
$(2,2)$ unit cell configuration. In addition to the unconstrained simulations,
we incorporate fully $SU(2)$-symmetric simulations of the model. By imposing
the symmetry, the simulated ground state is guaranteed to be in the spin-$0$
sector, i.e., a spin singlet. In contrast, an unconstrained simulation of the
ground state can spontaneously break $SU(2)$-symmetry, which would lower its
energy. An energy comparison is therefore another way to ascertain the nature
of the ground state. For large enough bond dimensions, the $SU(2)$-symmetric
simulations converge to the same energy as the unconstrained ones, confirming
a nonmagnetic VBC ground state with spin-$0$ of the model as previously
reported [11].

Within our iPEPS simulations, we use two ways to ascertain the nature of the
VBC ground state, (i) we prepare our initial state with the pinwheel and the
loop-six VBC patterns imprinted for a given low bond dimensions and
progressively increase $\chi_{B}$ in a manner which uses the converged state
vectors at any given $\chi_{B}$ as initial states for the simulation with one
higher bond dimension and (ii) an unconstrained optimization starting from a
random state. In procedure (i) we observe that while both the pinwheel VBC and
loop-six VBC patterns remain stable up to $\chi_{B}=12$, the latter is always
lower in energy at any given finite bond dimension [see Fig. 3 and Table 1].
The resulting spatial spin-spin correlation profiles at $\chi_{B}=12$ are
shown in Fig. 1. Further compelling evidence supporting a loop-six VBC ground
state scenario for a finite bond dimension is provided by the unconstrained
optimization which at higher bond dimensions $(\chi_{B}\geqslant 4)$ already
converges to the energy of the loop-six VBC [see Fig. 3 and Table 1]. Notice
that the pinwheel and loop-six pattern is explicitly imprinted in the
simulations at $\chi_{B}=[2,3]$, so that those points are not expressive. The
inset of Fig. 3 shows the meaningful, i.e., linear regime where the energy
differences between the two orders are small, highlighting the subtle
competition.

An extrapolation to the infinite bond dimension limit using a linear fit of
the three values of energy corresponding to the largest $\chi_{B}$ yields a
lower bound for the energy $E_{l}$. The last data point at $\chi_{B}=12$
provides an upper bound $E_{u}$, such that the true ground state energy lies
in the interval $[E_{l},E_{u}]$ [57]. To estimate the final ground state
energy, we compute $E_{0}=(E_{u}+E_{l})/2$ with an error of $\Delta
E=(E_{u}-E_{l})/2$, which results in

$\displaystyle\begin{split}E_{0}(\text{pinwheel VBC})&=-0.4408\pm 0.0001,\\\
E_{0}(\text{loop-six VBC})&=-0.4410\pm 0.0001\,,\end{split}$ (3)

which is lower than previous estimates of the ground state energy [11]. The
numerical values for the results in Fig. 3 are summarized in Table 1. Given
that the estimates of the ground state energy for the pinwheel and loop-six
VBC states evaluated in the limit $\chi_{B}\to\infty$ are very close,
variational iPEPS simulations have been employed to resolve which of these two
competing states wins in this limit. Until the largest reachable bond
dimension of $\chi_{B}=7$, the variational energies lie below the presented
simple update energies and reinforce the ground state to be a loop-six VBC
[51]. A direct comparison of the simple update and variational energies is
presented in Table 2.

$\chi_{B}$ | Unconstrained | Pinwheel | Loop-six
---|---|---|---
2 | -0.428020 | -0.405664 | -0.397792
3 | -0.435105 | -0.405664 | -0.397792
4 | -0.439242 | -0.438734 | -0.439242
5 | -0.439665 | -0.439242 | -0.439665
6 | -0.440058 | -0.439738 | -0.440058
7 | -0.440375 | -0.440091 | -0.440376
8 | -0.440700 | -0.440522 | -0.440700
9 | -0.440776 | -0.440584 | -0.440776
10 | -0.440859 | -0.440646 | -0.440859
11 | -0.440886 | -0.440667 | -0.440886
12 | -0.440908 | -0.440689 | -0.440908
Table 1: Energy comparison for different ground state configurations of the $(2,2)$ iPEPS. Note that the states at $\chi_{B}=[2,3]$ cannot be used, since the ground state pattern is already imprinted. $\chi_{B}$ | Simple update | Variational update
---|---|---
2 | -0.428020 | -0.433600
3 | -0.435105 | -0.437320
4 | -0.439242 | -0.439877
5 | -0.439665 | -0.440162
6 | -0.440058 | -0.440391
7 | -0.440375 | -0.440592
Table 2: Energy comparison between simple update and variational energies for
the ground state of the loop-six VBC. The variational update uses a two-tensor
checkerboard pattern on the coarse-grained Shuriken lattice. Figure 4:
Magnetization curve of the Heisenberg model for $\chi_{B}=10$. Upon tuning the
magnetic field, two magnetization plateaus at $1/3$ and $2/3$ of the saturated
magnetization $m_{S}=1/2$ appear. Additionally, we find the presence of a
small plateau at $m_{z}=0$ indicative of the gapped nature of the ground
state.

