# Partially disordered Heisenberg antiferromagnet with short-range stripe
correlations

G. G. Blesio  Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
F. T. Lisandrini  Physikalisches Institut, University of Bonn, Nussallee 12,
53115 Bonn, Germany  M. G. Gonzalez  Helmholtz-Zentrum Berlin für Materialien
und Energie, Hahn-Meitner Platz 1, 14109 Berlin, Germany  Dahlem Center for
Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin,
14195 Berlin, Germany

###### Abstract

Zero-point quantum fluctuations of a Néel order can produce effective
interactions between quasi-orphan spins weakly coupled to the lattice. On the
$\sqrt{3}\times\sqrt{3}-$distorted triangular lattice, this phenomenon leads
to a correlated partially disordered phase. In this article, we use matrix
product state methods to study a similar model: the $S=1/2$ stuffed square
lattice. Tunning the exchange amplitudes we go from a square lattice plus
orphan central spins at $J^{\prime}/J=0$, to the union jack lattice at
$J^{\prime}/J=1$, and a square lattice including all spins at
$J/J^{\prime}=0$. We calculate the complete antiferromagnetic phase diagram,
dominated by ferrimagnetic and Néel orders, and compare with existing results.
Most importantly, we find a partially disordered phase in the weakly
frustrated regime. In this phase, the Néel order from the square lattice is
unaffected, while the central spins form a collective state with exponentially
decaying double striped correlations. We also study the role of quantum
fluctuations by introducing an ordering staggered magnetic field on the square
sublattice, and find that the central spins order ferromagnetically when
fluctuations from the Néel order are suppressed.

##  I Introduction

Quantum spin systems provide a vast playground to study all kinds of
interesting phenomena. This is mainly due to the zero-point quantum
fluctuations which, enhanced by magnetic frustration, lead to the emergence of
novel collective many-body states. Over the last decade, a great deal of
interest has been devoted to the highly-frustrated Heisenberg
antiferromagnets, both theoretically and experimentally, in the search for
quantum spin liquids. These highly entangled states with fractional spinon
excitations have been found theoretically, for example, in two-dimensional
systems such as the kagome[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] or
triangular lattice with next-nearest neighbor interactions[13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23]. Experimentally, there are several spin-liquid
candidates (or compounds which show anomalous features and are thought to lie
close to one)[24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].

However, in some cases, fluctuations can counter-intuitively lead to ordered
states. For example, classical frustrated spin systems can exhibit an
accidental degeneracy in the classical limit at zero temperature, which is
usually caused by a vanishing coupling energy between subsystems. This happens
in the $J_{1}-J_{2}$ Heisenberg model on the square lattice at large $J_{2}$,
where the system decouples into two independent square lattices with two
independent Néel orders[35]. However, at finite temperatures the thermal
fluctuations correlate the angle between the two subsystems and the degeneracy
is lifted. This phenomenon has been studied in many classical cases and it is
usually referred to as order by disorder[36, 37, 38, 39, 40, 41]. Quantum
fluctuations at zero temperature can also break degeneracies from their
classical counterparts, playing a similar role as thermal fluctuations,
leading to order by quantum disorder[42, 43, 44, 45, 46, 47, 48, 49, 50, 51].

In frustrated systems, quantum fluctuations can also prevent different
subsystems from coupling. One example of this is the one-dimensionalization
effect which occurs when Heisenberg chains are coupled in a frustrated manner
forming a spatially anisotropic triangular lattice[52, 53, 54, 55, 56, 57,
58]. For spin $S=1/2$ systems, even though the chains are gapless and present
quasi-long range order, an inter-chain coupling of over 50$\%$ of the intra-
chain coupling is needed to break the one-dimensional character[54, 56]. For
spin $S=1$ systems, an interchain coupling of about the same magnitude as the
Haldane gap is needed in order to close it and develop a two-dimensional
incommensurate spiral order[57]. There are also several compounds in which
this mechanism is thought to play a key role in the effective reduction of the
dimension[59, 60, 61, 62, 63, 64].

Recently, it was proposed that zero-point quantum fluctuations above the
magnetic order can induce effective correlations between spins weakly coupled
to the lattice[65, 66]. This was studied on the
$\sqrt{3}\times\sqrt{3}-$distorted triangular lattice, a model proposed for
the LiZn2Mo3O8 compound, where the triangular lattice is deformed into an
emergent honeycomb lattice ($J$) coupled to central spins ($J^{\prime}$)[67].
Exact diagonalization[68] and matrix product state[65] calculations determined
that for $J^{\prime}>0.2J$ the center spins couple to the lattice, canting the
Néel order from the honeycomb subsystem into a ferrimagnet over the whole
triangular lattice. However, for $J^{\prime}<0.2J$ the center spins remain
disordered and decoupled from the lattice, forming a partially disordered
phase[65].

Furthermore, in the partially disordered phase, the center spins are
ferromagnetically correlated at short distances. These correlations are
originated from a Casimir-like effect, in the sense that they are mediated by
the zero-point quantum fluctuations of the Néel order of the honeycomb
lattice. The weakly-coupled nature of the partial disorder allows to integrate
out the degrees of freedom of the ordered sublattice. A second order
perturbation theory in $J^{\prime}$ and $1/S$ expansions resulted in an
effective model for the central spins with nearest neighbor ferromagnetic XY
interactions (which dominate at large $S$) and next-nearest neighbor
antiferromagnetic Ising interactions (dominant at small $S$)[66]. This model
exhibits a transition between a double stripe phase (dubbed $\langle
2\rangle$) and a ferromagnetic phase at $S_{c}=0.646$, where the magnetic
order is destabilized by quantum fluctuations. This is surprisingly close to
$S=1/2$, and could be the mechanism responsible of the partially disordered
phase. These results rise questions about the ubiquitousness of the partially
disordered phase in models with a subsystem of weakly coupled spins. In other
cases, the effective model governing the disordered subsystem will be
different and could either lead to other exotic disordered states, or the
system not hosting a partially disordered phase at all.

