# Exotic magnetization curves in classical square-kagomé spin lattices

Heinz-Jürgen Schmidt<EMAIL_ADDRESS>Fachbereich
Mathematik/Informatik/Physik, Universität Osnabrück, 49069 Osnabrück, Germany.
Johannes Richter  Institut für Physik, Universität Magdeburg, P.O. Box 4120,
D-39016 Magdeburg, Germany  Max-Planck-Institut für Physik Komplexer Systeme,
Nöthnitzer Straße 38, D-01187 Dresden, Germany

###### Abstract

Classical spin systems with non-coplanar ground states typically exhibit
nonlinear magnetization curves characterized by kinks and jumps. Our article
briefly summarizes the most important related analytical results. In a
comprehensive case study, we then address AF-square kagomé and AF/FM-square
kagomé spin lattices equipped with additional cross-plaquette interactions. It
is known that these systems have non-coplanar ground states that assume a
cuboctahedral structure in the absence of a magnetic field. When a magnetic
field $H$ is switched on, a rich variety of different phases develops from the
cuboctahedral ground state, which are studied in their dependence on $H$ and a
cross-plaquette coupling constant $J_{3}>0$. For the AF square-kagomé spin
lattice, we carefully identify and describe seven phases that appear in a
phase diagram with five triple points. The transitions between these phases
are predominantly discontinuous, although two cases exhibit continuous
transitions. In contrast, the phase diagram of the AF/FM square-kagomé model
shows only four phases with a single triple point, but these also lead to
exotic magnetization curves. Here, too, there are two types of phase
boundaries belonging to continuous and discontinuous transitions.

##  I Introduction

Classical Heisenberg spin models with exclusively non-coplanar ground states
are attractive for various reasons. These non-coplanar ground states break a
chiral $\mathbb{Z}_{2}$ symmetry, leading to phase transitions at finite
temperature, even in two space dimensions, while conventional magnetic order
is, for the Heisenberg models of interest here, subject to the Mermin-Wagner
theorem mermin1966absence . Another motif to study such systems is the
possible route to arriving at quantum spin liquids via frustration-induced
quantum melting of magnetic orders, see Hickey2017 .

Examples of spin systems with non-coplanar ground states can be found in the
recent literature. A non-coplanar ground state called cuboc2 was found in Ref.
Domenge2005_cuboc2  for the classical AF kagomé with ferromagnetic next-
neighbor exchange $J_{1}$ and antiferromagnetic $2$nd-neighbor exchange
$J_{2}$. The analogous cuboc1 phase was first reported in Refs.
janson2008modified ; janson2009intrinsic  for the classical AF kagomé with
antiferromagnetic next-neighbor exchange $J_{1}$ and $3$rd-neighbor exchange
$J_{d}$ along the diagonals of the hexagons. Later, these phases were
systematically analyzed in Ref. Messio_2011_clas_GS , and the notations cuboc1
and cuboc2 were introduced to be explained in more detail in the next
paragraph.

A similar spin system on a lattice with corner-sharing triangles is the
square-kagomé Heisenberg antiferromagnet (SKHAF), that was introduced about
$20$ years ago Sidd2001 ; schnack2001independent ; tomczak2003specific ;
richter2004-spin-peierls ; richter2009squago ; Sakai2013 ; Rousochatzakis2013
; derzhko2014square ; Rousochatzakis2015 ; Sakai2015 ; Derzhko2015review .
Over the past five years, this system has attracted increasing attention in
theoretical Hasegawa2018 ; Morita2018 ; Lugan2019 ; McClarty2020 ;
PhysRevB.102.241115 ; Iqbal2021 ; schmoll2022tensor ; squago_j1j2_2023 ;
squago_clas_2023 ; decorated_squago_2023  as well as experimental FMM:NC20 ;
YSK:IC21 ; Vasiliev2022 ; Vasiliev2023 ; Murtazoev-2023  investigations.

A recent study squago_clas_2023  focuses on the SKHAF with longer-range
Heisenberg couplings $J_{\times}$ and $J_{+}$ across the octagonal plaquettes,
see Fig. 1. The spin sites of the square-kagomé lattice can be divided into
two non-equivalent classes, class A sites, which form the squares, and class B
sites, which are in the middle of the bow ties connecting the squares, see
Fig. 1. The couplings between the A spins in the squares are denoted by
$J_{1}$ and the couplings between the A spins and the B spins by $J_{2}$. For
fully antiferromagnetic (AF) bonds, the spin vectors of the classical ground
state form a cuboctahedron, one of the $13$ Archimedean solids. This ground
state shows angles of $120^{\circ}$ between neighboring spin and is called a
cuboc1 state. By inverting the sign of $J_{2}$ one obtains a variant of the
square-kagomé lattice called AF/FM square-kagomé. Its ground state will be the
so-called cuboc3 state obtained by flipping the B spins of the cuboc1 state.

In this work, we will continue the investigations of squago_clas_2023  by
considering the ground states of the AF and AF/FM square-kagomé lattice in the
presence of a magnetic field $(0,0,H)$ giving rise to an additional Zeeman
term in the Hamiltonian. Important characteristics to be studied are the
magnetization curves $M(H)$ for different values of the coupling constants,
$S_{i}^{z}-$plots of the $z$ components of the ground state spins as a
function of $H$, and the phase diagrams showing different types of ground
states as a function of $H$ and the coupling constants. To keep the
representation simple, we have largely restricted ourselves to the case of
$J_{1}=1$, $J_{2}=\pm 1$ and $J_{\times}=J_{+}=:J_{3}>0$. We then have two
phase diagrams corresponding to $J_{2}=\pm 1$ in the $H-J_{3}-$ plane.

The reason why we expect interesting magnetization curves for a spin system
with a non-coplanar ground state is as follows. A classical spin system with a
coplanar ground state with vanishing magnetization $M=0$ at $H=0$ always has a
linear magnetization curve $M(H)$ with ground states approaching the fully
aligned FM state, similar to the struts of an umbrella when it is folded. For
a spin system with a non-coplanar ground state with $M=0$ for $H=0$, the
analogous construction would also lead to a linear magnetization curve, but
corresponding to a $4-$ dimensional and thus unphysical family of umbrella
states. The necessity of a physical restriction to at most $3-$ dimensional
ground states, on the other hand, leads to a non-linear magnetization curve
that typically exhibits jumps and kinks.

In fact, such “exotic" magnetization curves have so far been described mainly
for spin systems with non-coplanar ground states for $H=0$, see
Coffey_Trugman_92 ; SSS:PRL05 ; Kon:PRB05 ; Kon:PRB07 ; Konstantinidis2018 ;
Konstantinidis2023 . Recently in Ref. ZGZ:AP22  unconventional classical
nonlinear magnetization curves have been found for a frustrated spinel and the
$J_{1}$-$J_{d}$ AF kagomé.

The paper is organized as follows. In Section II we briefly describe the
numerical, semi-analytical and analytical methods used in the paper. Section
III is devoted to the study of the AF square-kagomé. We find seven phases,
additional to the cuboc1 state for $H=0$ and the FM phase for the saturation
regime, that are described in Section III.1 together with typical
magnetization curves and an $S_{i}^{z}-$plot for $J_{3}=0.1$. All phases are
non-coplanar like the cuboc1 ground state for $H=0$. There are discontinuous
as well as continuous phase transitions. Phase IV, which is adjacent to the FM
phase, can be described analytically by a parameter representation, as
explained in the Appendix C. For $J_{3}\lesssim 0.14$ there is a remarkable
phase VII, different from the other ones, with a large, possibly macroscopic
degeneracy.

The phase diagram for the AF/FM square-kagomé model is somewhat simpler,
consisting only of four phases, see Section IV. Also here all phases a non-
coplanar, except the analytical phase IV adjacent to the FM regime. Moreover,
the magnetization curve in phase IV is linear, whereas all other parts of
$M(H)$ are nonlinear.