##  IV Results on magnetization plateaus

Finally, we study the Heisenberg model on the Shuriken lattice in the presence
of an external magnetic field. We compute the average magnetization over all
the sites in the lattice along the field axis

$\displaystyle
m_{z}=\frac{1}{N}\sum_{i}\langle\psi_{0}(h)|\hat{S}_{i}^{z}|\psi_{0}(h)\rangle\
,$ (4)

where $|\psi_{0}(h)\rangle$ is the normalized ground state vector. In Fig. 4
we show the magnetization curve normalized to the saturation value of
$m_{S}=1/2$. The magnetization curve reveals the presence of three
magnetization plateaus, at $0$, $1/3$ and $2/3$ of the saturation value [18,
17, 58]. Furthermore, we observe a macroscopic jump from the $2/3$ plateau to
saturation magnetization as conventionally expected due to the presence of a
flat one-magnon band which leads to the appearance of localized multi-magnon
eigenstates [59, 60, 61, 62, 63, 64]. The plateau at $h\rightarrow 0$ is a
further indication of the fact that the ground state of the model at $h=0$ is
actually gapped. An estimate on the size of the spin gap $\Delta>0$ is given
by the width of the plateau $\Delta\sim 0.04J$, consistent with exact
diagonalization studies [16, 17]. The $1/3$ and $2/3$ plateaus can further be
characterized by the spatial pattern of spin-spin correlations and the
expectation values of the spin-$x$, -$y$ and -$z$ components. Those
expectation values are shown in Fig. 5 for both phases, at magnetic fields
$h=1.4$ and $h=2.8$ respectively. Interestingly, one observes that once the
magnetic field is turned on, a pattern governed by strong loop four resonances
emerges.

Within error-bars in the expectation values, the states at both magnetization
plateaus are invariant under translations of the original six-site
crystallographic unit cell and also under point group symmetries. It is also
visible that correlations are much stronger on the squares compared to the
triangle bonds in the lattice. For both plateau states the spins which are not
part of the squares are isolated and almost fully polarized [see Fig. 6],
implying that they are nearly aligned with the magnetic field. In contrast,
the spins on squares, despite a finite magnetization possess a nonzero singlet
density reminiscent of $h_{z}=0$ resonating plaquettes. This is also evidenced
by observing the different magnetization behaviours
$\langle\hat{S}_{i}^{z}\rangle$ of the two symmetry inequivalent sites over
the range of the magnetization as shown in Fig. 6. The three plateaus
appearing in Fig. 4 and the full saturation are shown with dotted lines. The
behaviour is in good agreement with previous numerical diagonalization studies
of the model on finite systems [17].

Figure 5: Ground state configurations of magnetic plateaus for $h=1.4$ (top)
and $h=2.8$ (bottom) at $\chi_{B}=10$. Both the $1/3$ plateau state and the
$2/3$ plateau state have strong spin-spin correlations on the squares, whose
spins appear to be in an entangled superposition. In contrast, spins on the
Shuriken sites are isolated and (almost) fully polarized.

Following the discussion of an analytic description for the magnetic plateau
states of the Heisenberg model on the Kagome lattice [65], one can similarly
construct the 2/3 plateau state on the Shuriken lattice. For the 2/3 case, one
can see in Fig. 5 and 6 that the expectation value
$\langle\hat{S}_{i}^{z}\rangle$ is roughly $0.25$ for the sites which are part
of the square and $0.5$ for the sites which are not part of the square. This
motivates the conclusion that the state vector for one unit cell consists of
an entangled state on the square and a product of this superposition with the
fully polarized non-square sites as
$\ket{\psi^{2/3}}=\ket{\uparrow}_{\leavevmode\hbox to3.96pt{\vbox
to6.56pt{\pgfpicture\makeatletter\hbox{\hskip 1.97829pt\lower-3.27988pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}\pgfsys@moveto{1.77829pt}{3.07988pt}\pgfsys@lineto{-1.77829pt}{-3.07988pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{3.07988pt}\pgfsys@lineto{1.77829pt}{-3.07988pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.85355pt}{0.0pt}\pgfsys@curveto{0.85355pt}{0.4714pt}{0.4714pt}{0.85355pt}{0.0pt}{0.85355pt}\pgfsys@curveto{-0.4714pt}{0.85355pt}{-0.85355pt}{0.4714pt}{-0.85355pt}{0.0pt}\pgfsys@curveto{-0.85355pt}{-0.4714pt}{-0.4714pt}{-0.85355pt}{0.0pt}{-0.85355pt}\pgfsys@curveto{0.4714pt}{-0.85355pt}{0.85355pt}{-0.4714pt}{0.85355pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\leavevmode\hbox to13.67pt{\vbox
to13.67pt{\pgfpicture\makeatletter\hbox{\hskip 6.83647pt\lower 35.84267pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}{{}} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{-3.5566pt}{39.12254pt}\pgfsys@lineto{-6.63647pt}{40.90083pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{-3.5566pt}{39.12254pt}\pgfsys@lineto{-1.7783pt}{36.04266pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{-3.5566pt}{46.23573pt}\pgfsys@lineto{-6.63647pt}{44.45744pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{-3.5566pt}{46.23573pt}\pgfsys@lineto{-1.7783pt}{49.31561pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{3.5566pt}{39.12254pt}\pgfsys@lineto{1.7783pt}{36.04266pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{3.5566pt}{39.12254pt}\pgfsys@lineto{6.63647pt}{40.90083pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{3.5566pt}{46.23573pt}\pgfsys@lineto{6.63647pt}{44.45744pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{3.5566pt}{46.23573pt}\pgfsys@lineto{1.7783pt}{49.31561pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{} {}{} {}{}
{}{}{}\pgfsys@moveto{-3.5566pt}{39.12254pt}\pgfsys@lineto{-3.5566pt}{46.23573pt}\pgfsys@lineto{3.5566pt}{46.23573pt}\pgfsys@lineto{3.5566pt}{39.12254pt}\pgfsys@lineto{-3.5566pt}{39.12254pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-3.5566pt}{39.12254pt}\pgfsys@moveto{-2.70305pt}{39.12254pt}\pgfsys@curveto{-2.70305pt}{39.59395pt}{-3.08519pt}{39.97609pt}{-3.5566pt}{39.97609pt}\pgfsys@curveto{-4.028pt}{39.97609pt}{-4.41014pt}{39.59395pt}{-4.41014pt}{39.12254pt}\pgfsys@curveto{-4.41014pt}{38.65114pt}{-4.028pt}{38.269pt}{-3.5566pt}{38.269pt}\pgfsys@curveto{-3.08519pt}{38.269pt}{-2.70305pt}{38.65114pt}{-2.70305pt}{39.12254pt}\pgfsys@closepath\pgfsys@moveto{-3.5566pt}{39.12254pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-3.5566pt}{46.23573pt}\pgfsys@moveto{-2.70305pt}{46.23573pt}\pgfsys@curveto{-2.70305pt}{46.70714pt}{-3.08519pt}{47.08928pt}{-3.5566pt}{47.08928pt}\pgfsys@curveto{-4.028pt}{47.08928pt}{-4.41014pt}{46.70714pt}{-4.41014pt}{46.23573pt}\pgfsys@curveto{-4.41014pt}{45.76433pt}{-4.028pt}{45.38219pt}{-3.5566pt}{45.38219pt}\pgfsys@curveto{-3.08519pt}{45.38219pt}{-2.70305pt}{45.76433pt}{-2.70305pt}{46.23573pt}\pgfsys@closepath\pgfsys@moveto{-3.5566pt}{46.23573pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{3.5566pt}{39.12254pt}\pgfsys@moveto{4.41014pt}{39.12254pt}\pgfsys@curveto{4.41014pt}{39.59395pt}{4.028pt}{39.97609pt}{3.5566pt}{39.97609pt}\pgfsys@curveto{3.08519pt}{39.97609pt}{2.70305pt}{39.59395pt}{2.70305pt}{39.12254pt}\pgfsys@curveto{2.70305pt}{38.65114pt}{3.08519pt}{38.269pt}{3.5566pt}{38.269pt}\pgfsys@curveto{4.028pt}{38.269pt}{4.41014pt}{38.65114pt}{4.41014pt}{39.12254pt}\pgfsys@closepath\pgfsys@moveto{3.5566pt}{39.12254pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{3.5566pt}{46.23573pt}\pgfsys@moveto{4.41014pt}{46.23573pt}\pgfsys@curveto{4.41014pt}{46.70714pt}{4.028pt}{47.08928pt}{3.5566pt}{47.08928pt}\pgfsys@curveto{3.08519pt}{47.08928pt}{2.70305pt}{46.70714pt}{2.70305pt}{46.23573pt}\pgfsys@curveto{2.70305pt}{45.76433pt}{3.08519pt}{45.38219pt}{3.5566pt}{45.38219pt}\pgfsys@curveto{4.028pt}{45.38219pt}{4.41014pt}{45.76433pt}{4.41014pt}{46.23573pt}\pgfsys@closepath\pgfsys@moveto{3.5566pt}{46.23573pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
}}}}\ket{\uparrow}_{\leavevmode\hbox to6.56pt{\vbox
to3.96pt{\pgfpicture\makeatletter\hbox{\hskip 3.27988pt\lower-1.97829pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}\pgfsys@moveto{-3.07988pt}{1.77829pt}\pgfsys@lineto{3.07988pt}{-1.77829pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-3.07988pt}{-1.77829pt}\pgfsys@lineto{3.07988pt}{1.77829pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.85355pt}{0.0pt}\pgfsys@curveto{0.85355pt}{0.4714pt}{0.4714pt}{0.85355pt}{0.0pt}{0.85355pt}\pgfsys@curveto{-0.4714pt}{0.85355pt}{-0.85355pt}{0.4714pt}{-0.85355pt}{0.0pt}\pgfsys@curveto{-0.85355pt}{-0.4714pt}{-0.4714pt}{-0.85355pt}{0.0pt}{-0.85355pt}\pgfsys@curveto{0.4714pt}{-0.85355pt}{0.85355pt}{-0.4714pt}{0.85355pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$. Using this ansatz,
it is straightforward to find the ground state vector