Motivated by these questions, in this article we study the stuffed square
lattice. This model is formed by a square lattice, $J$, connected to sites at
the center of each squares by $J^{\prime}$ (see Fig. 1). This allows us to
study the existence of the partial disorder phenomenon and the structure of
the inner correlations induced by quantum fluctuations of the Néel-ordered
square sublattice. A similar model has been proposed in the context of the
layered compound Sr2TcO4 compound, where frustrated center sites suppress
interplane couplings[69].

Figure 1: Stuffed square lattice where the exchange amplitude $J$ (full line)
couples the spins in the square lattice and $J^{\prime}$ (dashed line) couples
them to the spins at the center of each square. The three sublattices A, B,
and C are shown in colors. The plotted lattices are named $L_{y}\times
L_{x}=8\times 4$, and the number of sites is $N=8\times 4=32$. There are as
many sites C as sites AB. The red line and numbers on the right side indicate
the path chosen to map the system into a one-dimensional chain for the MPS
calculations.

We find that, for low values of $J^{\prime}$, the spins at the center of the
squares are disordered. However, their correlations present a double-stripped
pattern with exponentially decaying ferromagnetic correlations along one
direction and antiferromagnetic correlations to third-nearest neighbors in the
perpendicular direction. In analogy with the stuffed honeycomb lattice, we
argue that this is consistent with the expected effective model for the
central spins. We further study the role of quantum fluctuations in this phase
by introducing an ordering magnetic field on the square sublattice. For large
enough fields, quantum fluctuations are suppressed and the center sites become
ferromagnetically ordered.

The rest of the article is organized as follows: in Sec. II we introduce the
antiferromagnetic Heisenberg model for the stuffed square lattice and the
general considerations of our matrix product states (MPS) calculations. In
Sec. III, we show and discuss our results, first about the general quantum
phase diagram and finally we explore in more detail the weakly coupled limit.
Finally, in Sec. IV we present the summary and conclusions of our work.

##  II Model and Method

We define the antiferromagnetic Heisenberg Hamiltonian for the stuffed square
lattice as

$\mathcal{H}=J\sum_{\langle
ij\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+J^{\prime}\sum_{\left[ij\right]}\mathbf{S}_{i}\cdot\mathbf{S}_{j}$
(1)

where $J$ is the exchange interaction between nearest neighbors in the square
lattice $\langle ij\rangle$ and $J^{\prime}$ connects the center spins to the
square lattice $\left[ij\right]$. It is convenient to separate the lattice
into three sublattices (see Fig. 1). Sublattices A and B are always equivalent
and form the main square lattice, and the C sublattice is composed of the
spins at the center. This model has some well-known limits. For
$J^{\prime}/J=0$ it becomes the square lattice Heisenberg model (on the AB
sublattice, with completely orphan spins C). In the opposite limit, for
$J/J^{\prime}=0$, the system becomes again a square lattice, but with the
double of spins (in this limit the square lattice is formed by sublattices AB
and C). Finally, the case $J=J^{\prime}$ is known as the union jack
lattice[70, 71, 72, 73, 74, 75], a square lattice with only half of its next-
nearest neighbor interactions.

To solve the model in Eq. 1 we use MPS methods provided by the ITensor
libraries[76, 77]. In particular we use the density-matrix renormalization
group algorithm to find the ground state of the system, which requires a
transformation of the two-dimensional system into a one-dimensional chain. The
standard way to do this is with a zig-zag path from bottom up and from left to
right (shown by the red path in Fig. 1). Two-dimensional systems are usually
better represented by $L_{y}\times L_{x}$ cylinders with periodic boundary
conditions along $L_{y}$ and $L_{x}\geq L_{y}$[78]. However, in our case the
system is composed of two square lattices of size $L_{y}/2\times L_{x}$, so we
use $L_{x}\geq L_{y}/2$. The truncation procedure is controlled by the bond
dimension, $D$. Most results are obtained with $D=3000$, while in some cases
we have used up to $D=5000$ to ensure convergence of our results. Truncation
errors are always kept below $10^{-6}$.

##  III Results

Throughout the article we use the following parametrization for $J$ and
$J^{\prime}$:

$J=\cos\left(\alpha\frac{\pi}{2}\right)\qquad
J^{\prime}=\sin\left(\alpha\frac{\pi}{2}\right),$ (2)

in order to cover the whole range of antiferromagnetic exchange interactions
with $\alpha\in[0,1]$. This way, all limits are reached: for $\alpha=0$, $0.5$
and $1$, we get $J^{\prime}/J=0$, $1$, and $\infty$ ($J/J^{\prime}=0$),
respectively.

###  III.1 Phase diagram

Figure 2: Schematic phase diagram of the stuffed square lattice as obtained by
our MPS calculations. The phase diagram has three different phases: a
correlated partially disordered phase (PD) for low values of $\alpha$, a
ferrimagnetic phase (FI) for intermediate values, and a Néel order for large
values of $\alpha$.

Before getting into the detail of our calculations and results, we present the
phase diagram obtained, which is comprised of three clearly distinct phases
(see Fig. 2): a correlated partially disordered phase (PD), a
ferrimagnetically (FI) ordered phase, and a Néel ordered phase. For
$0\leq\alpha\lesssim 0.15$ the system exhibits a PD phase driven by quantum
fluctuations analogous to the one found in the stuffed honeycomb lattice[65].
In this phase, AB spins form a Néel order with algebraically decaying
correlations, while the C spins remain disordered with exponentially decaying
correlations. Nonetheless, the structure of correlations is nontrivial and it
is compatible with a double striped phase. This phase, albeit ordered, was
predicted for the effective model of the central spins of the stuffed
honeycomb lattice[66].

