# A semi-empirical analysis of the paramagnetic susceptibility of solid state
magnetic clusters

Lauro B. Braz, Fernando A. Garcia 1Instituto de Física da Universidade de São
Paulo (IFUSP), Universidade de São Paulo, São Paulo-SP, 05508-090, Brazil.

###### Abstract

Recent developments in the synthesis of new magnetic materials lead to the
discovery of new quantum paramagnets. Many of these materials, such as the
perovskites Ba4LnMn4O12 (Ln = Sc or Nb), Ba3Mn2O8, and Sr3Cr2O8 present
isolated magnetic clusters with strong intracluster interactions but weak
intercluster interactions, which delays the onset of order to lower
temperatures ($T$). This offset between the local energy scale and the
magnetic ordering temperature is the hallmark of magnetic frustration. At
sufficient high-$T$, the paramagnetic susceptibility ($\chi$) of frustrated
cluster magnets can be fit to a Curie-Weiss law, but the derived microscopic
parameters cannot in general be reconciled with those obtained from other
methods. In this work, we present an analytical microscopic theory to obtain
$\chi$ of dimer and trimer cluster magnets, the two most commonly found in
literature, making use of suitable Heisenberg-type Hamiltonians. We also add
intercluster interactions in a mean-field level, thus obtaining an expression
to the critical temperature of the system and defining a new effective
frustration parameter $f_{\text{eff}}$. Our method is exemplified by treating
the $\chi$ data of some selected materials.

##  I Introduction

The magnetism of solids encompass a really broad research field, ranging from
studies of magnets for applications to the investigation of the fundamentals
of electronic interactions in matter [1, 2]. Energy scales of the magnetic
interactions are set by the exchange constants $J$, which are mainly
determined by the nature of the interacting spins and the electronic structure
of the magnetic active atom coordination structure [3].

To estimate $J$ is an important step towards understanding the magnetism of a
particular material. The first approach to this problem is given by the Curie-
Weiss analysis of the material paramagnetic susceptibility $\chi$. For the
vast majority of magnetic solid state materials, $\chi$ as a function of
temperature ($T$) is fairly described by the Curie-Weiss expression (Equation
1)

$\chi(T)=\frac{C}{T-\theta_{\text{CW}}}$ (1)

where $C$ and $\theta_{\text{CW}}$ are the Curie and Curie-Weiss constants,
respectively. As is well known, $\theta_{\text{CW}}$ can be connected to $J$
and, in a mean field approach, to the magnetic ordering temperature of solids
[2], whereas the value of $C$ relates to the single ion spin configuration in
the solid. Exceptions, however, do exist for which the Curie-Weiss approach
cannot provide a physically meaningful set of parameters. Recently, the
magnetic properties of a large class of perovskite-type materials were
reviewed [4] providing many important examples where the set of the obtained
$C$ and $\theta_{\text{CW}}$ parameters do not connect well with the known
properties of the materials. This happens in respective of the apparent good
fittings of the high-$T$ $\chi$ data to the Curie-Weiss expression (Eq. 1).
Illustrative examples are provided by Ba4$Ln$Mn3O12 ($Ln=$ Sc or Nb) $\chi$
data [5, 6], for which the obtained $C$ values are much too low to be compared
with the expectation of $S=2$ spins from Mn3+ cations. If the obtained $C$
values are not reliable, it also raises questions about the obtained
$\theta_{\text{CW}}$ parameters.

The Ba4$Ln$Mn3O12 ($Ln=$ Sc or Nb) materials shared the same crystal structure
(see Figure 1) which host as building blocks isolated clusters of magnetic
cations, sitting within face sharing Oxygen octahedra. This building block
displays a large number of exchange paths which in turn contribute to large
$J$’s [3]. The local (within the cluster) magnetic interactions are strong but
the interclusters interactions are weak, since the clusters are well separated
in the structure. The onset of order, if observed, is thus delayed to low
temperatures. This is the hallmark of frustrated magnetism [7].

A feature of frustrated magnetism is that if the frustration is strong enough,
it may compete with quantum fluctuations for the system ground state, giving
rise to exotic types of quantum magnetism, such as quantum spin liquids [8, 9]
or Bose-Einstein condensates of spin excitations [10, 11]. Thus, materials
hosting this type of building blocks are good platforms to search for exotic
magnetism as noted in Ref. [4]. The usual way to quantify frustration is by
determining the frustration parameter $f$, or a lower bound to $f$, which
requires a trustworthy determination of the system’s energy scales.