The above-mentioned results are mainly obtained numerically or semi-
analytically. In the section V and in the appendix B we present some
analytical results for the area under the (normalized) magnetization curve
(“sum rule"), which are valid for general Heisenberg spin systems. Some of
these results are already known, but are scattered in the literature. For
example, this area $F$ is always equal to or larger than the area
${\scriptsize\frac{1}{2}}H_{\rm sat}$ given by the linear magnetization curve.
The latter case, i.e. $F>{\scriptsize\frac{1}{2}}H_{\rm sat}$, which is
typical for exotic magnetization curves, can be predicted by an analysis of
the matrix of coupling constants. Related analytical results on the saturation
field $H_{\rm sat}$ are contained in Appendix A. We close with a Summary and
Outlook in Section VI.

##  II Methods

Let us briefly describe the used methods. Similar as in Ref. squago_clas_2023
we use a Monte-Carlo like approach to get numerical data for the energy
$E_{0}$, the magnetization $M$ and spin-spin correlations
${\mathbf{s}}_{i}\cdot{\mathbf{s}}_{j}$ in the classical ground state.

We also use a semi-analytical approach introduced in squago_clas_2023 , which
works as follows: Starting from a ground state for a system of $N$ classical
spins ${\vec{s}}_{i}$, which we obtain numerically from our Monte Carlo
calculation, we form groups of spin vectors pointing approximately in the same
direction, i. e., satisfying, say, ${\mathbf{s}}_{i}\cdot{\mathbf{s}}_{j}\geq
0.995$. This gives a set of $M$ distinct spin directions, which is further
reduced by guessing their symmetry group. In the end, we have $K$ different
spin directions for the ground state, of which we obtain all others by
applying symmetry operations. Next, we calculate the energy (including the
Zeeman term) of the spin configuration
${\mathcal{K}}(\alpha_{1},\ldots,\alpha_{n})$ as a function of some parameters
$\alpha_{1},\ldots,\alpha_{n}$ describing the position of the remaining $K$
spin vectors. This energy is minimized numerically, starting with the initial
numerical values of the parameters corresponding to the $K$ spin vectors. It
should have become clear that the semi-analytical method cannot be applied
schematically, but requires a certain intuition. Faulty identifications of
closely neighboring spins usually show up in too high ground state energies.
Conversely, a slight lowering of the initial numerical energy is an indication
of a successful application of the method.

In special cases, typically in the last phase before saturation, it is
possible to solve the system of equations
$0=\partial{\mathcal{K}}/\partial\alpha_{i},\,i=1,\ldots,n$ analytically and
thus obtain explicit or parametric expressions for the magnetization curve.
Alternatively, we used a generalized Luttinger-Tisza method clas_GS_JPA_2022
to obtain analytical results for the latter case. This method provides
rigorous results and it can be used to confirm the semi-analytical results.

##  III AF square-kagomé

Figure 1: Left panel: The square-kagomé $J_{1}$-$J_{2}$ model with cross-
plaquette interactions $J_{+}$ and $J_{\rm x}$. We distinguish between sites A
forming the squares (black lines) and sites B at the center of the bow ties
(blue lines).
Right panel: Sketch of phase IV for the AF/FM square-kagomé lattice, see
Section IV.2.4. The black dots at sites B represent spin vectors of the form
${\mathbf{s}}_{i}=(0,0,1)$, and the red/green dots at sites A spin vectors of
the form ${\mathbf{s}}_{i}=(\pm\sqrt{1-z_{1}^{2}},0,z_{1})$, where $z_{1}$ is
given below, see Eq. (5).

Like the kagomé lattice the square-kagomé consists of corner-sharing
triangles, but enclosing squares instead of hexagons. Another difference to
the kagomé lattice is the existence of two non-equivalent sites A (forming the
squares) and B (center of the bow ties) as well as two non-equivalent nearest
neighbor bonds $J_{1}$ and $J_{2}$. The corresponding model is depicted in
Fig. 1, left panel. For numerical and some analytical calculations we have to
approximate the infinite square-kagomé lattice by a finite lattice model
${\mathcal{L}}$ with periodic boundary conditions. We may view
${\mathcal{L}}=\left({\mathcal{V}},{\mathcal{E}}\right)$ as an undirected
graph with vertex set ${\mathcal{V}}=\\{1,\ldots,N\\}$ representing the spin
sites and a set of edges

${\mathcal{E}}={\mathcal{S}Q}\cup{\mathcal{B}T}\cup{\mathcal{C}}\cup{\mathcal{P}}\;,$
(1)

representing, resp., the interacting pairs within the squares, bow ties, the
diagonal cross-plaquette pairs, and the vertical/horizontal ones. Then the
Hamiltonian ${\mathcal{H}}$ (without Zeeman term) can be written as

${\mathcal{H}}=\sum_{(i,j)\in{\mathcal{S}Q}}J_{1}\,{\mathbf{s}}_{i}\cdot{\mathbf{s}}_{j}+\sum_{(i,j)\in{\mathcal{B}T}}J_{2}\,{\mathbf{s}}_{i}\cdot{\mathbf{s}}_{j}+\sum_{(i,j)\in{\mathcal{C}}}J_{\times}\,{\mathbf{s}}_{i}\cdot{\mathbf{s}}_{j}+\sum_{(i,j)\in{\mathcal{P}}}J_{+}\,{\mathbf{s}}_{i}\cdot{\mathbf{s}}_{j}\;,$
(2)

see Fig. 1, left panel.

Recall that the zero-field cuboc ground-state phase is present in the entire
parameter region $J_{+},J_{\rm x}>0$ independent of the magnitudes of $J_{+}$
and $J_{\rm x}$, see Fig. 5 in Ref. squago_clas_2023 . For the sake of
simplicity we therefore consider only the symmetric case $J_{+}=J_{\rm
x}=J_{3}>0$ and further set $J_{1}=J_{2}=1$.

###  III.1 Phase diagram

Figure 2: Phase diagram for the AF square-kagomé model in the $H-J_{3}$-plane,
where $H$ denotes the magnetic field and $J_{3}$ the additional cross-
plaquette coupling. We have chosen $0\leq J_{3}\leq 2$ and $0\leq H\leq
12.8284...$. There are seven phases I - VII, additional to the cuboc1 ground
state for $H\to 0$ and the ferromagnetic ground state FM for $H\geq H_{\rm
sat}$ and five triple points (cyan color). The solid curves represent
discontinuous phase transitions and the dashed lines continuous ones. For the
region $1.5\leq H\leq 3$ and $0\leq J_{3}\leq 0.3$ we have added an
enlargement in the form of an inset. Typical common-origin plots for all seven
phases have been inserted such that the corresponding point in the
$H-J_{3}$-plane is indicated by a blue dot.

In the phase diagram (Fig. 2) we have chosen $0\leq J_{3}\leq 2$ and $0\leq
H\leq 10+2\sqrt{2}=12.8284...$ according to the value of $H_{\rm
sat}=4+3J_{3}+\sqrt{4+J_{3}^{2}}$, see (16), for $J_{3}=2$. There are seven
phases I - VII, additionally to the cuboc1 ground state for $H\to 0$ and the
ferromagnetic ground state FM for $H\geq H_{\rm sat}$. We obtain two
continuous transitions (additional to the transition IV-FM), corresponding to
the phase boundary between IV and VII and the phase boundary between II and
III. In both cases of continuous transitions, we have a subgroup relationship
between the symmetry groups of the phases involved, as it must necessarily be.
The magnetization jump $\Delta M$ at the phase boundary between I and II
varies over three decades and, for $J_{3}\gtrsim 1.6$, is hardly visible in
the numerically calculated magnetization curves. We observe five triple points
$T_{i},i=1,\ldots,5$, where three phase boundaries intersect and three phases
touch, see the cyan points in Fig. 2 and Table 1. As a consequence there are
six types of magnetization curves, see Table 1 and Fig. 3.