For $\alpha\geq 0.15$ a first-order phase transition to a FI canted state
occurs, where C spins are aligned ferromagnetically and A and B spins form an
angle $\pi\pm\phi$ with C. This phase is characterized by a non-zero total
magnetization. As $\alpha$ grows, the angle $\phi$ goes from $\simeq\pi/2$ to
$0$, where it transitions to a Néel ordered phase that recovers the U(1)
symmetry (see Fig. 2). This transition has been studied previously, and both
classical and linear spin-wave results show that $\phi=\arccos(J^{\prime}/2J)$
with a critical value $\alpha_{c}\simeq 0.7$ ($J^{\prime}=2J$)[70, 71]. On the
other hand, we find $\alpha_{c}\simeq 0.6$, which is in close agreement with
previous calculations with series expansions [$\alpha=0.633(5)$] and coupled
cluster methods [$\alpha=0.63(1)$][72, 73]. Finally, in the Néel phase the
spins in the AB sublattice are ferromagnetically aligned and opposite to the
spins in the C sublattice.

###  III.2 Ferro- and antiferromagnetic magnetizations

An important quantity to characterize these different phases is the static
spin structure factor

$S^{X}(\mathbf{q})=\frac{1}{N_{X}}\sum_{i,j\in X}\
\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle\
e^{i\mathbf{q}\mathbf{r}_{ij}}$ (3)

where $X$ refers to a given sublattice (A, B, C, or AB) or the whole lattice
(ABC); and $N_{X}$ is the number of sites in $X$. The wave-vector
$\mathbf{q}=(q_{x},q_{y})$ is discretized along the periodic direction
$q_{y}$, while it can take any value along $q_{x}$. In practice, however, both
can be taken as continuous as long as we remember that important features will
only appear at the values allowed by the cylindrical boundary conditions.
Orders related to forbidden wave-vectors are frustrated by the boundary
conditions. In this sense, it is important to note that all the sublattices
are square lattices (even though some are tilted with respect to the $x$ and
$y$ axes defined in Fig 1). We take $L_{x}$ and $L_{y}$ as even numbers in
order to be able to host the Néel orders at $\alpha=0$ and 1, as well as the
intermediate FI order.

As a first approach, we calculate the spin structure factor from Eq. 3 at the
points $\mathbf{q}=\mathbf{0}$ and
$\mathbf{q}={\boldsymbol{\pi}}=(\pm\pi,\pm\pi)$, corresponding to the
ferromagnetic and antiferromagnetic Néel magnetizations, respectively.
Explicitly, both magnetizations are written as

$m_{\text{FE}}^{X}=\sqrt{\frac{S^{X}(\mathbf{0})}{N_{X}}}\qquad
m_{\text{AF}}^{X}=\sqrt{\frac{S^{X}({\boldsymbol{\pi}})}{N_{X}}},$ (4)

where FE stands for ferromagnetic, AF for antiferromagnetic, and $X$ indicates
the sublattice. It is important to say that these are not true magnetizations,
since in two dimensions order can only exist in the thermodynamic limit[79]. A
further note of caution must be added: the peak in the spin structure factor
can only be associated to a semi-classical magnetization if there are no other
non-equivalent peaks. We will comment when this is, or not, the case. But, in
general, $m$ should just be considered as a convenient measure of the signal
coming from a given part of the spin structure factor.

Figure 3: Ferro- and antiferromagnetic magnetizations $m_{\text{FE}}^{X}$ and
$m_{\text{AF}}^{X}$ (see Eq. 4 in main text) as a function of
$\alpha=(2/\pi)\arctan(J^{\prime}/J)$. Different sublattices are indicated in
the legend, and different lattice sizes are indicated inside each panel.
Results for $X=B$ are not shown since $m^{B}=m^{A}$ always. The three
different phases are indicated by shaded areas and referenced in panel (b).
The gray dashed lines corresponds to the difference
$m_{\text{FE}}^{C}-m_{\text{AF}}^{ABC}$.

In Fig. 3, we show the values of sublattice magnetizations $m_{\text{AF}}^{X}$
and $m_{\text{FE}}^{X}$ for lattice sizes $4\times 12$, $8\times 12$ and
$12\times 14$ in panels (a), (b) and (c), respectively. The results are not
sensible to $L_{x}$ (not shown) and are qualitatively similar when changing
$L_{y}$. All lattices exhibit three distinguishable phases. The PD phase at
low values of $\alpha$ has the ferromagnetic magnetization from sublattices A
and B equal to the antiferromagnetic magnetization from the sublattice AB,
i.e. $m_{\text{FE}}^{A}=m_{\text{FE}}^{B}=m_{\text{AF}}^{AB}$. In this phase,
the previous magnetizations represent the only peaks in the corresponding spin
structure factors. All together, these features indicate that the Néel order
from the square lattice AB at $\alpha=0$ ($J^{\prime}=0$) extends to finite
values of $\alpha$. On the other hand, the sublattice C does not have either a
ferro- or antiferromagnetic magnetization and thus it cannot be characterized
by this analysis. However, it is important to note that the magnetization in
the AB sublattice is unaltered by the coupling of the $C$ sublattice. That is
to say, $m_{\text{AF}}^{AB}$ is almost constant in the PD phase. This also
happens in the stuffed honeycomb lattice in the weakly frustrated limit[65]
and it seems to be a common feature of models where orphan spins couple in a
frustrated manner to an unfrustrated lattice.