In this work, we propose to analyze the paramagnetic data of materials hosting
magnetic clusters adopting Heisenberg-type Hamiltonians [2] parametrized by
exchange constants $J_{n}$ , with the intercluster interactions taken into
account by a mean field approach. This is an alternative, semi-empirical
approach which, while already tested in the case of spin dimers [12, 13], is
not particularly explored in the case of spin trimers [5, 6].

In section II, we provide a general perspective on the proposed methodology
and then we illustrate our approach by fitting the $\chi$ data of a series of
materials hosting magnetic clusters in section III. Finally, we move to
discuss and summarize our results. Most interesting, we define a new
frustration parameter $f_{\text{eff}}$ in terms of the energy scale of the
effective intercluster interactions. In doing so, we show some frustrated
magnets are unlikely to display quantum many body states at further lower
temperatures, while other remain as strong candidates to host this type of
Physics.

##  II Models

Our approach is mainly intended to the description of the magnetic properties
of solids hosting magnetic clusters as exemplified by the crystal structures
in Figure 1$(a)\text{, }(d)\text{, }(f)\text{ and }(i)$. In all cases, the
magnetic cations are connected by multiple exchange paths, causing the local
magnetic interaction to be strong. We thus treat single cluster magnetic
properties by means of a microscopic Heisenberg-type Hamiltonian:

$\mathcal{H}=\sum_{ij}J_{ij}\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{j}$ (2)

where the indexes $i$ and $j$ run within the cluster sites,
$\boldsymbol{S}_{i}$ are the spin operators of the magnetic cations and
$J_{ij}$ is the exchange constant between spins $i$ and $j$. For instance,
applied to the case of the magnetic trimers, Equation 2 reads:

$\mathcal{H}_{\text{trim}}=J_{12}S_{1}\cdot S_{2}+J_{23}S_{2}\cdot
S_{3}+J_{13}S_{1}\cdot S_{3}.$ (3)

In this work, we shall also illustrate our approach with magnetic dimers, that
can be described by:

$\mathcal{H}_{\text{dim}}=J_{1}S_{1}\cdot S_{2}.$ (4)

Once the proper Hamiltonian is determined, we consider a weak magnetic field
$H$ and evaluate the system energy levels up to first order in the field:

$E_{n}=E_{n}^{(0)}+HE_{n}^{(1)}+\mathcal{O}(2)$ (5)

and the results are applied to calculate the system magnetic bare
susceptibility in the usual units of
$\text{emu}\cdot\text{mol}^{-1}\cdot\text{Oe}^{-1}$.

$\bar{\chi}(\beta)=N_{A}k_{B}\mu_{B}^{2}\frac{\sum_{n}e^{-E_{n}^{(0)}\beta}\left(E_{n}^{(1)}/\mu_{B}\right)^{2}}{\sum_{n}e^{-E_{n}^{(0)}\beta}}\beta$
(6)

where $N_{A}$ is the Avogadro number, $k_{\text{B}}$ is the Boltzmann
constant, $\mu_{\text{B}}$ is the Bohr magneton, $\beta=1/k_{B}T$ and
$E_{n}^{(0)}$and $E_{n}^{(1)}$ are, respectively, the zeroth and first order
eigenvalues of $\mathcal{H}$ (Equation 2) as defined by Equation 5. In many
situations of interest, Equations 3 and 4 can be diagonalized analytically
(see appendix A) and thus the perturbation theory to write down $\bar{\chi}$
can be carried out straightforwardly.

The expression obtained from Equation 6 contains the $J_{n}$ constants as free
parameters. The next step is to treat the cluster-cluster interaction in a
mean field approximation:

$\chi=\frac{\bar{\chi}}{1-\lambda\bar{\chi}}$ (7)

where $\lambda$ is the molecular field parameter, describing the inter-cluster
interaction. As we shall discuss, in many instances it is $\lambda$ which
should be adopted to estimate the values of the frustration parameter $f$.