Table 1: Coordinates of the five triple points $T_{i},\,i=1,\ldots,5$ in the phase diagram (Fig. 2) and the type of magnetization curves for values of $J_{3}$ between the triple points. When listing the jumps and kinks in the magnetization curve we always neglect the kink at $H=H_{\rm sat}$.  $i$ | $H=h_{i}$ | $J_{3}=j_{i}$ |  | interval | jumps (${\mathcal{J}}$) and kinks (${\mathcal{K}}$)
---|---|---|---|---|---
1 | 2.02593 | 0.80768 |  | $j_{1}<J_{3}$ | ${\mathcal{J}}\,{\mathcal{K}}\,{\mathcal{J}}$
2 | 2.1556 | 0.294157 |  | $j_{2}<J_{3}<j_{1}$ | ${\mathcal{J}}\,{\mathcal{J}}$
3 | 2.15822 | 0.246352 |  | $j_{3}<J_{3}<j_{2}$ | ${\mathcal{J}}$
4 | 2.26488 | 0.13962 |  | $j_{4}<J_{3}<j_{3}$ |  ${\mathcal{J}}$ ${\mathcal{J}}$
5 | 2.30494 | 0.116264 |  | $j_{5}<J_{3}<j_{4}$ |  ${\mathcal{J}}$ ${\mathcal{J}}$ ${\mathcal{K}}$
|  |  |  | $0<J_{3}<j_{5}$ |  ${\mathcal{J}}$ ${\mathcal{J}}$ ${\mathcal{J}}$ ${\mathcal{K}}$

###  III.2 Description of single phases

Here we describe the seven phases of the phase diagram in Fig. 2. The
corresponding “common-origin plots" in this Figure provide a graphical
illustration of their spin structure. In a common-origin plot, the entire
ground state configuration of spins in real space is concentrated into a point
cloud on a single unit sphere. We also show one representative example of the
field dependence of the $z$-components of the spin vectors of the ground state
configuration in Fig. 4.

####  III.2.1 Phase I

We observe $14$ different spin vectors, two of which occupy the north and
south pole of the unit sphere and the remaining $12$ are distributed on three
different circles with constant $z$ in groups of four. These groups form
rectangles with parallel edges. Hence the symmetry group of this phase is
$D_{2}$, the symmetry group of the rectangle. For $H\to 0$ the points of the
center rectangle merge in pairs, and the resulting $12$ spin vectors form a
cuboctahedron. We show an example corresponding to $H=1.5$ and $J_{3}=1$, see
Fig. 2. In every finite model of the square-kagomé lattice with $N$ spin
sites, $N=6\,L^{2},\;L$ even, $10$ spins occupy sub-lattices with $N/12$ sites
and four spins occupy sub-lattices with $N/24$ sites. The partial merging for
$H\to 0$ results in the $12$ sublattices of the cuboc1 state, each with $N/12$
sites.

####  III.2.2 Phase II

We observe $14$ different spin vectors, which are pairwise distributed on six
different circles with constant $z$ and two further spin vectors with
different $z$, see the example in Figure 2 for $H=1.5$ and $J_{3}=1$. There is
a reflection symmetry which can be described as $y\leftrightarrow-y$ upon a
suitable choice of the coordinate system and hence the symmetry group is
$D_{1}$.

####  III.2.3 Phase III

If $H$ is increased beyond a critical value, the eight circles of phase II
merge into four circles with constant $z_{i},\,i=1,\ldots,4,$ and thus form a
continuous transition to another phase called III. Three of these circles
contain rectangles with parallel edges formed by spin vectors and the fourth
circle contains a pair of antipodal spin vectors, also parallel to the edges
of the rectangles, see the example in Figure 2 for $H=2.4$ and $J_{3}=1$. This
spin configuration has thus $D_{2}$-symmetry. Since $z_{1}$ and $z_{3}$ are
very close to each other, the corresponding circles cannot be distinguished by
eye.

####  III.2.4 Phase IV

Further increasing $H$ (e. g. beyond $H=2.8$ at $J_{3}=1.0$) we obtain a new
phase denoted by IV. It consists of eight different spin vectors, which are
distributed on two circles with constant $z=z_{i},\,i=1,2$, see the example in
Figure 2 for $H=8$ and $J_{3}=1$. The two groups of spin vectors form two
squares twisted by $45^{\circ}$. Hence this ground state has a $D_{4}$
symmetry.

This phase can be analytically described by a parameter representation.
Relevant quantities as magnetic field, $z_{1}$, $z_{2}$, magnetization and
energy can be expressed as functions of a parameter $w$ that varies between
$1$ and $w_{max}(J_{3})$. These functions are specified in Appendix C.

####  III.2.5 Phase V

For small values of $J_{3}$, by increasing $H$ we enter a new phase V directly
from phase I (e. g. for $J_{3}=0.2$ at $H=2.10188$), where phase V is similar
to phase II. We observe $14$ different spin vectors, which are pairwise
distributed on seven different circles with constant $z$, see the example in
Figure 2 for $H=2.15$ and $J_{3}=0.2$. There is a $D_{1}$ reflection symmetry
which can be described as $y\leftrightarrow-y$ upon a suitable choice of the
coordinate system.

####  III.2.6 Phase VI

Below the triple point $T_{5}$ phase V gives way for a new phase IV (e. g. for
$J_{3}=0.1$ at $H=2.387$). We observe $14$ different spin vectors, which are
pairwise distributed on seven different circles with constant $z$ forming
pairs, see the example in Figure 2 for $H=2.3$ and $J_{3}=0.1$. The symmetry
of this phase is the rotation of the $x-y$-plane with $180^{\circ}$, or
$(x,y)\leftrightarrow(-x,-y)$ and hence its symmetry group of order $2$ can be
chosen as $C_{2}$ in order to distinguish it from the reflection symmetry
group $D_{1}$ of phase II.

Figure 3: According to Table 1 and the phase diagram in Figure 2 there are six
different types of magnetization curves, of which we show typical sections
that were calculated numerically. From top to bottom, $J_{3}$ assumes the
values $J_{3}=1.5,0.5,0.25,0.2,0.12,0.05$. In the cases where the
magnetization jumps or kinks are not visible, we have inserted enlargements in
the form of insets. The two insets for $J_{3}=0.12$ and $J_{3}=0.05$ are based
on semi-analytical calculations.

####  III.2.7 Phase VII

For $J_{3}=0.1$ and $2.387<H<2.47456$ we obtain a new, disordered phase
denoted by VII. It has a large degeneracy, as shown by Figure 4, reminiscent
of the spin liquid for $J_{3}=H=0$, see squago_clas_2023 . For the semi-
analytical calculation we have randomly picked out a variety of ground state
configurations of this phase and found eight different spin vectors, such that
four of them are arbitrarily positioned on the unit sphere and the remaining
four spin vectors forming antipodal pairs, see the example in Figure 2 for
$H=2.45$ and $J_{3}=0.1$. The symmetry group of this phase is thus reduced to
the identity. Both groups of spin vectors have strictly separated $z$-values.
As $H$ increases and approaches the phase boundary between phase VII and IV,
the $z$ values of the first group (with uncorrelated $z$-values) converge
towards some value $z_{I}>0$, while the $z$ values of the second group
(forming antipodal pairs) converge towards $z_{II}=0$, see Figure 4. This
numerical finding suggests the approach of analytically determining the phase
boundary VII/IV by the condition that one of the two $z$-values in phase IV is
equal to $0$. The result

$J_{3}=1-\frac{2}{\sqrt{8-\frac{H^{2}}{2}}}$ (3)

agrees very well with the semi-analytically calculated phase boundary. It
follows that the first, non-zero limit value for $z$ is

$z_{I}=\frac{\sqrt{\left(J_{3}-2\right)J_{3}+\frac{1}{2}}}{1-J_{3}}\;.$ (4)

For $J_{3}=0.1$ used in Fig. 4 we have $z_{I}=0.61864$. The corresponding
phase transition is continuous.