$\displaystyle\ket{\framebox[12.0925pt]{\raisebox{-4.26773pt}{{
\leavevmode\hbox to13.67pt{\vbox
to13.67pt{\pgfpicture\makeatletter\hbox{\hskip 6.83647pt\lower 35.84267pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}{{}} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{-3.5566pt}{39.12254pt}\pgfsys@lineto{-6.63647pt}{40.90083pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{-3.5566pt}{39.12254pt}\pgfsys@lineto{-1.7783pt}{36.04266pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{-3.5566pt}{46.23573pt}\pgfsys@lineto{-6.63647pt}{44.45744pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{-3.5566pt}{46.23573pt}\pgfsys@lineto{-1.7783pt}{49.31561pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{3.5566pt}{39.12254pt}\pgfsys@lineto{1.7783pt}{36.04266pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{3.5566pt}{39.12254pt}\pgfsys@lineto{6.63647pt}{40.90083pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{3.5566pt}{46.23573pt}\pgfsys@lineto{6.63647pt}{44.45744pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{}{}
{}{}{}\pgfsys@moveto{3.5566pt}{46.23573pt}\pgfsys@lineto{1.7783pt}{49.31561pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{} {}{} {}{}
{}{}{}\pgfsys@moveto{-3.5566pt}{39.12254pt}\pgfsys@lineto{-3.5566pt}{46.23573pt}\pgfsys@lineto{3.5566pt}{46.23573pt}\pgfsys@lineto{3.5566pt}{39.12254pt}\pgfsys@lineto{-3.5566pt}{39.12254pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-3.5566pt}{39.12254pt}\pgfsys@moveto{-2.70305pt}{39.12254pt}\pgfsys@curveto{-2.70305pt}{39.59395pt}{-3.08519pt}{39.97609pt}{-3.5566pt}{39.97609pt}\pgfsys@curveto{-4.028pt}{39.97609pt}{-4.41014pt}{39.59395pt}{-4.41014pt}{39.12254pt}\pgfsys@curveto{-4.41014pt}{38.65114pt}{-4.028pt}{38.269pt}{-3.5566pt}{38.269pt}\pgfsys@curveto{-3.08519pt}{38.269pt}{-2.70305pt}{38.65114pt}{-2.70305pt}{39.12254pt}\pgfsys@closepath\pgfsys@moveto{-3.5566pt}{39.12254pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-3.5566pt}{46.23573pt}\pgfsys@moveto{-2.70305pt}{46.23573pt}\pgfsys@curveto{-2.70305pt}{46.70714pt}{-3.08519pt}{47.08928pt}{-3.5566pt}{47.08928pt}\pgfsys@curveto{-4.028pt}{47.08928pt}{-4.41014pt}{46.70714pt}{-4.41014pt}{46.23573pt}\pgfsys@curveto{-4.41014pt}{45.76433pt}{-4.028pt}{45.38219pt}{-3.5566pt}{45.38219pt}\pgfsys@curveto{-3.08519pt}{45.38219pt}{-2.70305pt}{45.76433pt}{-2.70305pt}{46.23573pt}\pgfsys@closepath\pgfsys@moveto{-3.5566pt}{46.23573pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{3.5566pt}{39.12254pt}\pgfsys@moveto{4.41014pt}{39.12254pt}\pgfsys@curveto{4.41014pt}{39.59395pt}{4.028pt}{39.97609pt}{3.5566pt}{39.97609pt}\pgfsys@curveto{3.08519pt}{39.97609pt}{2.70305pt}{39.59395pt}{2.70305pt}{39.12254pt}\pgfsys@curveto{2.70305pt}{38.65114pt}{3.08519pt}{38.269pt}{3.5566pt}{38.269pt}\pgfsys@curveto{4.028pt}{38.269pt}{4.41014pt}{38.65114pt}{4.41014pt}{39.12254pt}\pgfsys@closepath\pgfsys@moveto{3.5566pt}{39.12254pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{3.5566pt}{46.23573pt}\pgfsys@moveto{4.41014pt}{46.23573pt}\pgfsys@curveto{4.41014pt}{46.70714pt}{4.028pt}{47.08928pt}{3.5566pt}{47.08928pt}\pgfsys@curveto{3.08519pt}{47.08928pt}{2.70305pt}{46.70714pt}{2.70305pt}{46.23573pt}\pgfsys@curveto{2.70305pt}{45.76433pt}{3.08519pt}{45.38219pt}{3.5566pt}{45.38219pt}\pgfsys@curveto{4.028pt}{45.38219pt}{4.41014pt}{45.76433pt}{4.41014pt}{46.23573pt}\pgfsys@closepath\pgfsys@moveto{3.5566pt}{46.23573pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}}}$
$\displaystyle=\frac{1}{2}\left(\ket{\downarrow\uparrow\uparrow\uparrow}-\ket{\uparrow\downarrow\uparrow\uparrow}-\ket{\uparrow\uparrow\downarrow\uparrow}+\ket{\uparrow\uparrow\uparrow\downarrow}\right)$
(5)