Figure 1: $(a)$ Structural model of the Ba3Mn2O8 and Sr3Cr2O8 materials
(symmetry group R$\bar{3}$m). The spheres represent Mn/Cr (purple; magnetic),
Ba (green), and O (red) atoms. $(b)$ Ba3Mn2O8 $\chi$ data (blue dots) and
theoretical fit (red line) $(c)$ Sr3Cr2O8 $\chi$ data with blue as yellow dots
representing, respectively, data obtained with the field applied parallel and
perpendicular to the sample’s $ab$ plane. The red and black lines represent
the respective theoretical fittings. $(d)$ Sr3Co3S2O3 structural model. The
spheres represent Co (blue; magnetic), Sr (green), O (red), and S (yellow)
atoms. $(e)$ As in $(c)$ but for the Sr3Co3S2O3 material. $(f)$ Structural
model of the Ba4ScMn3O12 and Ba4NbMn3O12 materials (symmetry group R$3$m). The
spheres represent Mn (purple/yellow; magnetic), Ba (green), O (red), and Nb/Sc
(blue) atoms. $(g)-(h)$ Susceptibility data (blue dots) and respective fitting
(red line) for the Ba4ScMn3O12, and Ba4NbMn3O12 materials, respectively. $(i)$
Structural model of BaTi0.5Mn0.5O3. The spheres represent Mn (purple;
magnetic), Ti (light blue), O (red), and Ba (orange) atoms. $(j)$
Susceptibility data (blue dots) and fittings (red line) for BaTi0.5Mn0.5O3.

##  III Results

Now that we have developed the appropriate tools to model clusters, the next
step is to apply the theory to better understand some model systems. We start
taking spin dimers into account, as a way to illustrate our approach in an
already familiar environment, and then we move to investigate spin trimers in
some selected hexagonal perovskites [4]. These low symmetry systems are prone
to exhibit rotated Oxygen octahedra coordinating transition metal cations,
which will precisely give rise to the multiplicity of the exchange paths that
cause the formation of strongly coupled magnetic clusters. We end this section
analyzing a system exhibiting both spin dimers and spin trimers.

###  III.1 Dimers

We select the Ba3Mn2O8 [14, 15, 16], Sr3Cr2O8 [11], and Sr3Co3S2O3 [17] spin
dimer materials to our analysis. The same trigonal structure, described by the
space group R$\bar{3}$m, are shared by the two former materials and is
depicted in Figure 1$(a)$. The Sr3Co3S2O3 structure is described by the space
group Pbam and is presented in Figure 1$(d)$.

Focusing on the trigonal materials, charge balance in the systems dictate that
we have Mn5+and Cr5+ cations which carry, respectively, $S=1$ and $S=1/2$
spins. The Ba3Mn2O8 total $\chi$ is modeled by adopting one exchange constant
$J_{1}$ and one $\lambda$ molecular field parameter plus a constant
diamagnetic contribution $\chi_{0}$ as in Equation 8:

$\chi(T)=q_{\text{dim}}\chi_{\text{dim}}+\chi_{0}$ (8)

where $q_{\text{dim}}$ is the molar fraction of the dimer specimen in the
sample and $\chi_{\text{dim}}$ is obtained from 7 (with $\bar{\chi}$ as
suitable to case of dimers, see Equation 13) and carries the $J_{1}$ and
$\lambda$ parameters. The fitting and experimental data are presented in
Figure 1$(b)$ and the obtained parameters are $J_{1}/k_{B}=17.66(7)$ K and
$\lambda=-16.35(9)$ Oe$\cdot$mol$\cdot$emu-1.

In the case of the Sr3Cr2O8 material [11], we model to Equation 8 an orphan
spin contribution ($C/T$) to account for the observed rise in $\chi$ at low
temperatures. The fitting and experimental data are presented in Figure 1$(c)$
and the obtained parameters are $J_{1}/k_{B}=61.5(2)$ K, $\lambda=-5(1)$
Oe$\cdot$mol$\cdot$emu-1, $C=4.06(9)\cdot 10^{-3}$ emu$\cdot$K/mol, and
$\chi_{0}=1.9(1)\cdot 10^{-4}$ emu/mol Cr.

Lastly, as discussed in Ref. [17], the Sr3Co3S2O3 compound was expected to
hold an effective spin $S=3/2$ because of the Co2+ valence, which is again
deduced from charge balance. Due to the octahedral crystal field, however,
Co2+ cations carry low $S=1/2$ spins. The total $\chi$ is modeled by Equation
8 and the obtained parameters for the FC (ZFC) susceptibility data are
$J_{1}/k_{B}=97.9(2)$ K ($J_{1}/k_{B}=99.6(2)$ K), $\lambda=2.50(16)$
Oe$\cdot$mol$\cdot$emu-1 ($\lambda=0.94(11)$ Oe$\cdot$mol$\cdot$emu-1), and
$\chi_{0}=0.00829(3)$ emu/mol Co ($\chi_{0}=0.00824(3)$ emu/mol Co).Slightly
distinct values are reported in Ref. [17] because here we included the
intercluster interactions, which we found to be ferromagnetic ($\lambda>0$).
The fitting and experimental data are presented in Figure 1$(e)$.