Figure 4: The $z$-components of the spin vectors of the ground state
configuration semi-analytically calculated for $J_{1}=J_{2}=1$, $J_{3}=0.1$
and $1.5\leq H\leq 3.0$. We can distinguish all five phases I,V,VI,VII,IV,
similarly as in the magnetization curve, see Figure 3, lower right panel. Note
the large degeneracy of the ground states of phase VII for $2.387<H<2.47456$,
of which we have only drawn a selection.

##  IV AF/FM square-kagomé

The AF/FM square-kagomé is obtained for $J_{1}>0$ and $J_{2}<0$, i. e. for AF
couplings within the squares and FM couplings within the bow ties connecting
the squares. If the magnetic field vanishes, the AF/FM case can be reduced to
the AF case by simultaneously inverting the sign of $J_{2}$ and flipping the
spin vectors at the sites B (sitting in the center of the bow ties), leaving
the Hamiltonian unchanged. Consequently, the ground state in the quadrant
$J_{\times},J_{+}>0$ will be the cuboc3 spin configuration obtained from the
cuboc1 ground state of the AF square-kagomé by flipping the spins at sites B,
see squago_clas_2023 . However, the Zeeman term is not invariant under this
transformation, so that the magnetization of the AF/FM square kagomé must be
investigated separately. For this we set $J_{1}=1$ and $J_{2}=-1$ and restrict
ourselves to the diagonal $J_{\times}=J_{+}=:J_{3}>0$.

###  IV.1 Phase diagram

Figure 5: Phase diagram for the AF/FM square-kagomé model in the
$H-J_{3}$-plane, where $H$ denotes the magnetic field and $J_{3}$ the
additional cross-plaquette coupling. We have chosen $0\leq J_{3}\leq 2$ and
$0\leq H\leq 6$. In addition to the cuboc3 ground state for $H\to 0$ and the
ferromagnetic ground state FM for $H\geq H_{\rm sat}$ there are four phases I
- IV and one triple point (cyan color). The solid curves represent
discontinuous phase transitions and the dashed ones continuous phase
transitions.

In the phase diagram, see Fig. 5, we have chosen $0\leq J_{3}\leq 2$ and
$0\leq H\leq 6$ according to the value of $H_{\rm sat}=2(1+J_{3})$, see (18),
for $J_{3}=2$. There are four phases I - IV, additionally to the cuboc3 ground
state for $H\to 0$ and the ferromagnetic ground state FM for $H\geq H_{\rm
sat}$. We found two continuous transitions (additional to the transition IV -
FM) corresponding to the phase boundaries between II and III and between III
and IV. In both cases there is a symmetry group inclusion of the form
$D_{1}\subset D_{2}$. We obtain one triple point at
$H=h_{1}=0.974276,\,J_{3}=j_{1}=1.52014$, where the three phases I,II, III
touch, see the cyan point in Fig. 5. As a consequence there are two types of
magnetization curves, one with one jump and one kink for $J_{3}>j_{1}$ and one
with one jump and two kinks for $J_{3}<j_{1}$, see Fig. 6. Two typical plots
of the $z$-components of all ground state spin vectors depending on $H$ are
shown in Fig. 7.

Figure 6: According to the phase diagram in Figure 5, there are two different
types of magnetization curves. We show typical $M(H)$ curves for the values
$J_{3}=1.0$ and $J_{3}=1.75$ that were calculated numerically. In the case
$J_{3}=1.75$ we have added an enlargement in the form of an inset to make the
jump and the kink in the magnetization curve more clearly visible.  Figure 7:
The $z$-components of the spin vectors of the ground state configuration semi-
analytically calculated for $J_{1}=-J_{2}=1$, $0\leq H\leq H_{\rm sat}$. and
two different values of $J_{3}$.
Left panel: $J_{3}=0.5$. We can distinguish the phases I-III-IV, similarly as
in the left magnetization curve of Fig. 6.
Right panel: $J_{3}=1.95$. Here also the phase II is visible and the
continuous transition to phase III, see the enlargement in the inset.

###  IV.2 Description of single phases

Here we describe the four phases of the phase diagram in Figure 5. The
corresponding common-origin plots in this Figure provide a graphical
illustration of their spin structure.

####  IV.2.1 Phase I

For small $H$ we observe a ground state configuration, called phase I,
consisting of $14$ spin vectors with four different $z$ values
$z_{i},\,i=1,\ldots,4$. They form three rectangles with parallel edges that
can be chosen parallel to the $x$\- and $y$-axis and a pair of the form
$(\pm\sqrt{1-z_{4}^{2}},0,z_{4})$. The symmetry group of phase I is hence
$D_{2}$. For $H\to 0$, one of the rectangles degenerates into a pair of spin
vectors and the resulting $12$ vectors form the cuboc3 ground state mentioned
above.

####  IV.2.2 Phase III

This phase is adjacent to phase I below the triple point, i. e., when
$J_{3}<j_{1}=1.52014$ and has the same structure as phase I, i. e., consisting
of three rectangles and one pair, also with the symmetry group $D_{2}$, but a
different energy. When approaching the boundary to phase IV, see below, one of
the rectangles and the pair of spin vectors converge towards the north pole
$(0,0,1)$ of the unit sphere and the other two rectangles approach two vectors
of the form $(\pm\sqrt{1-z_{1}^{2}},0,z_{1})$, resulting in a continuous
transition to phase IV.

####  IV.2.3 Phase II

Above the triple point, i. e., when $J_{3}>j_{1}=1.52014$, there exists
another phase, called phase II, located between phase I and phase III. It
consists of $14$ spin vectors forming six pairs related by the symmetry
$y\leftrightarrow-y$, if we choose the coordinated system correspondingly, and
two further spin vectors with $y=0$. The symmetry group is thus $D_{1}$.

####  IV.2.4 Phase IV

This phase is the only one with coplanar spin structure and consequently the
corresponding part of the magnetization curve is linear. It can be
analytically described and consists of three different sublattices of spin
vectors of the form ${\mathbf{s}}_{i}=(0,0,1)$ and
${\mathbf{s}}_{i}=(\pm\sqrt{1-z_{1}^{2}},0,z_{1})$ distributed on the spin
lattice according to Fig. 1, right panel. Hence phase IV has thus the same
symmetry group $D_{1}$ as phase II. The various quantities as $z_{1}$,
magnetization $M/N$ and energy $E$ are given as functions of $H$ and $J_{3}$
by

$\displaystyle z_{1}$ $\displaystyle=$
$\displaystyle\frac{H+2}{2\left(J_{3}+2\right)}\;,$ (5) $\displaystyle M/N$
$\displaystyle=$ $\displaystyle\frac{H+J_{3}+4}{3\left(J_{3}+2\right)}\;,$ (6)
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{(H+2)\left(H+2J_{3}+6\right)}{6\left(J_{3}+2\right)}\;.$
(7)

For $H\to H_{\rm sat}=2(1+J_{3})$ it follows that $z_{1}\to 1$ and $M/N\to 1$,
i. e., the ground state approaches the FM state.

##  V Sum rule for magnetization curves

In this section we will formulate some results on the area below the
magnetization curves for the square-kagomé systems that we consider in this
article and which are of relevance for the exotic character of these curves.
More general results that hold for general classical Heisenberg spin systems
are proven in Appendix B.

For a linear magnetization curve $M/N=\frac{H}{H_{\rm sat}}$ (dotted line in
Figure 8) the area $F$ below the graph of $M/N$ is $\frac{1}{2}H_{\rm sat}$.
In general, one can show that $F\geq\frac{1}{2}H_{\rm sat}$, see Appendix B.
Hence $F>\frac{1}{2}H_{\rm sat}$ implies that the magnetization curve cannot
be completely linear and may show exotic features as jumps or kinks.