for the square terms, consisting of pairwise singlets on the four bonds. Since
the individual unit cells are not entangled in our ansatz it is easy to check
with exact diagonalization (ED), that the full state is the ground state of
the subspace we have chosen. The per-site energy of the analytic construction
is $-2h/6$. This perfectly fits our simple update results, where for $h=2.8$
we find an energy of $-0.933331$ while the analytic result is $-0.933333$.
This construction built out of one-magnon states on the squares and fully
polarized spins on Shuriken sites is the well-known magnon crsytal state [66,
18, 59].

Figure 6: On-site magnetization of the triangle and square sites versus the
magnetic field $h$. Dotted vertical lines represent the four magnetization
plateaus.

For the 1/3 plateau the same approach is not successful. If we limit ourselves
to the subspace with fully polarized spins on the non-square sites and the
superposition on the square sites, we can find a ground state from ED but this
time the analytic energy for $h=1.4$ of $-0.566667$ is clearly above the
result $-0.592928$ of the SU calculations. This is expected however, since as
one can see in Fig. 6 the spin expectation values
$\langle\hat{S}_{i}^{z}\rangle$ are neither exactly $0.5$ for the polarized
nor zero for the spins on the square. Therefore, a more sophisticated ansatz
would be needed to describe this state but this beyond the scope of this work.

##  V Conclusions

In this work we have studied the ground state properties of the $S=1/2$
Heisenberg anti-ferromagnet on the Shuriken lattice using two-dimensional
tensor network techniques of iPEPS and iPESS. We find that the incorporation
of non-local correlations proves indispensable in accurately capturing the
nature of the ground state, which is shown to be a loop-six VBC at any given
finite bond dimension up until $\chi_{B}=12$. Indeed, here we are faced with a
scenario featuring a delicate energetic competition between states governed by
different stabilization mechanisms. In particular, we have (i) a pinwheel VBC
favored by energy gain from short-range loop resonances and (ii) a loop-six
VBC favored by strong resonances over longer-length loops which are amplified
by the dressing of virtual singlets on top of the nearest-neighbor basis. This
is precisely what we aim to address by employing the TN framework which
naturally contains both these key ingredients and allows us to accurately
investigate their interplay towards accurately determining the nature of the
ground state. Our estimate of the ground state energy per site in the
thermodynamic limit obtained by extrapolating $\chi_{B}\to\infty$ is given by
$E_{0}=-0.4410\pm 0.0001$.

We have also investigated the effect of an external magnetic field in the
model and obtained its magnetization curve which shows three magnetization
plateaus at $0$, $1/3$ and $2/3$ of the saturation magnetization. The width of
the magnetization plateau at zero-field gives us an estimate of the spin gap
$\Delta\sim 0.04J$ consistent with exact diagonalization studies [16, 17]. The
nature of the phases at these plateaus are not only polarized but also show
strong signature of singlet correlations on four-site plaquettes. These states
are found to respect the spatial symmetries (both translation and point group)
of the Shuriken lattice, in contrast to the pinwheel VBC state.