The passage to the FI phase is marked by a first-order phase transition in the
C sublattice. At the phase transition, C spins develop a sudden ferromagnetic
order accompanied by an almost classical value of $m_{\text{FE}}^{C}\simeq
0.5$ (that does not seem to decrease with the lattice size). At the same time,
$m_{\text{FE}}^{A}=m_{\text{FE}}^{B}\neq m_{\text{AF}}^{AB}$, indicating that
A and B sublattices no longer form a Néel order in AB. Instead, sublattice AB
develops a small ferromagnetic magnetization that grows with $\alpha$,
signalling that the spins A and B are losing the collinearity from the Néel
order. However, sublattices A and B are still purely ferromagnetic and there
is a three-sublattice order (together with C). All of these features are
strongly suggesting a FI order in the whole lattice, as predicted by previous
calculations[72, 73]. A FI phase is characterized by a total non-zero
magnetization. We will use this fact to probe the FI order and calculate the
angle between the three sublattices.

Before moving on to the Néel phase, let us remark again the similarities to
the stuffed honeycomb lattice model[65]. First, there is a small increase of
the magnetization in sublattice A and B when the sublattice C orders
ferromagnetically. This can be interpreted as a consequence of the reduction
of quantum fluctuations caused by the sublattice C acting as a magnetic field
on the sublattice AB[80, 65]. Secondly, the magnetization sublattice C is
always higher or equal than the magnetization in A and B (clearly seen in Fig.
3 for the $L_{y}=12$ lattices). This result is typical from lattices with
inequivalent sites, where higher values of magnetization are observed for
sites with lower effective coordination number[81].

To complete the analysis of Fig. 3 we turn to the other end of the phase
diagram (large values of $\alpha$). For the Néel phase, the ferromagnetic
magnetizations of sublattices C and AB are the same as the antiferromagnetic
magnetization on the whole lattice,
$m_{\text{FE}}^{C}=m_{\text{FE}}^{AB}=m_{\text{AF}}^{ABC}$. This implies that
the antiferromagnetic order in the whole lattice ABC is formed by two
ferromagnetic sublattices C and AB (both have the same number of sites).
Exactly in the limit $\alpha=1$ the system becomes a tilted square lattice
with only nearest-neighbor interactions $J^{\prime}$ ($J=0$). This phase
extends to finite values of $J/J^{\prime}$. To better visualize the phase
transition, in Fig. 3 we plot the difference
$m_{\text{FE}}^{C}-m_{\text{AF}}^{ABC}$ in gray dashed lines. From our
calculations we can place the phase transition at $\alpha_{c}=0.60(5)$, marked
by the separation of the relevant magnetizations, $m_{\text{FE}}^{C}\neq
m_{\text{AF}}^{ABC}$, i.e. $m_{\text{FE}}^{C}-m_{\text{AF}}^{ABC}\neq 0$. The
critical point coincides with the value of $\alpha$ for which the lowest
magnetization is observed for all lattices, $\alpha=0.6$. This critical point
is in close agreement with the value obtained from the series expansions and
CCM calculations, $\alpha=0.63(1)$[72, 73].

###  III.3 Total magnetization and magnetic order

Figure 4: Energy in each subspace $S^{z}$ minus the ground-state energy at
$S^{z}=0$ for different values of $\alpha$ as a function of $S^{z}/(SN_{c})$.
This normalization is chosen so that 1 indicates the fully polarized C
sublattice. Both panels show the same results in different scales. The arrows
indicate the maximum value of the $S^{z}$ subspace where the ground state can
be found. Calculations correspond to the $8\times 12$ lattice.

As mentioned above, the FI phase is characterized by a non-zero total
magnetization. This means that the ground state of the system can be found in
several $S^{z}$ subspaces. We define the subspace $S^{z}_{\text{max}}$ as the
one with the largest $S^{z}$ that still contains the ground state. Also,
$E(S^{z})$ refers to the energy of the lowest-lying state in the subspace
$S^{z}$. This implies that $E(S^{z}=0)=...=E(S^{z}_{\text{max}})$ (equivalent
to $E(-S^{z}_{\text{max}})$ due to inversion symmetry). In contrast, a ground
state with zero total magnetization can only be found on the $S^{z}=0$
subspace. Therefore, $E(S^{z})$ is a good quantity to differentiate the phases
of our model. We show these results in Fig. 4. The axis is normalized by the
highest possible magnetization value of the C sublattice, $SN_{\text{C}}$
where $S=1/2$. Also, to be able to compare different values of $\alpha$, we
always subtract the ground-state energy $E_{0}=E(S^{z}=0)$. The results shown
correspond to the $8\times 12$ lattice, but all other lattices show equivalent
behaviors. The top and bottom panels show two different energy scales in order
to show more clearly all features.

For $\alpha=0$, the C spins are completely decoupled and therefore do not
contribute to the total energy. Thus, $E(S^{z}=0)=...=E(SN_{\text{C}})$ and
the first real excited state (hosted by the AB sublattice) can be found in the
$S^{z}=SN_{\text{C}}+1$ subspace (see blue curves in Fig. 4). This first
excited state becomes gapless only in the thermodynamic limit, representing
the magnonic excitations corresponding to the Goldstone modes. When
$\alpha\neq 0$ but small, in the PD phase, we observe that the ground state
has $S^{z}=0$ and all other subspaces represent excited states. This means
that, even though the C spins do not affect the magnetic order in the AB
square lattice (remember that AB magnetization does not change and the order
remains a Néel order, see Fig. 3), they are not trivially decoupled as in
$\alpha=0$. Instead, they form a state which harbors the lowest energy
excitations of the system. This is understandable in this regime, where
$J^{\prime}$ is much smaller than $J$. From this calculations, it is not
possible to predict what happens to the excitations of the C sublattice in the
thermodynamic limit (whether the disordered state is gapped or gapless).

For values of $\alpha$ corresponding to the FI phase, we find indeed a non-
zero total magnetization. This is shown in Fig. 4, where the ground-state
energy is found on several consecutive subspaces. The value
$S^{z}_{\text{max}}$ (indicated by black arrows) is the highest close to the
transition point to the PD phase, and then decreases when $\alpha$ increases
approaching the Néel phase. At some point it becomes zero, indicating the
passing to the Néel order in the whole lattice ABC.