###  III.2 Trimers

Two materials presenting magnetic trimers were selected: Ba4ScMn3O12 and
Ba4NbMn3O12. The materials share the same crystal structure which is depicted
in Figure 1$(f)$. In both cases, we assume the presence of Mn4+cations
carrying $S=3/2$ spins and Hamiltonian 3 is adopted to describe the trimer
magnetism. It should be clear, however, that we can adopt
$J_{12}=J_{23}=J_{1}$ and $J_{13}=J_{2}=\alpha J_{1}$, where $0\leq\alpha\leq
1$. This assumption make it possible to perform an exact diagonalization of 3.
In principle, one could think of $\alpha$ as a small number, but the large
multiplicity of exchange paths cause $\alpha$ to be $>0.5$ for magnetic
cations within face sharing octahedra. Intertrimer interactions are described
by just one molecular field parameter $\lambda$. We adopt the model
susceptibility:

$\chi(T)=q_{\text{trim}}\chi_{\text{trim}}+\chi_{0}$ (9)

where $q_{\text{trim}}$ is the molar fraction of the trimer specimen in the
sample and $\chi_{\text{trim}}$ is obtained from equation 9 (see Equation 15
for more details). One should note that $\chi_{\text{trim}}$ carries three
parameters: the exchange constants $J_{1}$ and $J_{2}$ and $\lambda$.

In the case of Ba4ScMn3O12, the following parameters are obtained:
$J_{1}/k_{B}=199.44(2)$ K, $\alpha=0.60041(1)$, $\chi_{0}=-2.67(4)\cdot
10^{-5}$ emu$\cdot$K/mol/Oe, and a negligible value of $\lambda$. As for
Ba4NbMn3O12 , we obtain: $J_{1}/k_{B}=172(1)\text{ K}$, $\alpha=0.7651(1)$,
$\lambda=77.4(5)$ mol$\cdot$Oe/emu/K, and $\chi_{0}=-0.00263(3)$
emu$\cdot$K/mol/Oe. The fittings results are compared to the data in Figure
1$(f)-(g)$, respectively. The significant value of $\lambda$ found in the
later case is in agreement with the observed critical temperature of $\approx
32$ K determined for Ba4NbMn3O12.

###  III.3 Dimers and trimers

We now turn to the challenging case offered by the BaTi0.5Mn0.5O3 [18, 19]
material, which presents spin dimers, trimers, and orphans. Its crystal
structure is depicted in Figure 1$(i)$. The proposed model to $\chi$ assumes
the following form:

$\chi(T)=q_{\text{trim}}\chi_{\text{trim}}+q_{\text{dim}}\chi_{\text{dim}}+q_{\text{orp}}\chi_{\text{orp}}+\chi_{0}$
(10)

where $q_{\text{orp}}$ is the molar fraction of the orphan spins in the sample
and $\chi_{\text{orp}}$ is the orphan spin susceptibility. As discussed
elsewhere [18, 19], the fractions are statistically determined to be:
$q_{\text{trim}}=q_{\text{orp}}=1/16$, and $q_{\text{dim}}=2/16$. We adopt a
Curie-Weiss form to $\chi_{\text{orp }}$ (with $C_{\text{orp}}$ and
$\theta_{\text{orp}}$ parameters). The obtained fitting is compared to the
data in Figure 1$(j)$ and the obtained cluster parameters are:
$J_{1}/k_{B}=201(2)$ K, $\alpha=0.85(2)$, and $\lambda=48(15)$
mol$\cdot$Oe/emu/K (calculated considering only timer-trimer interactions),
while the orphan spin parameters are: $\theta_{\text{orp}}=14.4(5)$ K and
$C_{\text{orp}}=0.217(2)$ emu$\cdot$K/mol/Oe.