Our work paves the way for future investigation of the Heisenberg model on the
Shuriken lattice in more general settings. It would be interesting to study
the anisotropic model with different couplings on the square and the triangle
bonds, or longer range couplings which could potentially be of relevance in
describing the recently studied materials [6, 8, 9]. An investigation of the
excitation spectrum would be another promising route to revealing diverse
manifestation of frustration [67]. Similarly, the corresponding model for
higher spins, e.g., $S=1$, could be explored, which could be host to a
trimerized ground state and display a wealth of magnetization plateaus hosting
exotic phases. This perspective seems even more interesting as it seems
plausible to recreate frustrated systems in Shuriken lattices under the
precisely controlled conditions of _quantum simulations_ [68] involving ultra-
cold atoms, giving rise to the interesting situation of benchmarking quantum
and classical simulations against each other.

##  VI Acknowledgments

We thank Arnaud Ralko, Ronny Thomale and Erik Weerda for insightful
discussions and helpful comments on the manuscript. Y. I. acknowledges support
from the Department of Science and Technology (DST), India through the MATRICS
Grant No. MTR/2019/001042, CEFIPRA Project No. 64T3-1, the ICTP through the
Associates Programme and from the Simons Foundation through grant number
284558FY19. This research was supported in part by the National Science
Foundation under Grant No. NSF PHY-1748958, IIT Madras through the Institute
of Eminence (IoE) program for establishing the QuCenDiEM group (Project No.
SB20210813PHMHRD002720), the International Centre for Theoretical Sciences
(ICTS), Bengaluru, India during a visit for participating in the program
“Frustrated Metals and Insulators” (Code: ICTS/frumi2022/9). Y. I.
acknowledges the use of the computing resources at HPCE, IIT Madras. The
Berlin team has been supported by the BMBF (MUNIQC-ATOMS), the DFG (CRC 183 on
“Entangled states of matter”, project No. 277101999), and the Helmholtz Center
Berlin. The authors would like to thank the HPC Service of ZEDAT, Freie
Universität Berlin for computing time [69].

*

##  Appendix A Tensor networks

###  A.1 Tensor network structures and algorithms

In this section, we present further technical details about the tensor network
structures and algorithms employed in the main article. We will start with the
_infinite projected entangled simplex state_ (iPESS) ansatz [70], which can be
straightforwardly extended from the original formulation on the Kagome lattice
to the Shuriken lattice. To this end, we consider the dual lattice (the so-
called $(4,8^{2})$ Archimedean lattice), where the spin-$1/2$s are located on
the lattice links instead of the lattice sites. This lattice is visualized in
Fig. 7 in green.

Figure 7: Shuriken lattice shown in black and its dual lattice (the so-called
$(4,8^{2})$ Archimedean lattice) shown in green. Since the spin-$1/2$s live on
the links of the dual lattice, additional three-index simplex tensors are
introduced on the vertices for the iPESS ansatz.

In order to connect the spins, additional purely virtual three-index simplex
tensors have to be introduced (shown in gray). An elementary iPESS unit cell
therefore consists of six lattice site tensors carrying the physical degree of
freedom, and four simplex tensors connecting them.

As an alternative approach we consider a modified version of the _infinite
projected entangled pair state_ (iPEPS). It is constructed by a partial
coarse-graining of the original lattice to a square lattice with missing
bonds. To this end we merge the four spins on each square configuration into a
single site, which is modeled by a tensor with physical dimension $p^{4}$. The
two remaining sites per unit cell are left unchanged. This mapping is shown in
Fig. 8.

Figure 8: Shuriken lattice shown on the left and iPEPS ansatz shown on the
right. The four sites on each square are coarse-grained into an effective
site, highlighted by the green dotted area. The resulting structure is a
square lattice with missing links.

One advantage that both TN structures share is the fact that one virtual bond
in the TN corresponds to only two links in the original Shuriken lattice.
Since the maximal entanglement shared between neighbouring lattice sites is
limited by the bond dimension of the TN ansatz, it is favourable to keep this
number small. Due to this property, we can directly compare results with the
same bond dimension. Note that a coarse-graining of the six spins per unit
cell in the Shuriken lattice directly results in a regular square lattice.

###  A.2 Simple update

In order to obtain an approximation of the ground state wave function, we
employ the simple update technique [48] in both TNs. This method is based on
an imaginary-time evolution under the Hamiltonian [24], in which all the
tensors in the networks are updated sequentially. For a sufficiently long
evolution, the ground state is projected out. For the iPESS network we employ
a regular three-site update of the different simplex configurations [70]. In
order to restore the individual tensors after each update, a higher-order
singular value decomposition (SVD) is used. During this process, the singular
values are truncated to the fixed bulk bond dimension $\chi_{B}$ to keep the
simulations computationally feasible. The process is illustrated in Fig. 9 for
the update of a down simplex, denoted $\bigtriangledown$, together with the
three connected lattice tensors.

Figure 9: Simple update in the iPESS simulations of the Shuriken lattice. The
Trotterized three-body Hamiltonian gate is absorbed into a triangle
configuration. A truncated higher-order SVD is used to decompose the resulting
six-index tensor back into the separate iPESS tensors. The same procedure is
applied to the other simplex tensors along with the different lattice tensors.

The simple update for the iPEPS ansatz on the deformed square lattice follows
the same spirit. However, instead of only updating three sites (along with one
simplex tensor) as in the iPESS ansatz, we choose a six-site update across
corners in the lattice. A single update step is presented in Fig. 10.

Figure 10: Simple update in the iPEPS simulations of the Shuriken lattice. The
Trotterized six-body Hamiltonian gate is absorbed into a bottom-left corner of
the deformed square lattice. The individual tensors are restored using two
successive SVDs with truncation to a fixed bond dimension. The update of a
top-right corner is performed similarly.

Again, the virtual links of the network are kept at a maximal bond dimension
$\chi_{B}$, which is achieved by two successive SVDs. Notice that we omit to
show additional diagonal tensors carrying the singular values on each virtual
link for more clarity both in Fig. 9 and Fig. 10.