Regarding the nature of the transition between the PD and the FI phases, in
Fig. 4 the ferromagnetic states are already visible as excited states before
the transition, i.e., in the plateau at $E-E_{0}\simeq 0.06$ for $\alpha=0.2$.
Namely, the spin structure factor of the C spins in the plateau presents a
ferromagnetic peak. These ferrimagnetic states lower their energy as $\alpha$
increases until they become the true ground state in the FI phase. The energy-
level crossing along with the abrupt change in the magnetization observed in
Fig. 3 confirms a first-order phase transitions of the C spins. We note that
the small deviations around the plateau come from the difficulty in obtaining
well-converged results when two completely different phases have very similar
energies.

Figure 5: Semiclassical angle between spins A and B as a function of $\alpha$
for different lattice sizes (see Eq. 6 in main text). The spin-wave (same as
the classical) solution is shown in black line[70]. The inset shows the
semiclassical picture of the magnetic order.

We can further characterize the FI phase by using the semiclassical picture of
the magnetic order in Ref. [65], shown in the inset of Fig. 5. This picture
assumes that all spins within a given sublattice (A, B, and C) point in the
same direction with an effective size given by $m^{X}$ for each sublattice.
This is not true for finite-size systems, which conserve SU(2) symmetry and
therefore $\langle S_{i}^{\gamma}\rangle=0$ for $\gamma=x$, $y$, $z$. However,
because the structure factor from each sublattice shows only a ferromagnetic
peak, we can still use the semiclassical picture as an approximation.

We know the values of $m_{\text{FE}}^{X}$ from the $S^{z}=0$ calculations
shown in Fig. 3 (for which we recall that $m^{\text{A}}$ and $m^{\text{B}}$
are always the same). In the subspace $S^{z}_{\text{max}}$, the expectation
values of the $S^{z}_{i}$ operator on each site of the sublattice C are close
to the value of the magnetization $m_{\text{FE}}^{C}$ from the $S^{z}=0$
subspace (i.e. $\langle S_{i}^{z}\rangle\simeq m_{\text{FE}}^{C}$); and for
the A and B spins, the expectation values are approximately $\langle
S_{i}^{z}\rangle\simeq-m_{\text{FE}}^{A}\cos\phi$. From the total
magnetization $\sum_{i\in ABC}\langle S_{i}^{z}\rangle=\sum_{i\in C}\langle
S_{i}^{z}\rangle+\sum_{i\in AB}\langle S_{i}^{z}\rangle$, we then get

$S^{z}_{\text{max}}=N_{\text{C}}\ m_{\text{FE}}^{C}-N_{\text{AB}}\
m_{\text{FE}}^{A}\ \cos\phi,$ (5)

where $2\phi$ is the angle between spins A and B (see inset of Fig. 5). Thus,
we can calculate the angles in the FI phase using

$2\phi=2\arccos\left(\frac{m_{\text{FE}}^{C}-\frac{S^{z}_{\text{max}}}{N_{\text{C}}}}{m_{\text{FE}}^{A}}\right)$
(6)

where we used the fact that $N_{\text{AB}}=N_{\text{C}}$ in all our lattices.
It is important to note that even if this formula is strictly correct only for
the FI phase, it is also gives the proper results for the PD and Néel phases.
In the Néel phase, all sublattices A, B and C show only ferromagnetic peaks.
In that case, we get $S^{z}_{\text{max}}=0$ and
$m_{\text{FE}}^{C}=m_{\text{FE}}^{AB}$ (which should replace
$m_{\text{FE}}^{A}$), and thus $2\phi=0$ indicating that the spins A and B
point all in the same direction and antiparallel to those in sublattice C.
Again, this is consistent with the Néel order over the whole lattice ABC. In
the PD phase, we have to consider $m_{\text{FE}}^{C}=0$. Since
$S^{z}_{\text{max}}=0$ and $m_{\text{FE}}^{A}$ is finite, the formula results
in $2\phi=\pi$, indicating that spins at A and B are antiparallel and
consistent with the AB Néel order.

We show in Fig. 5 the results for the angle obtained from Eq. 6 with the
considerations mentioned above. We can see that the behavior of the angle
between spins A and B is very similar to the classical and semiclassical
solutions, in the sense that there is an almost linear part at low $\alpha$ to
later drops abruptly at large values of $\alpha$. Close to the phase
transition at $\alpha=0.6$, $\cos(\phi)$ takes values around $1$, leading to a
large uncertainty in the calculation of $\phi$. This can be seen in Eq. 6
where small deviation of $S^{z}_{\text{max}}$ from 0 lead to large changes in
$2\phi$.

The key difference between our calculations and the semiclassical result is
that the quantum fluctuations push the semiclassical behavior to higher values
of $\alpha$ and leave a region where the AB sublattice has the same order as
in $\alpha=0$. That is, the orphan spins do not couple to the lattice and form
a collective state. This effect of the zero-point quantum fluctuations is also
observed in the stuffed honeycomb lattice[65]. From our calculations it is
difficult to determine if the angle changes smoothly or not at the critical
point between PD and FI phases. However, the first-order phase transition may
be accompanied by a small discontinuity in the angle. As the C spins order
ferromagnetically with almost classical magnetization close to the critical
point and $\alpha\neq 0$, the AB spins are expected to see this change deviate
by some finite angle from the Néel order.

###  III.4 Structure factor and correlations

So far we have completed the analysis and characterization of the FI phase for
intermediate values of $\alpha$ and the Néel order for larger values of
$\alpha$. We also know that at low values of $\alpha$, in the PD phase, the
Néel order in the sublattice AB remains stable regardless of the strength with
which the C spins are coupled. All it remains, then, is to characterize the
state formed by the C spins in the PD phase.