For the AF square-kagomé with $J_{1}=J_{2}=1$ and $J_{+}=J_{\times}=:J_{3}$ we
have the following result:

$\Delta F:=F-\frac{1}{2}\,H_{\rm
sat}=-1+\frac{1}{2}\sqrt{J_{3}^{2}+4}+\frac{J_{3}}{6}>0\;,$ (8)

for all $J_{3}>0$ thus indicating exotic magnetization curves. To give a
numerical example: For $J_{3}=1$, corresponding to Figure 8, we have $H_{\rm
sat}=12.8284$ and the numerical integration of the magnetization curve yields
$\Delta F=0.2847003$, very close to the analytical value of $\Delta
F=\frac{1}{6}\left(3\sqrt{5}-5\right)=0.2847006554\ldots$.

For AF/FM square-kagomé systems with $J_{1}=-J_{2}=1$ and $0\leq
J_{+}=J_{\times}=:J_{3}<2$ the analogous result reads:

$\Delta F:=F-\frac{1}{2}\,H_{\rm sat}=\frac{1}{2}\left(2-J_{3}\right)>0\;,$
(9)

To give an example: For $J_{3}=1/2$ we have $H_{\rm sat}=3$ according to
$H_{\rm sat}=2\left(1+J_{3}\right)$, see (18). The numerical integration of
the magnetization curve yields $\Delta F=0.50002$, very close to the
analytical value of $\Delta F=1/2$.

Figure 8: Magnetization curve for the AF square-kagomé and coupling constants
$J_{1}=J_{2}=J_{3}=1$, the area $F$ below the magnetization curve and the
complementary area $G$. The area $F\approx 4.90273$ is larger than the area
$\frac{1}{2}H_{\rm sat}\approx 4.618$ below the linear magnetization curve
(dotted line).

##  VI Summary and Outlook

Spin systems with non-coplanar ground states are expected to have non-linear
magnetization curves with kinks and jumps. In general, magnetization curves
can only be studied numerically or semi-analytically, but some features can be
determined analytically, e.g. the saturation field, the area under the
magnetization curve and sometimes the last part of the curve just before
saturation. We have compiled the corresponding analytical results in this
paper.

As an extended case study, we have further investigated the AF square-kagomé
and AF/FM square-kagomé lattices which are known to have exclusive non-
coplanar ground states with cuboctahedral structure when the magnetic field
vanishes. Indeed we found a diverse spectrum of ground state phases depending
on the magnetic field $H$ and a certain coupling constant $J_{3}>0$. The
latter describes the strength of the additional cross-plaquette interactions.

For the AF square-kagome, we were able to identify seven phases with five
triple points and describe their structure in detail. The transitions between
these phases are mostly discontinuous, but in two cases also continuous. They
can be characterized more precisely with so-called $S_{i}^{z}-$plots showing
the $z$-components of all ground state spin vectors as a function of $H$. In
contrast, the AF/FM square kagome has only four phases with one triple point,
but also shows exotic magnetization curves with continuous as well as
discontinuous phase transitions.

The question can be raised whether our results are typical for spin systems
with cuboc ground states. As mentioned in the Introduction, cuboc phases were
originally found in variants of the AF kagomé lattice. Therefore, the
investigation of magnetization curves etc. for these models would be an
obvious extension of the present work. Corresponding studies are in
preparation.

## Acknowledgment

We thank Martin Gembé, Ciarán Hickey, Yasir Iqbal and Simon Trebst for
instructive discussions in connection with a common project on the square-
kagomé spin lattices, from which the topic of the present paper developed.

##  Appendix A Saturation fields

When the magnetic field $H$ reaches or exceeds a certain saturation value
$H_{\rm sat}$, all spins align parallel to the direction of the magnetic
field. In calculating $H_{\rm sat}$ for the systems considered in this work we
have followed the approach presented in HJS_classical_I ; HJS_classical_IV ;
clas_GS_JPA_2022 , which will be briefly recapitulated here. This approach is
limited to finite spin systems but it turns out that the results for $H_{\rm
sat}$ are independent of the system size (provided $N$ is not too small) and
hence also hold for the infinite square-kagomé spin lattice.

Thus we consider the matrix ${\mathbbm{J}}$ of coupling constants $J_{ij}$
(with vanishing diagonal entries) and introduce the dressed coupling matrix
${\mathbbm{J}}({\boldsymbol{\lambda}})$ defined by

${\mathbbm{J}}({\boldsymbol{\lambda}})_{ij}:=J_{ij}+\lambda_{i}\,\delta_{ij}\quad\mbox{for}\quad
i,j=1,\ldots,N\;,$ (10)

with variable diagonal entries $\lambda_{i}$ subject to the condition
$\sum_{i}\lambda_{i}=0$. The choice of ${\boldsymbol{\lambda}}$ will be often
referred to as a gauge. The energy of the spin configuration is independent of
the gauge. The minimal eigenvalue of ${\mathbbm{J}}({\boldsymbol{\lambda}})$
will be denoted by $j_{\rm min}({\boldsymbol{\lambda}})$ and the absolute
ground state energy is given by

$E_{\rm min}/N=j_{\rm min}(\widehat{\boldsymbol{\lambda}})\;,$ (11)

where $\widehat{\boldsymbol{\lambda}}$ is the ground state gauge. Another
useful gauge is the homogeneous gauge $\widetilde{\boldsymbol{\lambda}}$
defined as follows. Consider the mean row sum of the undressed $J$-matrix
${\mathbbm{J}}$

$\tilde{\jmath}:=\frac{1}{N}\sum_{ij}J_{ij}\;,$ (12)

and define

$\widetilde{\lambda}_{i}:=\tilde{\jmath}-\sum_{j}J_{ij}\quad\mbox{for}\quad
i=1\ldots,N\;.$ (13)

It follows that $\tilde{\jmath}$ will be an eigenvalue of
${\mathbbm{J}}(\widetilde{\boldsymbol{\lambda}})$ corresponding to the
eigenvector $(1,1,\ldots,1)$ representing a fully aligned spin configuration.

Then the saturation field is given by

$H_{\rm sat}=2\left(\tilde{\jmath}-j_{\rm
min}(\widetilde{\boldsymbol{\lambda}})\right)\;,$ (14)

see HJS_classical_IV . Especially, if $\tilde{\jmath}=j_{\rm
min}(\widetilde{\boldsymbol{\lambda}})$ and hence $H_{\rm sat}=0$ the spin
system will assume the ferromagnetic ground state
$\uparrow\uparrow\ldots\uparrow$ even for vanishing fields and hence has been
called “ferromagnetic" in HJS_classical_IV .

As an example we consider the AF square-kagomé with $J_{1}=J_{2}=1$ and
$J_{\times},J_{+}\geq 0$. It can be shown that

$H_{\rm
sat}=\left\\{\begin{array}[]{r@{\quad:\quad}l}2(3+J_{\times})&J_{\times}\geq
2\,J_{+},\\\
4+J_{\times}+2J_{+}+\sqrt{\left(J_{\times}-2J_{+}\right){}^{2}+4}&J_{\times}\leq
2\,J_{+},\\\ \end{array}\right.\;.$ (15)

For the special case $J_{+}=J_{\times}=:J_{3}\geq 0$ considered in this paper
this equation specializes to

$H_{\rm sat}=4+3J_{3}+\sqrt{4+J_{3}^{2}}\;.$ (16)

The analogous result for the AF/FM square-kagomé with $J_{1}=-J_{2}=1$ and
$J_{+},J_{\times}\geq 0$ reads

$H_{\rm
sat}=\left\\{\begin{array}[]{r@{\quad:\quad}l}2(1+J_{\times})&J_{\times}\geq
2\,J_{+}-2,\\\
{J_{\times}}+2{J_{+}}-2+\sqrt{({J_{\times}}-2{J_{+}}+2)^{2}+4}&J_{\times}\leq
2\,J_{+}-2,\\\ \end{array}\right.\;.$ (17)

For the special case $0\leq J_{+}=J_{\times}=:J_{3}\leq 2$ considered in the
examples in this paper this equation specializes to

$H_{\rm sat}=2\left(1+J_{3}\right)\;.$ (18)

##  Appendix B Sum rule for the magnetization curve of general Heisenberg spin
systems

In this section we will formulate and prove some generalization of the results
in Section V for general classical Heisenberg spin systems. Some of these
results already appear in HJS_classical_IV . The notation to be introduced
only applies to this section. Analogous results apply to quantum spin systems,
but here the magnetization curve is a step function and its area can be
expressed as a sum. This explains the wording “sum rule".