Besides the two presented TN approaches, we also implemented a simple update
scheme on the original Shuriken lattice, using two-body gates on neighbouring
sites to evolve the wave function. Similarly as for the iPESS, the update is
very local and could not resolve all the magnetization plateaus present in the
model, so that we rejected the simulations.

###  A.3 Environments and expectation values

In order to compute accurate expectation values for the wave functions
obtained by the simple update, we employ a _corner transfer matrix
renormalization group_ (CTMRG) [52, 53, 54, 55] procedure to compute the
effective environments. To this end, the six lattice sites per unit cell are
coarse-grained into a single iPEPS site with local Hilbert space dimension
$2^{6}=64$. This maps the Shuriken lattice to a regular square lattice, for
which a directional CTMRG procedure can directly compute the approximate
contraction of the infinite lattice.

Figure 11: A CTMRG routine is used to approximate the contraction of the
infinite square lattice by a set of fixed-point environment tensors.

As shown in Fig. 11, the contraction of the infinite square lattice is
approximated by a set of eight fixed-point tensors surrounding every iPEPS
tensor in the unit cell. Expectation values can then be computed
straightforwardly by evaluating local operators
$\langle\psi|\hat{O}|\psi\rangle/\langle\psi|\psi\rangle$, where the
environment around the sites on which the operator acts and the norm of the
wave function is approximated by the CTMRG tensors.