Figure 6: Spin structure factor $S^{\text{X}}(\mathbf{q})$ normalized by the
value at the antiferromagnetic peak $S^{\text{AB}}(\boldsymbol{\pi})$ for
$\alpha=0.1$. Lattice sizes are $4\times 12$ and $8\times 12$ (top and bottom,
respectively), and different sublattices are AB and C (left and right,
respectively). The first Brillouin zone is delimited by white dashed lines.

To do this, we start by calculating the complete spin structure factors of the
AB and C sublattices, shown on the left and right panels of Fig. 6,
respectively. The results for the $4\times 12$ and $8\times 12$ lattices are
shown in the top and bottom panels, respectively. For both lattices,
$S(\mathbf{q})$ is normalized by the antiferromagnetic peak in the AB
sublattice $S^{\text{AB}}(\boldsymbol{\pi})$. On one hand, the peaks in the AB
sublattice appear at $S(\boldsymbol{\pi})$ and get sharper when the lattice
size increases, signalling the magnetic order towards the thermodynamic limit.
On the other hand, the features in the C sublattice appear at
$(\pm\frac{\pi}{2},0)$ and $(0,\pm\frac{\pi}{2})$ in the first Brillouin zone.
These peaks are consistent with a double-striped order, normally dubbed
$\langle 2\rangle$. Surprisingly enough, $\langle 2\rangle$ is one of the
competing phases predicted for the effective model of C spins interacting due
to quantum fluctuations of the Néel order in the stuffed honeycomb
lattice[66]. However, in the present case the peaks get lower and diffuse when
the lattice size increases, indicating a disordered state with strong quantum
effects. For the $L_{y}=12$ lattices (not shown), the points
$(0,\pm\frac{\pi}{2})$ are not allowed in the Brillouin zone due to the
cylindrical boundary conditions. Peaks at $(\pm\frac{\pi}{2},0)$ could still
appear, but this does not happen. Instead, the results show that stripes tend
to align along the long and non-periodic axis, but are frustrated. The next
available lattice that can host this kind of peaks has $L_{y}=16$. For this
case we are not able to obtain quantitatively precise results and the C
sublattice shows a well-established $\langle 2\rangle$ phase that spuriously
breaks the SU(2) symmetry by having finite and almost classical $\langle
S^{z}_{i}\rangle$. The latter decrease with increasing MPS bond dimension, but
not nearly enough to obtain accurate results. This behavior of the MPS method
is expected when bond dimensions are not large enough to precisely simulate
the given state.

Figure 7: The effect of excitations in the AB sublattice (i.e., two-spin flip,
representing a coherent two-magnon excitation) leading to effective second
nearest neighbors interactions between C spins (green). In (a), the case of
the stuffed honeycomb lattice taken from Ref. [66] and in (b) the analogous
excitations for the present case of the stuffed square lattice. Excited AB
spins are shown in darker colors, and effective field over C spins is shown by
the shaded figures.

The similarity between the stuffed honeycomb and square lattices can be
understood in terms of the same physical processes (in Fig. 7 we illustrate
the similarities). In Ref. [66], the $xy$ ferromagnetic interactions between C
spins arise from the transverse quantum fluctuations of the ordered sublattice
AB. This can be qualitatively understood considering the second order
processes involving a virtual spin-flip of one AB spin connected through the
$J^{\prime}$ bonds to two C spins. In our case, the same process is realized
between first- and second-nearest neighbors in the square C sublattice. On the
other hand, in the stuffed honeycomb lattice, antiferromagnetic Ising
interactions arise from coherent two-magnon processes that connect two C spins
through two AB spins [illustrated Fig. 7(a)]. In our present case, these
processes would lead to the same kind of interactions between third- and
fourth-nearest neighbors in the square lattice C [illustrated in Fig. 7(b)].
Even though this is not a formal derivation, it is to be expected that the
effective Hamiltonian of the C spins in this case is one with $xy$
ferromagnetic interactions to first- and second-nearest neighbors and Ising
antiferromagnetic interactions to third- and fourth-nearest neighbors.

To complement the spin structure factor analysis, we show in Fig. 8 the real-
space correlations to first-, second- and third-nearest neighbors in the
square C sublattice. Ferro- and antiferromagnetic correlations are indicated
by blue and orange, respectively, whereas the strength is indicated by the
thickness of the lines (correlations with absolute values bellow 0.005 are not
shown for simplicity). The average values and standard deviation for each type
of correlation are shown in the numbers between the two lattices. For both the
$4\times 12$ and the $8\times 12$ lattices ($2\times 12$ and $4\times 12$ C
sublattices), the largest correlations are ferromagnetic to nearest neighbors
in one direction and antiferromagnetic to third-nearest neighbors in the
perpendicular direction. This is consistent with our previous qualitative
picture of the effective interactions. Furthermore, as it happens for the case
of the stuffed honeycomb lattice in the low-$S$ limit[66], the
antiferromagnetic correlations are the dominant ones. All these features
reinforce the similarities between the two lattices, and strongly suggest that
these effective interactions are generally originated from a Casimir-like
effect due to zero-point quantum fluctuations of a magnetically ordered
sublattice. However, these weak effects may only be seen in these kind of
models where C spins do not interact directly with one another. As it has been
suggested for the stuffed honeycomb lattice[82], small antiferromagnetic
interactions between the spins in the square sublattice C may be sufficient to
order them and wash away these effects.

Figure 8: First- to third-nearest neighbor correlations in the C sublattice
corresponding to the $4\times 12$ (top) and $8\times 12$ (bottom) lattices for
$\alpha=0.1$. Blue and orange lines indicate ferro- and antiferromagnetic
correlations $\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle$, respectively.
The thickness is proportional to
$|\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle|$. The numbers indicate the
average and standard deviation of the type of correlation along the lattice
(indicated by arrows).