Table 1: Summary of experimentally obtained parameters within our model paramagnetic susceptibility.  Material |  $J_{1}/k_{B}$ (K) |  $\alpha$ ($J_{2}/J_{1}$) |  $\lambda$ (Oe$\cdot$mol/emu) | $s_{\text{eff}}$ |  $T_{\text{C,eff}}$ (K) | $f_{\text{eff}}$ | $f_{\text{CW}}$
---|---|---|---|---|---|---|---
Ba3Mn2O8 | $17.66(7)$ | - | $-16.35(9)$ | $1$ | $16.36(9)$ | $9.14$ | $18.7$
Sr3Cr2O8 | $61.5(2)$ | - | $-5(1)$ | $1/2$ | $1.7(4)$ | $0.90$ | $26.2$
Sr3Co3S2O3 | $99.6(2)$ | - | $1.82(17)$ | $1/2$ | $2.4(2)$ | $0.63$ | $42.8$
Ba4ScMn3O12 | $199.44(2)$ | $0.60041(1)$ | $-0.05543(5)$ | $1/2$ | $0.0208(2)$ | $0.01$ | $0.41$
Ba4NbMn3O12 | $172(1)$ | $0.7651(1)$ | $77.4(5)$ | $1/2$ | $29.0(1)$ | $0.91$ | $0.02$
BaTi0.5Mn0.5O3 | $201(2)$ | $0.85(2)$ | $48(15)$ | $1/2$ | $18(6)$ | $180$ | $1024$

##  IV Discussion

We would like to discuss two aspects or our results: $i)$ the set of obtained
parameters and $ii)$ the interpretation of $\lambda$ and its relation to the
frustration parameter $f$, which is of great relevance to the characterization
of putative quantum phases associated to these materials. We start with the
later.

By comparing the high temperature ($\beta\rightarrow 0$) limit of Equation 7
to the the Curie-Weiss law (Equation 1), one reaches the expression
$\bar{\theta}_{\text{CW}}=\lambda C$, which in turn can be applied to define
an effective exchange constant $J_{\text{eff }}$for the intercluster
interaction as
$\lambda\equiv\frac{J_{\text{eff}}k_{B}}{g^{2}\mu_{B}^{2}N_{A}}$. This
expression gives a well defined meaning to $\lambda$ as the parameter
describing long range intercluster interactions.

Indeed, in forming the clusters, most of the magnetic degrees of freedom are
frozen at low temperatures and the remaining degrees of freedom interact with
an energy scale characterized by $\bar{\theta}_{\text{CW}}$. Therefore, the
correct estimate to the system frustration should be given by a new effective
frustration parameter
$f_{\text{eff}}\equiv|\frac{\bar{\theta}_{\text{CW}}}{T_{\text{L}}}|$ where
$T_{\text{L}}$ is the lowest temperature at which the system is still in the
paramagnetic state. If order is not observed, $T_{\text{L}}$ is given by the
lowest experimentally achieved temperature and the $f_{\text{eff}}$ parameter
thus obtained is a lower bond to $f_{\text{eff}}$. Moreover, we have now a way
to estimate the temperature for long range magnetic order that will be
triggered by the intercluster interactions. We shall name this the effective
critical temperature, $T_{\text{C,eff}}$. Adopting the overall mean field
relation $|\theta_{\text{CW}}|=T_{\text{C }}$, where $T_{\text{C }}$ is the
magnetic critical temperature, we write
$|\bar{\theta}_{\text{CW}}|=T_{\text{C,eff}}$ and then
$|\bar{\theta}_{\text{CW}}|=T_{\text{C,eff}}=\lambda C$. To obtain $C$, we
consider that the remaining degrees of freedom of the magnetic clusters should
be associated with an effective spin $s_{\text{eff}}$ , thus we can write:

$T_{\text{C,eff}}=|J_{\text{eff}}|\frac{s_{\text{eff}}(s_{\text{eff}}+1)}{3}$
(11)

We propose that $s_{\text{eff}}$ should be calculated as either the
expectation value of the $S_{z}$ operator of the cluster ground state spin
state in the case of trimers, or by the first excited total spin state of the
cluster, in the case of dimers. Concerning the later, the necessity of
considering the first excited state in the case of dimers can be understood as
follows: the first order energy correction for the spin excitation in a given
magnetic field $H$ is of the type $E_{0}^{(1)}=-g\mu_{B}Hs_{\text{eff}}$. For
an antiferromagnetic dimer, $s_{\text{eff}}$ is always $0$ if one adopts the
spin ground state to calculate $s_{\text{eff}}$. Our assumption about
$s_{\text{eff}}$ thus means that the system magnetism at low-$T$ is dominated
only by the ground state spin configuration except when it turns out be null,
when one should peak the first excited state. In the appendix B we show how to
determine $s_{\text{eff}}$ for trimers.