## References

  * Savary and Balents [2016] L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2016).
  * Mendels  _et al._ [2007] P. Mendels, F. Bert, M. A. de Vries, A. Olariu, A. Harrison, F. Duc, J. C. Trombe, J. S. Lord, A. Amato, and C. Baines, Quantum magnetism in the paratacamite family: Towards an ideal Kagomé lattice, Phys. Rev. Lett. 98, 077204 (2007).
  * Han  _et al._ [2012] T.-H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm, and Y. S. Lee, Fractionalized excitations in the spin-liquid state of a Kagome-lattice antiferromagnet, Nature 492, 406 (2012).
  * Khuntia  _et al._ [2020] P. Khuntia, M. Velazquez, Q. Barthélemy, F. Bert, E. Kermarrec, A. Legros, B. Bernu, L. Messio, A. Zorko, and P. Mendels, Gapless ground state in the archetypal quantum Kagome antiferromagnet ZnCu3(OH)6Cl2, Nat. Phys. 16, 469 (2020).
  * Fåk  _et al._ [2012] B. Fåk, E. Kermarrec, L. Messio, B. Bernu, C. Lhuillier, F. Bert, P. Mendels, B. Koteswararao, F. Bouquet, J. Ollivier, A. D. Hillier, A. Amato, R. H. Colman, and A. S. Wills, Kapellasite: A Kagome quantum spin liquid with competing interactions, Phys. Rev. Lett. 109, 037208 (2012).
  * Yakubovich  _et al._ [2021] O. V. Yakubovich, L. V. Shvanskaya, G. V. Kiriukhina, A. S. Volkov, O. V. Dimitrova, and A. N. Vasiliev, Hydrothermal synthesis and a composite crystal structure of Na6Cu7BiO4(PO4)4[Cl(OH)]3 as a candidate for quantum spin liquid, Inorg. Chem. 60, 11450 (2021).
  * Pohle  _et al._ [2016] R. Pohle, O. Benton, and L. D. C. Jaubert, Reentrance of disorder in the anisotropic Shuriken Ising model, Phys. Rev. B 94, 014429 (2016).
  * Fujihala  _et al._ [2020] M. Fujihala, K. Morita, R. Mole, S. Mitsuda, T. Tohyama, S.-i. Yano, D. Yu, S. Sota, T. Kuwai, A. Koda, H. Okabe, H. Lee, S. Itoh, T. Hawai, T. Masuda, H. Sagayama, A. Matsuo, K. Kindo, S. Ohira-Kawamura, and K. Nakajima, Gapless spin liquid in a square-Kagome lattice antiferromagnet, Nat. Commun. 11, 3429 (2020).
  * Liu  _et al._ [2022] B. Liu, Z. Zeng, A. Xu, Y. Sun, O. Yakubovich, L. Shvanskaya, S. Li, and A. Vasiliev, Low-temperature specific-heat studies on two square-Kagome antiferromagnets, Phys. Rev. B 105, 155153 (2022).
  * Wolf  _et al._ [1972] W. P. Wolf, B. Schneider, D. P. Landau, and B. E. Keen, Magnetic and thermal properties of dysprosium aluminum garnet. II. Characteristic parameters of an Ising antiferromagnet, Phys. Rev. B 5, 4472 (1972).
  * Astrakhantsev  _et al._ [2021] N. Astrakhantsev, F. Ferrari, N. Niggemann, T. Müller, A. Chauhan, A. Kshetrimayum, P. Ghosh, N. Regnault, R. Thomale, J. Reuther, T. Neupert, and Y. Iqbal, Pinwheel valence bond crystal ground state of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the Shuriken lattice, Phys. Rev. B 104, L220408 (2021).
  * Lugan  _et al._ [2019] T. Lugan, L. D. C. Jaubert, and A. Ralko, Topological nematic spin liquid on the square Kagome lattice, Phys. Rev. Research 1, 033147 (2019).
  * Morita and Tohyama [2018] K. Morita and T. Tohyama, Magnetic phase diagrams and magnetization plateaus of the spin-1/2 antiferromagnetic Heisenberg model on a square-Kagome lattice with three nonequivalent exchange interactions, J. Phys. Soc. Jpn. 87, 043704 (2018).
  * Hasegawa  _et al._ [2018] Y. Hasegawa, H. Nakano, and T. Sakai, Metamagnetic jump in the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the square Kagome lattice, Phys. Rev. B 98, 014404 (2018).
  * Ralko and Rousochatzakis [2015] A. Ralko and I. Rousochatzakis, Resonating-valence-bond physics is not always governed by the shortest tunneling loops, Phys. Rev. Lett. 115, 167202 (2015).
  * Rousochatzakis  _et al._ [2013] I. Rousochatzakis, R. Moessner, and J. van den Brink, Frustrated magnetism and resonating valence bond physics in two-dimensional Kagome-like magnets, Phys. Rev. B 88, 195109 (2013).
  * Nakano and Sakai [2013] H. Nakano and T. Sakai, The two-dimensional S=1/2 Heisenberg antiferromagnet on the Shuriken lattice – a lattice composed of vertex-sharing triangles, J. Phys. Soc. Jpn. 82, 083709 (2013).
  * Richter  _et al._ [2009] J. Richter, J. Schulenburg, P. Tomczak, and D. Schmalfuß, The Heisenberg antiferromagnet on the square-Kagomé lattice, Condens. Matter Phys. 12, 507 (2009).
  * Tomczak and Richter [2003] P. Tomczak and J. Richter, Specific heat of the spin- Heisenberg antiferromagnet on Squagome lattice, J. Phys. A 36, 5399 (2003).
  * Orús [2014] R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  * Eisert [2013] J. Eisert, Entanglement and tensor network states, (2013), arXiv:1308.3318 .
  * Eisert  _et al._ [2010] J. Eisert, M. Cramer, and M. B. Plenio, Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010).
  * Verstraete  _et al._ [2008] F. Verstraete, J. I. Cirac, and V. Murg, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).
  * Jordan  _et al._ [2008] J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac, Classical simulation of infinite-size quantum lattice systems in two spatial dimensions, Phys. Rev. Lett. 101, 250602 (2008).
  * Verstraete and Cirac [2004] F. Verstraete and I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, (2004), arXiv:cond-mat:0407066 .
  * Liao  _et al._ [2017] H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z. Huang, B. Normand, and T. Xiang, Gapless spin-liquid ground state in the $S=1/2$ Kagome antiferromagnet, Phys. Rev. Lett. 118, 137202 (2017).
  * Picot  _et al._ [2016] T. Picot, M. Ziegler, R. Orús, and D. Poilblanc, Spin-$s$ kagome quantum antiferromagnets in a field with tensor networks, Phys. Rev. B 93, 060407 (2016).
  * Picot and Poilblanc [2015] T. Picot and D. Poilblanc, Nematic and supernematic phases in kagome quantum antiferromagnets under the influence of a magnetic field, Phys. Rev. B 91, 064415 (2015).
  * Kshetrimayum  _et al._ [2016] A. Kshetrimayum, T. Picot, R. Orús, and D. Poilblanc, Spin-$\frac{1}{2}$ Kagome XXZ model in a field: Competition between lattice nematic and solid orders, Phys. Rev. B 94, 235146 (2016).
  * Schmoll and Orús [2020] P. Schmoll and R. Orús, Benchmarking global $SU(2)$ symmetry in two-dimensional tensor network algorithms, Phys. Rev. B 102, 241101 (2020).
  * Boos  _et al._ [2019] C. Boos, S. P. G. Crone, I. A. Niesen, P. Corboz, K. P. Schmidt, and F. Mila, Competition between intermediate plaquette phases in ${\mathrm{SrCu}}_{2}$(${\mathrm{BO}}_{3}{)}_{2}$ under pressure, Phys. Rev. B 100, 140413 (2019).
  * Kshetrimayum  _et al._ [2020a] A. Kshetrimayum, C. Balz, B. Lake, and J. Eisert, Tensor network investigation of the double layer Kagome compound Ca10Cr7O28, Ann. Phys. (N.Y.) 421, 168292 (2020a).
  * Schmoll  _et al._ [2022] P. Schmoll, C. Balz, B. Lake, J. Eisert, and A. Kshetrimayum, Finite temperature tensor network algorithm for frustrated two-dimensional quantum materials, (2022), arXiv:2211.00121 .