Finally, we show in Fig. 9 the decay of correlations. Taking a leftmost site,
we calculate all the correlations with same-type spins along the $L_{x}$
direction. The results do not change qualitatively if a site at the center of
the system is taken, showing that the edges do not play an important role.
Shown results correspond to the $8\times 12$ lattice, and the differences with
values for different $L_{y}$ positions are all smaller than symbol size. The
correlations used are the same on the left and right panel. On the left panel,
the linear axis allows to identify exponential decays. The C sublattice shows
a clear and strong exponential decay of the correlations through a good linear
fit. This indicates short-range correlations in the C sublattice along the
stripes and a gapped state. This, together with the scaling behavior of the
peaks in the static structure factor, is consistent with a disordered phase.
On the right panel, the logarithmic $x$ axis allows to see power-law decaying
correlations. Ordered magnets are expected to show power-law decaying
correlations. In this panel we see that the AB sublattice shows a good linear
behavior at shorter distances, indicating an ordering tendency towards the
thermodynamic limit. At large distances, the correlations deviate from the
power law and agree better with an exponential decay. This change of behavior
is expected for MPS methods and it is connected to the finite bond
dimension[83]. The difference in the decay law of the correlations for the two
sublattices is a strong indication of a PD phase.

Figure 9: Correlations along the $L_{x}$ direction for the $8\times 12$
lattice at $\alpha=0.1$. Blue and orange indicate the AB and C sublattices,
respectively. Log-lin (left) and log-log (right) scales allow to differentiate
between exponential and power-law decaying correlations. For each case, a
linear fit is performed and shown in lines of the corresponding colors.

However, the antiferromagnetic correlations between third-nearest neighbors in
the perpendicular direction are strong (see Fig. 8). In the $L_{y}=8$ lattices
it is not possible to study the decay of such correlations, because the third-
nearest neighbor along $L_{y}$ is the same in the two directions due to the
cylindrical boundary conditions. This could be the reason for such a high
value of correlations, $-0.6$, which is rather close the $-0.75$ from a
singlet. On the other hand, for the $L_{y}=4$ lattices, the strong
antiferromagnetic correlations to third-nearest neighbors are along the
$L_{x}$ direction. This lattice is too small and not very representative of
the two-dimensional limit to jump to conclusions, but no decay of these
correlations with the distance is observed. Also, ferromagnetic correlations
along $L_{y}$ of exactly $0.25$ could be a signal of a triplet forming between
sites. Altogether, it is difficult to determine the ground state in the
thermodynamic limit from these calculations, and other methods suitable for
larger lattices may be needed. However, we have collected enough evidence
pointing towards a disordered but correlated state. This is consistent with a
correlated PD at low valus of $\alpha$. As in the case of the stuffed
honeycomb lattice[65], short-range correlations inside the disordered
sublattice are mostly ferromagnetic. But here we were able to characterize the
structure of the correlations as a double stripe.

###  III.5 Stability of the partially disordered phase and classical limit

To complete our analysis, we study the stability of the PD phase. In Ref.
[66], an effective Hamiltonian for the central spins in the stuffed honeycomb
lattice was derived. Such Hamiltonian and its corresponding ground state
depend on the value of the spin $S$. The authors found that for small values
of $S$ the ground state is a double stripe $\langle 2\rangle$, whereas for
large $S$ the solution is ferromagnetic. Between the two phases, the
fluctuations increase the corrections to the magnetization, indicating a
possible disordered phase at $S_{c}=0.646$ (close to the quantum case
$S=1/2$). Our solution for $S=1/2$ shows a disordered structure whose
correlations show signatures of the $\langle 2\rangle$ phase. Studying the
same system at higher values $S$ should lead to a ferromagnetic state for the
C sublattice. However, such study is beyond the scope of this article.
Instead, we approach the classical limit by applying a staggered magnetic
field $h$ on the open edges of the AB sublattice, reinforcing the Néel order.
For large enough magnetic fields, the quantum fluctuations above the Néel
order become small, and should induce a ferromagnetic in-plane order
(considering a magnetic field in the $z$ direction).

We show in Fig. 10 the magnetization as a function of the staggered magnetic
field $h$ applied on the edges of the AB sublattice. The full lines show the
antiferromagnetic magnetization $m^{\text{AB}}_{\text{AF}}$ for the AB
sublattice and the ferromagnetic magnetization $m^{\text{C}}_{\text{FE}}$ for
the C sublattice. The results shown correspond to the $8\times 8$ lattice at
$\alpha=0.1$. As $h$ grows, the magnetization of the AB sublattice grows,
getting closer to the classical value; while the ferromagnetic magnetization
of C spins remains close to 0. Eventually, for $h\geq 0.3\,J$, the C spin
structure factor develops a ferromagnetic peak and the ferromagnetic
magnetization $m^{C}_{\text{FE}}$ grows notoriously. At $h=0$, the system
preserves the SU(2) symmetry from the Hamiltonian and the in-plane
contribution represents $2/3$ of the total structure factor, indicating an
equal contribution of every spin direction. However, for $h\neq 0$ the
Hamiltonian has only U(1) symmetry around the $z$ spin axis. For the
sublattice AB, this means that the signal in the $z$ structure factor
increases slowly as $h$ increases (while the signal in $xy$ decreases). On the
other hand, above $h\geq 0.3\,J$, we observe that the C sublattice forms a
ferromagnetic state where the in-plane contributions accounts for almost all
the signal at the ferromagnetic peak (see dashed lines in Fig. 10). This
indicates that the ferromagnetic moment of the C sublattice is indeed in the
$xy$-plane, perpendicular to the Néel order in the AB sublattice, in agreement
with predictions for the effective model on the stuffed honeycomb lattice[66].