We consider the magnetization per site $\mu=M/N$ as a function $\mu(h)$ of the
magnetic field $h$ for $0\leq h\leq h_{\rm sat}$ and its inverse function by
$h(\mu)$. The minimal energy per site (without Zeeman term) for given $\mu$
will be denoted by $\epsilon(\mu):=E_{\rm min}(M)/N$. It follows that

$h(\mu)=\frac{\partial\epsilon(\mu)}{\partial\mu}\;.$ (19)

The total energy per site (with Zeeman term) will be denoted by

${\sf H}(h):=\epsilon(\mu(h))-\mu(h)\,h\;.$ (20)

Hence ${\sf H}(h)$ is the Legendre transform of $\epsilon(\mu)$, see
HJS_classical_IV , and

$\frac{\partial{\sf H}(h)}{\partial h}=-\mu(h)\;.$ (21)

Cum grano salis, this also applies if $\mu(h)$ is not differentiable for a
finite number of points, as in the examples of exotic magnetization curves
considered in this paper, see Figure 8.

The area $F$ below the graph of $\mu(h)$ is given by

$\displaystyle F$ $\displaystyle=$ $\displaystyle\int_{0}^{h_{\rm
sat}}\mu(h)\,dh\stackrel{{\scriptstyle(\ref{Hmu})}}{{=}}-\int_{0}^{h_{\rm
sat}}\frac{\partial{\sf H}(h)}{\partial h}\,dh={\sf H}(0)-{\sf H}(h_{\rm
sat})$ (22) $\displaystyle=$
$\displaystyle\epsilon(\mu(0))-\epsilon(\mu(h_{\rm sat}))+\mu(h_{\rm
sat})\,h_{\rm sat}=\epsilon(\mu(0))-\epsilon(1)+h_{\rm sat}\;,$ (23)

since $\mu(h_{\rm sat})=1$. The complementary area $G$ above the graph of
$\mu(h)$, see Figure 8, is given by

$G:=\int_{\mu(0)}^{1}h(\mu)\,d\mu=h_{\rm
sat}-F=-\epsilon(\mu(0))+\epsilon(1)\;.$ (24)

Recall the definitions and results given in Appendix A. Then the absolute
ground state energy is given by

$\epsilon(\mu(0))=j_{\rm min}(\widehat{\boldsymbol{\lambda}})\;,$ (25)

where $\widehat{\boldsymbol{\lambda}}$ is the ground state gauge. Moreover,
for the fully aligned FM state we obtain

$\epsilon(1)=\frac{1}{N}\sum_{ij}J_{ij}=\tilde{\jmath}\;,$ (26)

the mean row sum of the undressed $J$-matrix. Recall that the saturation field
is given by

$h_{\rm sat}=2\left(\tilde{\jmath}-j_{\rm
min}(\widetilde{\boldsymbol{\lambda}})\right)\;,$ (27)

see (27). This entails

$\displaystyle F$ $\displaystyle\stackrel{{\scriptstyle(\ref{F2})}}{{=}}$
$\displaystyle\epsilon(\mu(0))-\epsilon(1)+h_{\rm
sat}\stackrel{{\scriptstyle(\ref{gsenergy},\ref{meanrs},\ref{hsata})}}{{=}}j_{\rm
min}(\widehat{\boldsymbol{\lambda}})-\tilde{\jmath}+2\left(\tilde{\jmath}-j_{\rm
min}(\widetilde{\boldsymbol{\lambda}})\right)$ (28) $\displaystyle=$
$\displaystyle\frac{1}{2}\,h_{\rm sat}+\left(j_{\rm
min}(\widehat{\boldsymbol{\lambda}})-j_{\rm
min}(\widetilde{\boldsymbol{\lambda}})\right)\;.$ (29)

In the case $j_{\rm min}(\widehat{\boldsymbol{\lambda}})=j_{\rm
min}(\widetilde{\boldsymbol{\lambda}})$ it follows that
$F=G=\frac{1}{2}\,h_{\rm sat}$. Especially this holds in the “parabolic" case
where $\mu(h)$ is linear, i. e., $\mu(h)=\frac{h}{h_{\rm sat}}$ (dotted line
in Figure 8). In general, we have $F\geq\frac{1}{2}\,h_{\rm sat}$, since
$j_{\rm min}(\widehat{\boldsymbol{\lambda}})\geq j_{\rm
min}(\widetilde{\boldsymbol{\lambda}})$, see HJS_classical_I . The condition
$j_{\rm min}(\widehat{\boldsymbol{\lambda}})>j_{\rm
min}(\widetilde{\boldsymbol{\lambda}})$, or, equivalently,
$F>\frac{1}{2}\,h_{\rm sat}$ is hence an indication for an exotic
magnetization curve, but, strictly speaking, it is neither necessary nor
sufficient.

###  B.1 Sum rule for the AF square-kagomé

In order to apply this to the AF square-kagomé with $J_{1}=J_{2}=1$ and
$J_{\times},J_{+}\geq 0$ we have to make a case distinction.

####  B.1.1 Case of $J_{\times}<2J_{+}$

We have the following results:

$\displaystyle j_{\rm min}(\widehat{\boldsymbol{\lambda}})$ $\displaystyle=$
$\displaystyle-1-\frac{1}{3}\left(J_{\times}+J_{+}\right),$ (30)
$\displaystyle j_{\rm min}(\widetilde{\boldsymbol{\lambda}})$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left(-3\sqrt{(2J_{+}-J_{\times})^{2}+4}-4J_{+}-J_{\times}\right),$
(31) $\displaystyle\Delta F$ $\displaystyle:=$ $\displaystyle
F-\frac{1}{2}\,h_{\rm
sat}=-1+\frac{1}{2}\sqrt{(2J_{+}-J_{\times})^{2}+4}+\frac{1}{6}\left(2J_{+}-J_{\times}\right)>0\;,$
(32)

thus indicating exotic magnetization curves for all $J_{\times}<2\,J_{+}$.
(30) follows from eq. (9) in Ref. squago_clas_2023  and (25).

####  B.1.2 Case of $J_{\times}\geq 2J_{+}$

We have the following results:

$\displaystyle j_{\rm min}(\widehat{\boldsymbol{\lambda}})$ $\displaystyle=$
$\displaystyle-1-\frac{1}{3}\left(J_{\times}+J_{+}\right),$ (33)
$\displaystyle j_{\rm min}(\widetilde{\boldsymbol{\lambda}})$ $\displaystyle=$
$\displaystyle-1+\frac{1}{3}\left(J_{+}-2J_{\times}\right),$ (34)
$\displaystyle\Delta F$ $\displaystyle:=$ $\displaystyle F-\frac{1}{2}\,h_{\rm
sat}=\frac{1}{3}\left(J_{\times}-2\,J_{+}\right)\geq 0\;,$ (35)

thus indicating exotic magnetization curves for all $J_{\times}>2\,J_{+}$.

###  B.2 Sum rule for the AF/FM square-kagomé

To state the analogous results for the AF/FM square-kagomé with
$J_{1}=-J_{2}=1$ and $J_{\times},J_{+}\geq 0$ we make the following case
distinction.