All obtained parameters are shown in table 1. We also list $f_{\text{CW}}$
parameters which are the frustration parameters obtained from the usual Curie-
Weiss analysis. The values of $J_{n}$ clearly distinguish between the exchange
constants due to magnetic interactions mediated by corner sharing octahedra,
the case of the selected dimer based materials, and face sharing octahedra,
the case of selected trimer based materials. Intercluster interactions of both
FM and AFM types are present, but are definitely small energy scales when
compared to the intracluster interactions. Notwithstanding, these are the
energy scales that will control the onset of magnetic order at sufficient low
temperatures.

Comparing $f_{\text{eff}}$ and $f_{\text{CW}}$, one can conclude that all
dimer based materials are at most moderately frustrated magnetic systems (the
case of Ba3Mn2O8) if the intercluster energy scale is considered. It is
therefore unlikely that a quantum magnetic state will emerge from these
materials. Interesting glassy behavior, however, could be expected.

The case of the trimer based materials is even more revealing. Ba4NbMn3O12
order at $T\approx 32$ K and is not a frustrated material. Ba4ScMn3O12, on the
other hand, is expected to order only about $T\approx 0.02$ K, a temperature
at which quantum effects could tune the system into an exotic ground state.
The intercluster interactions, however, are rather weak making it unlikely.
The mixed dimer/trimer material BaTi0.5Mn0.5O3 remain characterized as a
strongly frustrated magnet with relatively large intercluster interactions. As
already observed [19], the system disorder is large, which may hinder the
appearance of a quantum spin liquid, although the physics will be that of a
correlated disordered quantum magnet. Thus, further investigations at low $T$
are invited.

##  V Conclusion and perspectives

The Curie-Weiss law has great use to analyze the magnetism of very general
systems. It fails, however, in giving a microscopic view of the problem. We
found the later to be particularly critical to determine the energy scale of
the magnetic interaction of materials hosting magnetic clusters. Therefore, in
this work we proposed a methodology to study the susceptibility of cluster
magnets which considers intracluster and intercluster interactions.

Our methods were applied to a series of materials and the obtained parameters
were discussed and put into perspective. Our semiempirical approach to the
problem could identify potential material candidates to low temperature
explorations in the search for quantum many body ground states. Our main
proposal is that intercluster interactions must be taken into account while
discussing the level of frustration in the system. This is the energy scale
that correctly sets the odds for a system to display a quantum many body
state.

In particular, the trimmer system Ba4ScMn3O12, which present a very small
inter-cluster interaction, with a predicted ordering temperature of $\approx
0.02$ K, is unlikely to display this type of physics, but the case of mixed
dimer/trimmer system BaTi0.5Mn0.5O3 remain open for further experimental
investigation.

Lastly, one could argue that our model, in comparison to Curie-Weiss analysis,
is only adding fitting parameters. We point out, however, that our approach
starts from a microscopic model and then scales to the description of
macroscopic susceptibility data. The adopted parameter set are thus not
arbitrary. In fact, the present method proposes a more meaningful analysis of
magnetic susceptibility data of systems presenting magnetic clusters.

##  VI Acknowledgments

We thank H. Tanaka and collaborators, A. Aczel and collaborators Aczel  _et
al._ [11], K. To Lai and M. Valldor Lai and Valldor [17] and E. Komleva and
collaborators Komleva  _et al._ [20] who provided us with the Ba3Mn2O8,
Sr3Cr2O8, Sr3Co3S2O8, and Ba4NbMn3O12 data, respectively. The financial
support from Fundação de Amparo a Pesquisa do Estado de São Paulo is
acknowledged by L.B.B. (Grant No. 2019/27555-9) and F.A.G. (Grant No.
2019/25665-1).

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##  VII Appendix

##  Appendix A Exact diagonalization

In the following development we present analytical solutions to the problems
of dimer and trimer clusters interactions, and also to inter-cluster
interactions. Our approach to study interactions consists in find the
eigenenergies of the respective Heisenberg Hamiltonian,
$\mathcal{H}=\sum_{ij}J_{ij}S_{i}\cdot S_{j}$, in which $S_{i}$ are spin
operators and $J_{ij}$ are the exchange constants (usually in units of K),
related to the probability of electronic hopping from site $i$ to site $j$.

###  A.1 Dimers

Let us consider a dimer system consisting of two sites with electrons which
may hop to the neighbor and interact effectively with total spins $S_{1}$ and
$S_{2}$, and exchange constant $J_{1}$. The Hamiltonian treating the
interaction is the Heisenberg Hamiltonian showed in Equation 12.