Figure 10: The antiferromagnetic magnetization $m^{\text{AB}}_{\text{AF}}$ for
the AB sublattice and the ferromagnetic magnetization
$m^{\text{C}}_{\text{FE}}$ for the C sublattice as a function of the staggered
magnetic field $h$ applied on the AB sublattice. The full lines correspond to
the total magnetization, while the dashed lines correspond to the in-plane
$xy$ and out-of-plane $z$ contributions for the C sublattice. Results
correspond to the $8\times 8$ lattice with $\alpha=0.1$.

##  IV Conclusions

In this article we have studied the stuffed square lattice, which consists of
a square lattice ($J$) with extra spins at the center of each square. These
central spins are connected to the spins in the square lattice by
$J^{\prime}$. Using a parametrization $J=\cos\left(\alpha\pi/2\right)$ and
$J^{\prime}=\sin\left(\alpha\pi/2\right)$, we covered the whole range of
antiferromagnetic interactions with $\alpha\in[0,1]$. Using MPS on ladders
from $L_{y}=4$ to $12$, and $L_{x}$ up to $14$, we found three different
phases in the corresponding quantum phase diagram. These are represented
schematically in Fig. 2.

For low values of $\alpha$, we found a partially disordered but correlated
phase, in analogy to the previously studied stuffed honeycomb lattice[65, 66].
In this phase, the square lattice presents a Néel magnetic order that is
unaffected by the coupling of the central spins $J^{\prime}$. The central
spins, on the other hand, are decoupled from the square lattice but correlated
between themselves even though there is no direct exchange coupling between
them. These effective correlations are induced by a Casimir-like effect, due
to the zero-point quantum fluctuations of the Néel order in the square
lattice, which acts as the medium in which the central spins are submerged.
The effective Hamiltonian of the central spins seems to be dominated by short-
range ferromagnetic (first and second nearest neighbors) and long-range
antiferromagnetic interactions (third and fourth nearest neighbors). The
latter are the dominant correlations, but are not sufficient to develop a
long-range magnetic order in the central-spins sublattice. Instead, a
structure of double-striped correlations appears, which decay exponentially
along the stripes. For finite systems, the ground state in this phase has
total $S^{z}=0$ with finite and very low energy excitations (of order
$J^{\prime}$) belonging to the central-spins sublattice. In the thermodynamic
limit, the square lattice would host gapless magnon excitations corresponding
to the Goldstone modes, which would be the lowest-lying excitations of the
system.

When $\alpha$ grows, there is a first-order phase transition to a
ferrimagnetic state. The exact point of the transition, however, seems to
depend on the system size. In this sense, it is important to note that the
double stripe correlations along the $x$-direction are frustrated in
$L_{y}=12$ because of the periodic boundary conditions, causing the critical
value of $\alpha$ to be particularly lower for this system size (the
ferrimagnetic phase is favored). To confirm the existence of the partially
disordered phase in the thermodynamic limit, we have done calculations for
systems with $L_{y}=16$. Even though these results are not entirely reliable,
we are able to detect a strong jump in the ferromagnetic magnetization of the
sublattice C, signalling a phase transition at $\alpha_{C}=0.175(25)$. All in
all, we can say that the partially disordered phase should exist for
$\alpha\lesssim 0.15$ in the thermodynamic limit.

For intermediate values of $\alpha$, the central spins order ferromagnetically
and induce a canting angle on the square lattice. The resulting ground state
is ferrimagnetic with total spin different from 0. Contrary to most
conventional ferrimagnetic systems, here the total non-zero ferromagnetic spin
emerges solely from antiferromagnetic interactions in a system of equal spins.
The value of $S^{z}_{\text{max}}$ is maximum close to the transition point to
the partially disordered phase, and decreases to $0$ as $\alpha$ grows. At
$\alpha=0.6$, there is a phase transition to a Néel order over the whole
lattice, which is a tilted square lattice dominated by $J^{\prime}$. This
value is in agreement with the predicted phase transition by series expansions
and coupled cluster method calculations[72, 73], $\alpha=0.63$.

Finally, inspired by the calculations in Ref. [66] for the effective
Hamiltonian of the central spins in the stuffed honeycomb lattice, we have
studied the effect of reducing the quantum fluctuations. We did so by applying
a staggered magnetic field reinforcing the magnetic Néel order of the square
lattice. We obtained that a phase transition occurs when quantum fluctuations
are damped, into a phase where the central spins are ferromagnetically
aligned. Furthermore, the ferromagnetic C correlations exist only in the plane
perpendicular to the Néel order in the AB sublattice. This result is
consistent with the large-S results of the effective Hamiltonian for the
stuffed honeycomb lattice.

The most interesting part of the phase diagram lies in the weakly-frustrated
regime at low values of $\alpha$ or $J^{\prime}/J$, as it is the only phase
that does not appear in the semiclassical solution of the model. And it is,
therefore, of purely quantum origin. This adds up to the previous calculations
on the stuffed honeycomb lattice, and shows that the partially disordered
phase is common in these kind of systems. Also, the disordered sublattice is
not trivially disordered and the inner correlations of the central spins
develop a double stripe pattern, originated by the quantum fluctuations from
the ordered sublattice. In conclusion, these kind of systems with weakly
coupled spins to a magnetically ordered lattice provide an interesting
playground to study exotic quantum states.

## Acknowledgments

The authors would like to thank Luis O. Manuel, Urban F. P. Seifert, and
Johannes Reuther for the fruitful discussions. GGB is supported by Slovenian
Research Agency (ARRS) under Grant no. P1-0044 and J1-2458. Part of
computation were performed on the supercomputer Vega at the Institute of
Information Science (IZUM) in Maribor, Slovenia. FTL acknowledges funding from
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in
particular under Germany’s Excellence Strategy – Cluster of Excellence Matter
and Light for Quantum Computing (ML4Q) EXC 2004/1 – 390534769.

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