####  B.2.1 Case of $J_{\times}>2J_{+}-2$

We have the following results:

$\displaystyle j_{\rm min}(\widehat{\boldsymbol{\lambda}})$ $\displaystyle=$
$\displaystyle-1-\frac{1}{3}\left(J_{\times}+J_{+}\right),$ (36)
$\displaystyle j_{\rm min}(\widetilde{\boldsymbol{\lambda}})$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(-5-2J_{\times}+J_{+}\right),$ (37)
$\displaystyle\Delta F$ $\displaystyle:=$ $\displaystyle F-\frac{1}{2}\,h_{\rm
sat}=\frac{1}{3}\left(J_{\times}-2\,J_{+}+2\right)>0\;,$ (38)

thus indicating exotic magnetization curves for all $J_{\times}>2\,J_{+}-2$.
(36) is identical to (30) due to the fact that the AF and the AF/FM square-
kagomé have the same ground state energy (without Zeeman term).

####  B.2.2 Case of $J_{\times}\leq 2J_{+}-2$

We have the following results:

$\displaystyle j_{\rm min}(\widehat{\boldsymbol{\lambda}})$ $\displaystyle=$
$\displaystyle-1-\frac{1}{3}\left(J_{\times}+J_{+}\right),$ (39)
$\displaystyle j_{\rm min}(\widetilde{\boldsymbol{\lambda}})$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left(-3\sqrt{(-{J_{\times}}+2{J_{+}}-2)^{2}+4}-J_{\times}-4{J_{+}}+2\right),$
(40) $\displaystyle\Delta F$ $\displaystyle:=$ $\displaystyle
F-\frac{1}{2}\,h_{\rm
sat}=\frac{1}{6}\left(3\sqrt{(J_{\times}-2{J_{+}}+2)^{2}+4}-J_{\times}+2{J_{+}}-8\right)\geq
0\;,$ (41)

thus indicating exotic magnetization curves for all $J_{\times}<2\,J_{+}-2$.

##  Appendix C AF square-kagomé phase IV

It is possible to describe phase IV analytically for all values of $J_{3}>0$
if we assume that the ground state has the fxorm described in the main text
depending on two $z$-values $z_{1}$ and $z_{2}$. Its energy
$E(J_{3},H,z_{1},z_{2})$ is given by

$E(J_{3},H,z_{1},z_{2})=\frac{1}{3}\left(-H\left(z_{1}+2z_{2}\right)+2{J_{3}}\left(z_{1}^{2}+z_{2}^{2}-1\right)+2z_{2}^{2}+4z_{1}z_{2}-2\sqrt{2-2z_{1}^{2}}\sqrt{1-z_{2}^{2}}\right)\;.$
(42)

Ground states of this phase satisfy $\partial E/\partial z_{1}=\partial
E/\partial z_{2}=0$. These equations admit a parametric solution of the
following form:

$\displaystyle z_{1}$ $\displaystyle=$
$\displaystyle\frac{w\left(\sqrt{2}\left(J_{3}-1\right)+w\right)}{\sqrt{2}\left(2J_{3}-1\right)w+2}\,W$
(43) $\displaystyle z_{2}$ $\displaystyle=$ $\displaystyle W$ (44)
$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\left(5J_{3}-3\right)w+w^{2}+4}{3\sqrt{2}\left(2J_{3}-1\right)w+6}\,W$
(45) $\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{2\left(\sqrt{2}J_{3}w^{2}+\left(2J_{3}\left(J_{3}+1\right)-1\right)w+\sqrt{2}\left(J_{3}+1\right)\right)}{\left(2J_{3}-1\right)w+\sqrt{2}}\,W,\mbox{
where }$ (46) $\displaystyle W$ $\displaystyle:=$
$\displaystyle\frac{\sqrt{\left(w^{2}-1\right)\left(\left(2J_{3}-1\right)w\left(\left(2J_{3}-1\right)w+2\sqrt{2}\right)+2\right)}}{w\sqrt{\frac{1}{2}\left(8\left(J_{3}-1\right)J_{3}+1\right)w^{2}+\sqrt{2}\left(3J_{3}-1\right)w-\left(J_{3}-2\right)J_{3}+1}}\;,$
(47)

and $w$ is a parameter satisfying $1\leq w\leq w_{max}(J_{3})$. For example,
$w_{max}(J_{3}=1)=\frac{1+\sqrt{5}}{\sqrt{2}}=2.28825\ldots$.