$\mathcal{H}_{\text{dim}}=J_{1}S_{1}\cdot S_{2}$ (12)

The basis which diagonalizes the operators
$S_{1}^{2},S_{2}^{2},S_{d}^{2}=(S_{1}+S_{2})^{2}$, and $S_{z}$ (of dimension
$(2s_{1}+1)(2s_{2}+1)$) has eigenvectors $|s_{d},m_{d}\rangle$, where
$S_{d}^{2}|s_{d},m_{d}\rangle=\hbar s_{d}(s_{d}+1)|s_{d},m_{d}\rangle$, and
$S_{z}|s_{d},m_{d}\rangle=\hbar m_{d}|s_{d},m_{d}\rangle$. All expected values
in this subsection are taken with respect to these eigenkets.

With this basis, it is possible to diagonalize the Hamiltonian in Equation 12
for any values of spin operators $S_{1}$ and $S_{2}$, and find $E=\langle
s_{d},m_{d}|\mathcal{H}|s_{d},m_{d}\rangle=J_{1}\langle S_{1}\cdot
S_{2}\rangle$. The trick here is to define a dimer quantum number such that
$\langle S_{d}^{2}\rangle=\langle(S_{1}+S_{2})^{2}\rangle=\langle
S_{1}^{2}+S_{2}^{2}-2S_{1}\cdot S_{2}\rangle=s_{d}(s_{d}+1)\implies\langle
S_{1}\cdot
S_{2}\rangle=\frac{1}{2}[s_{d}(s_{d}+1)-s_{1}(s_{1}+1)-s_{2}(s_{2}+1)]$. The
quantum number $s_{d}$ runs from $|s_{1}-s_{2}|$ through $s_{1}+s_{2}$ in
steps of 1. Therefore, Equation 13 holds for the eigenenergies of the dimer
system.

$E_{d}(s_{d})=\frac{J_{1}}{2}[s_{d}(s_{d}+1)-s_{1}(s_{1}+1)-s_{2}(s_{2}+1)].$
(13)

Adding a Zeeman-type perturbation $\mathcal{H}_{Z}=-g\mu_{B}HS_{z}$ (in units
of $\hbar$) in the z-direction can be done straightforwardly:
$E_{n}^{(1)}=-g\mu_{B}Hm_{d}$.

###  A.2 Trimers

Differently of dimers, the susceptibility of trimers does not goes to zero
$T\rightarrow 0$ , because each trimer will still present a net magnetic
moment.

Now the Heisenberg Hamiltonian of a trimer cluster is written as in Equation
3. Here, we specialize in the case $J_{12}=J_{23}=J_{1}$ and
$J_{13}=J_{2}=\alpha J_{1}$, where $0\leq\alpha\leq 1$, which is adequate for
systems of magnetic trimers forming as in the cases depicted in figure 1.
Thus:

$\mathcal{H}_{\text{trim}}=J_{1}(S_{1}\cdot S_{2}+S_{2}\cdot S_{3}+\alpha
S_{1}\cdot S_{3})$ (14)

To diagonalize $\mathcal{H}_{\text{trim}}$ is similar to the case of the
dimers. The basis containing good quantum numbers to the problem will be the
one which couples three angular momenta (of dimension
$(2s_{1}+1)(2s_{2}+1)(2s_{3}+1)$), and therefore diagonalizes simultaneously
the operators
$S_{i}^{2},S_{d}^{2}=(S_{1}+S_{2})^{2},S^{2}=(S_{1}+S_{2}+S_{3})^{2}$, and
$S_{z}$. Therefore, the eigenstates are denoted $|s_{d},s,m\rangle$, where
$S_{d}^{2}|s_{d},s,m\rangle=\hbar s_{d}(s_{d}+1)|s_{d},s,m\rangle$,
$S^{2}|s_{d},s,m\rangle=\hbar s(s+1)|s_{d},s,m\rangle$, and
$S_{z}|s_{d},s,m\rangle=\hbar m|s_{d},s,m\rangle$. All expected values in this
subsection are taken with respect to these eigenkets.