  * Kshetrimayum  _et al._ [2017] A. Kshetrimayum, H. Weimer, and R. Orús, A simple tensor network algorithm for two-dimensional steady states, Nat. Commun. 8, 1291 (2017).
  * Czarnik  _et al._ [2012] P. Czarnik, L. Cincio, and J. Dziarmaga, Projected entangled pair states at finite temperature: Imaginary time evolution with ancillas, Phys. Rev. B 86, 245101 (2012).
  * Czarnik and Dziarmaga [2015] P. Czarnik and J. Dziarmaga, Variational approach to projected entangled pair states at finite temperature, Phys. Rev. B 92, 035152 (2015).
  * Czarnik  _et al._ [2016] P. Czarnik, M. M. Rams, and J. Dziarmaga, Variational tensor network renormalization in imaginary time: Benchmark results in the Hubbard model at finite temperature, Phys. Rev. B 94, 235142 (2016).
  * Kshetrimayum  _et al._ [2019] A. Kshetrimayum, M. Rizzi, J. Eisert, and R. Orús, Tensor network annealing algorithm for two-dimensional thermal states, Phys. Rev. Lett. 122, 070502 (2019).
  * Mondal  _et al._ [2020] S. Mondal, A. Kshetrimayum, and T. Mishra, Two-body repulsive bound pairs in a multibody interacting Bose-Hubbard model, Phys. Rev. A 102, 023312 (2020).
  * Czarnik  _et al._ [2019] P. Czarnik, J. Dziarmaga, and P. Corboz, Time evolution of an infinite projected entangled pair state: An efficient algorithm, Phys. Rev. B 99, 035115 (2019).
  * Hubig and Cirac [2019] C. Hubig and J. I. Cirac, Time-dependent study of disordered models with infinite projected entangled pair states, SciPost Phys. 6, 31 (2019).
  * Kshetrimayum  _et al._ [2020b] A. Kshetrimayum, M. Goihl, and J. Eisert, Time evolution of many-body localized systems in two spatial dimensions, Phys. Rev. B 102, 235132 (2020b).
  * Kshetrimayum  _et al._ [2021] A. Kshetrimayum, M. Goihl, D. M. Kennes, and J. Eisert, Quantum time crystals with programmable disorder in higher dimensions, Phys. Rev. B 103, 224205 (2021).
  * Dziarmaga [2021] J. Dziarmaga, Time evolution of an infinite projected entangled pair state: Neighborhood tensor update, Phys. Rev. B 104, 094411 (2021).
  * Dziarmaga [2022] J. Dziarmaga, Time evolution of an infinite projected entangled pair state: A gradient tensor update in the tangent space, Phys. Rev. B 106, 014304 (2022).
  * Cirac  _et al._ [2021] J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  * Xie  _et al._ [2014a] Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, and T. Xiang, Tensor renormalization of quantum many-body systems using projected entangled simplex states, Phys. Rev. X 4, 011025 (2014a).
  * Jiang  _et al._ [2008] H. C. Jiang, Z. Y. Weng, and T. Xiang, Accurate determination of tensor network state of quantum lattice models in two dimensions, Phys. Rev. Lett. 101, 090603 (2008).
  * Corboz  _et al._ [2010] P. Corboz, R. Orús, B. Bauer, and G. Vidal, Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states, Phys. Rev. B 81, 165104 (2010).
  * Bruognolo  _et al._ [2021] B. Bruognolo, J.-W. Li, J. von Delft, and A. Weichselbaum, A beginner’s guide to non-Abelian iPEPS for correlated fermions, SciPost Phys. Lect. Notes , 25 (2021).
  * [51] J. Naumann, E. Weerda, M. Rizzi, J. Eisert, and P. Schmoll, in preparation.
  * Nishino and Okunishi [1996] T. Nishino and K. Okunishi, Corner transfer matrix renormalization group method, J. Phys. Soc. Jap. 65, 891 (1996).
  * Nishino and Okunishi [1997] T. Nishino and K. Okunishi, Corner transfer matrix algorithm for classical renormalization group, J. Phys. Soc. Jap. 66, 3040 (1997).
  * Orús and Vidal [2009] R. Orús and G. Vidal, Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction, Phys. Rev. B 80, 094403 (2009).
  * Orús [2012] R. Orús, Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems, Phys. Rev. B 85, 205117 (2012).
  * Schmoll  _et al._ [2020] P. Schmoll, S. Singh, M. Rizzi, and R. Orús, A programming guide for tensor networks with global $SU(2)$ symmetry, Ann. Phys. (N.Y.) 419, 168232 (2020).
  * Corboz [2016] P. Corboz, Improved energy extrapolation with infinite projected entangled-pair states applied to the 2D Hubbard model, Phys. Rev. B 93, 045116 (2016).
  * Richter  _et al._ [2022] J. Richter, O. Derzhko, and J. Schnack, Thermodynamics of the spin-half square Kagome lattice antiferromagnet, Phys. Rev. B 105, 144427 (2022).
  * Schnack  _et al._ [2001] J. Schnack, H. J. Schmidt, J. Richter, and J. Schulenburg, Independent magnon states on magnetic polytopes, Eur. Phys. J. B 24, 475 (2001).
  * Schulenburg  _et al._ [2002] J. Schulenburg, A. Honecker, J. Schnack, J. Richter, and H.-J. Schmidt, Macroscopic magnetization jumps due to independent magnons in frustrated quantum spin lattices, Phys. Rev. Lett. 88, 167207 (2002).
  * Derzhko and Richter [2004] O. Derzhko and J. Richter, Finite low-temperature entropy of some strongly frustrated quantum spin lattices in the vicinity of the saturation field, Phys. Rev. B 70, 104415 (2004).
  * Zhitomirsky and Tsunetsugu [2004] M. E. Zhitomirsky and H. Tsunetsugu, Exact low-temperature behavior of a Kagomé antiferromagnet at high fields, Phys. Rev. B 70, 100403 (2004).
  * Derzhko  _et al._ [2007] O. Derzhko, J. Richter, A. Honecker, and H.-J. Schmidt, Universal properties of highly frustrated quantum magnets in strong magnetic fields, Low Temp. Phys. 33, 745 (2007), https://doi.org/10.1063/1.2780166 .
  * Mizoguchi  _et al._ [2021] T. Mizoguchi, Y. Kuno, and Y. Hatsugai, Flat band, spin-1 Dirac cone, and Hofstadter diagram in the fermionic square Kagome model, Phys. Rev. B 104, 035161 (2021).
  * Capponi  _et al._ [2013] S. Capponi, O. Derzhko, A. Honecker, A. M. Läuchli, and J. Richter, Numerical study of magnetization plateaus in the spin-$\frac{1}{2}$ Kagome Heisenberg antiferromagnet, Phys. Rev. B 88, 144416 (2013).
  * Richter  _et al._ [2004] J. Richter, O. Derzhko, and J. Schulenburg, Magnetic-field induced Spin-Peierls instability in strongly frustrated quantum spin lattices, Phys. Rev. Lett. 93, 107206 (2004).
  * McClarty  _et al._ [2020] P. A. McClarty, M. Haque, A. Sen, and J. Richter, Disorder-free localization and many-body quantum scars from magnetic frustration, Phys. Rev. B 102, 224303 (2020).
  * Glaetzle  _et al._ [2014] A. W. Glaetzle, M. Dalmonte, R. Nath, I. Rousochatzakis, R. Moessner, and P. Zoller, Quantum spin-ice and dimer models with Rydberg atoms, Phys. Rev. X 4, 041037 (2014).
  * Bennett  _et al._ [2020] L. Bennett, B. Melchers, and B. Proppe, Curta: A general-purpose high-performance computer at ZEDAT, Freie Universität Berlin (2020).
  * Xie  _et al._ [2014b] Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, and T. Xiang, Tensor renormalization of quantum many-body systems using projected entangled simplex states, Phys. Rev. X 4, 011025 (2014b).