## References

  * (1) N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models,” _Phys. Rev. Lett._ , vol. 17, no. 22, p. 1133, 1966. [Online]. Available: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.17.1133
  * (2) C. Hickey, L. Cincio, Z. Papic̀, and A. Paramekanti, “Emergence of chiral spin liquids via quantum melting of noncoplanar magnetic orders,” _Phys. Rev. B_ , vol. 96, p. 115115, 2017. [Online]. Available: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.115115
  * (3) J.-C. Domenge, P. Sindzingre, C. Lhuillier, and L. Pierre, “Twelve sublattice ordered phase in the ${J}_{1}-{J}_{2}$ model on the kagomé lattice,” _Phys. Rev. B_ , vol. 72, p. 024433, Jul 2005. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.72.024433
  * (4) O. Janson, J. Richter, and H. Rosner, “Modified kagome physics in the natural spin-1/2 kagome lattice systems: Kapellasite ${C}u_{3}{Z}n({O}{H})_{6}{C}l_{2}$ and Haydeeite ${C}u_{3}{M}g({O}{H})_{6}{C}l_{2}$,” _Phys. Rev. Lett._ , vol. 101, no. 10, p. 106403, 2008. [Online]. Available: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.106403
  * (5) ——, “Intrinsic peculiarities of real material realizations of a spin-1/2 kagomé lattice,” in _J. Phys.: Conf. Ser._ , vol. 145, no. 1. IOP Publishing, 2009, p. 012008. [Online]. Available: https://iopscience.iop.org/article/10.1088/1742-6596/145/1/012008/meta
  * (6) L. Messio, C. Lhuillier, and G. Misguich, “Lattice symmetries and regular magnetic orders in classical frustrated antiferromagnets,” _Phys. Rev. B_ , vol. 83, p. 184401, May 2011. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.83.184401
  * (7) R. Siddharthan and A. Georges, “Square kagome quantum antiferromagnet and the eight-vertex model,” _Phys. Rev. B_ , vol. 65, p. 014417, Dec 2001. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.65.014417
  * (8) J. Schnack, H.-J. Schmidt, J. Richter, and J. Schulenburg, “Independent magnon states on magnetic polytopes,” _Eur. Phys. J. B_ , vol. 24, no. 4, pp. 475–481, 2001. [Online]. Available: https://link.springer.com/article/10.1007/s10051-001-8701-6
  * (9) P. Tomczak and J. Richter, “Specific heat of the spin-$1/2$ Heisenberg antiferromagnet on squagome lattice,” _J. Phys. A: Math. Gen._ , vol. 36, no. 20, p. 5399, 2003. [Online]. Available: https://doi.org/10.1088/0305-4470/36/20/304
  * (10) J. Richter, O. Derzhko, and J. Schulenburg, “Magnetic-field induced spin-peierls instability in strongly frustrated quantum spin lattices,” _Phys. Rev. Lett._ , vol. 93, p. 107206, Sep 2004. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.93.107206
  * (11) J. Richter, J. Schulenburg, P. Tomczak, and D. Schmalfuß, “The Heisenberg antiferromagnet on the square-kagome lattice,” _Cond. Matter Phys._ , vol. 12, no. 3, pp. 507–517, 2009. [Online]. Available: https://doi.org/10.5488/CMP.12.3.507
  * (12) 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._ , vol. 82, p. 083709, 2013\. [Online]. Available: https://journals.jps.jp/doi/abs/10.7566/JPSJ.82.083709
  * (13) I. Rousochatzakis, R. Moessner, and J. v. d. Brink, “Frustrated magnetism and resonating valence bond physics in two-dimensional kagome-like magnets,” _Phys. Rev. B_ , vol. 88, no. 19, p. 195109, nov 2013. [Online]. Available: http://dx.doi.org/10.1103/PhysRevB.88.195109
  * (14) O. Derzhko, J. Richter, O. Krupnitska, and T. Krokhmalskii, “The square-kagome quantum Heisenberg antiferromagnet at high magnetic fields: The localized-magnon paradigm and beyond,” _Low Temp. Phys._ , vol. 40, no. 6, pp. 513–520, 2014. [Online]. Available: https://doi.org/10.1063/1.4881184
  * (15) A. Ralko and I. Rousochatzakis, “Resonating-valence-bond physics is not always governed by the shortest tunneling loops,” _Phys. Rev. Lett._ , vol. 115, p. 167202, Oct 2015. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.115.167202
  * (16) H. Nakano, Y. Hasegawa, and T. Sakai, “Magnetization jump in the magnetization process of the spin-1/2 Heisenberg antiferromagnet on a distorted square-kagome lattice,” _J. Phys. Soc. Jpn._ , vol. 84, no. 11, p. 114703, 2015. [Online]. Available: https://doi.org/10.7566/JPSJ.84.114703
  * (17) O. Derzhko, J. Richter, and M. Maksymenko, “Strongly correlated flat-band systems: The route from heisenberg spins to hubbard electrons,” _Int. J. Mod. Phys. B_ , vol. 29, no. 12, p. 1530007, 2015. [Online]. Available: https://www.worldscientific.com/doi/abs/10.1142/S0217979215300078
  * (18) 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_ , vol. 98, p. 014404, Jul 2018. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.98.014404
  * (19) 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._ , vol. 87, no. 4, p. 043704, 2018. [Online]. Available: https://doi.org/10.7566/JPSJ.87.043704
  * (20) T. Lugan, L. D. C. Jaubert, and A. Ralko, “Topological nematic spin liquid on the square kagome lattice,” _Phys. Rev. Research_ , vol. 1, p. 033147, Dec 2019. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevResearch.1.033147
  * (21) P. A. McClarty, M. Haque, A. Sen, and J. Richter, “Disorder-free localization and many-body quantum scars from magnetic frustration,” _Phys. Rev. B_ , vol. 102, p. 224303, Dec 2020. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.102.224303
  * (22) Y. Kuno, T. Mizoguchi, and Y. Hatsugai, “Flat band quantum scar,” _Phys. Rev. B_ , vol. 102, p. 241115(R), Dec 2020. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.102.241115
  * (23) 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_ , vol. 104, p. L220408, Dec 2021. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.104.L220408
  * (24) P. Schmoll, A. Kshetrimayum, J. Naumann, J. Eisert, and Y. Iqbal, “Tensor network study of the spin-$1/2$ Heisenberg antiferromagnet on the shuriken lattice,” 2022. [Online]. Available: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.064406
  * (25) J. Richter and J. Schnack, “Magnetism of the $s=\frac{1}{2}$ $J_{1}$-$J_{2}$ square-kagome lattice antiferromagnet,” _Phys. Rev. B_ , vol. 107, p. 245115, Jun 2023. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.107.245115
  * (26) M. Gembé, H.-J. Schmidt, C. Hickey, J. Richter, Y. Iqbal, and S. Trebst, “Noncoplanar magnetic order in classical square-kagome antiferromagnets,” _Phys. Rev. Res._ , vol. 5, p. 043204, Jun 2023. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevResearch.5.043204
  * (27) N. Niggemann, N. Astrakhantsev, A. Ralko, F. Ferrari, A. Maity, T. Müller, J. Richter, R. Thomale, T. Neupert, J. Reuther, Y. Iqbal, and H. O. Jeschke, “Quantum paramagnetism in the decorated square-kagome antiferromagnet Na6Cu7Bio4(PO4)4Cl3,” _Phys. Rev. B_ , vol. 108, p. L241117, Dec 2023. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.108.L241117
  * (28) 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._ , vol. 11, p. 3429, 2020. [Online]. Available: https://doi.org/10.1038/s41467-020-17235-z
  * (29) 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,” _Inorganic Chemistry_ , vol. 60, no. 15, pp. 11 450–11 457, 2021. [Online]. Available: https://doi.org/10.1021/acs.inorgchem.1c01459
  * (30) 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_ , vol. 105, p. 155153, Apr 2022. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.105.155153
  * (31) M. Markina, P. Berdonosov, T. Vasilchikova, K. Zakharov, A. Murtazoev, V. Dolgikh, A. Moskvin, V. Glazkov, A. Smirnov, and A. Vasiliev, “Static and resonant properties of decorated square kagome lattice compound KCu7(TeO4)(SO4)5Cl,” 2022. [Online]. Available: https://arxiv.org/abs/2212.11623
  * (32) A. F. Murtazoev, K. A. Lyssenko, M. M. Markina, V. A. Dolgikh, A. N. Vasiliev, and P. S. Berdonosov, “New Nabokoite-like Phases $A$Cu7TeO4(SO4)5Cl ($A$ = Na, K, Rb, Cs) with Decorated and Distorted Square Kagome Lattices,” _ChemPhysChem_ , vol. 24, p. e202300111, 2023. [Online]. Available: https://chemistry-europe.onlinelibrary.wiley.com/doi/abs/10.1002/cphc.202300111
  * (33) D. Coffey and S. A. Trugman, “Magnetic properties of undoped ${C}_{60}$,” _Phys. Rev. Lett._ , vol. 69, p. 176, 1992. [Online]. Available: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.69.176
  * (34) C. Schröder, H.-J. Schmidt, J. Schnack, and M. Luban, “Metamagnetic phase transition of the antiferromagnetic Heisenberg icosahedron,” _Phys. Rev. Lett._ , vol. 94, p. 207203, 2005. [Online]. Available: http://link.aps.org/abstract/PRL/v94/e207203
  * (35) N. P. Konstantinidis, “Antiferromagnetic Heisenberg model on clusters with icosahedral symmetry,” _Phys. Rev. B_ , vol. 72, p. 064453, 2005. [Online]. Available: http://link.aps.org/abstract/PRB/v72/e064453
  * (36) ——, “Unconventional magnetic properties of the icosahedral symmetry antiferromagnetic Heisenberg model,” _Phys. Rev. B_ , vol. 76, p. 104434, 2007. [Online]. Available: http://link.aps.org/abstract/PRB/v76/e104434
  * (37) N. Konstantinidis, “Zero-temperature magnetic response of small fullerene molecules at the classical and full quantum limit,” _JMMM_ , vol. 449, pp. 55–62, 2018. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0304885317308235
  * (38) ——, “Origin of the classical magnetization discontinuities of the dodecahedron,” 2023. [Online]. Available: https://arxiv.org/pdf/2311.17205.pdf
  * (39) M. E. Zhitomirsky, M. V. Gvozdikova, and T. Ziman, “Noncoplanar multi-k states in frustrated spinel and kagome magnets,” _Ann. Phys._ , vol. 447, p. 169066, 2022. [Online]. Available: https://doi.org/10.1016/j.aop.2022.169066
  * (40) H.-J. Schmidt and J. Richter, “Classical ground states of spin lattices,” _J. Phys. A: Math. Theor._ , vol. 55, no. 46, p. 465005, nov 2022. [Online]. Available: https://dx.doi.org/10.1088/1751-8121/aca36d
  * (41) H.-J. Schmidt, “Theory of ground states for classical Heisenberg spin systems I,” 2017. [Online]. Available: https://arxiv.org/abs/1701.02489v2
  * (42) ——, “Theory of ground states for classical Heisenberg spin systems IV,” 2017. [Online]. Available: https://arxiv.org/abs/1710.00318