As the objective is now obtain three dot products between spin operators we
first define the same dimer quantum number $s_{d}$ in order to obtain $\langle
S_{1}\cdot S_{2}\rangle$ and $\langle S_{2}\cdot S_{3}\rangle$. Next we
consider $S=S_{1}+S_{2}+S_{3}$ which results in a total spin quantum number
$\langle S^{2}\rangle=s(s+1)$. Working out $\langle S^{2}\rangle$ we isolate
$\langle S_{1}\cdot S_{3}\rangle$ and use our definitions for $S_{d}$. Finally
with all dot products in hand we substitute them in $\langle E\rangle$ and
obtain Equation 15 for the eigenenergies of the system:

$\begin{split}E_{t}(s,s_{d})=\frac{J}{2}\Big{\\{}s(s+1)-s_{3}(s_{3}+1)+s_{d}(s_{d}+1)(\alpha-1)\\\
-\alpha\big{[}s_{1}(s_{1}+1)+s_{2}(s_{2}+1)\big{]}\Big{\\}}\end{split}$ (15)

Now the quantum numbers $s_{d}$ and $s$ admit values as follows. The dimer
quantum number $s_{d}$ is a result of the coupling of two spin angular
momenta, with total spins $s_{1}$, and $s_{2}$, so its minimum value is
$|s_{1}-s_{2}|$, and its maximum is $s_{1}+s_{2}$, varying in steps of $1$. In
its turn, the quantum number $s$, and its multiplicity $m$ are a result of the
coupling between $S_{d}$, and $S_{3}$, therefore the minimum of $s$ is
$|s_{d}-s_{3}|$, while its maximum is $s_{d}+s_{3}$, with the multiplicity $m$
running from $m=-s$ to $m=s$. The trick here is to make the multiplicity
$m_{3}$ of the third spin vary for each value of $s_{d}$, giving the total
spin quantum number, $s=s_{d}+m_{3}$, within the range of values $s$ is
defined.

For example, take $s_{1}=s_{2}=s_{3}=1/2$, so $s_{d}=\\{1,0\\}$, while the
minimum of $s$ is $|s_{d}-s_{3}|=|1-1/2|=1/2$ and its maximum is
$s_{d}+s_{3}=1+1/2=3/2$; there are $(2s_{1}+1)(2s_{2}+1)(2s_{3}+1)=8$ states
in total. In this way the possible energy states are $E_{t}(3/2,1)$
($s_{d}=1,m_{3}=1/2,s=s_{d}+m_{3}=3/2$), which is $4$-degenerated because of
$m$, $E_{t}(1/2,1)$ ($s_{d}=1,m_{3}=-1/2,s=s_{d}+m_{3}=1/2$), $2$-degenerated
because of $m$, and $E_{t}(1/2,0)$ ($s_{d}=0,m_{3}=1/2,s=s_{d}+m_{3}=3/2$),
which is $2$-degenerated because of $m$.

Adding a Zeeman-type perturbation $\mathcal{H}_{Z}=-g\mu_{B}HS_{z}$ (in units
of $\hbar$) in the z-direction can be done straightforwardly:
$E_{n}^{(1)}=-g\mu_{B}Hm$.

##  Appendix B Cluster effective spin

To apply Equation 11 to estimate the critical temperature resulting from
inter-cluster interactions, we need to determine the effective spins of the
clusters in our system. Despite the system may be in several spin states
running from $0$ through $s_{\text{tot}}$ (total sum of the cluster spins),
excited states do not take part in the low temperature magnetic response. We
thus adopt, in the case of trimers, as $s_{\text{eff}}$: we are thus
interested in the expected value of the $S_{z}$ operator with respect to the
cluster ground state.

Let us take the Heisenberg Hamiltonian of our cluster system $\mathcal{H}_{0}$
(Equations 4 or 3) and a Zeeman term $\mathcal{H}^{\prime}$ such that
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}^{\prime}$. We then numerically
diagonalize it and determine the ground state $|GS\rangle$, which in this case
is a vector of length $2s_{\text{tot}}+1$. In its turn, the $S_{z}$ operator
is a $(2s_{\text{tot}}+1)\times(2s_{\text{tot}}+1)$ matrix so that the
expected value of equation 16 is just a contraction of vectors with a matrix.

$s_{\text{eff}}=\langle GS|S_{z}|GS\rangle.$ (16)

Here we considered some usual cases for total spin as a function of the
relation between the first and, in the case of trimers, second neighbors
interactions. In Figure 2 we show the phase space $J_{2}\times J_{1}$ for
trimers of spins $s=1/2$, $s=1$, $s=3/2$, and $s=2$. We identify a discrete
behavior of the ground state of the system as a function of the
$\alpha=J_{2}/J_{1}$ relation.

Figure 2: $(a)-(d)$ The $J_{2}$ vs. $J_{1}$ trimers’ ground state (colors)
phase diagrams of $s=1/2$, $s=1$, $s=3/2$, and $s=2$ spin trimers,
respectively . In all panels the upper straight lines denote
$\alpha=J_{1}/J_{2}=1$, while the bottom ones denote $\alpha=1/2$. Since in
real systems $\alpha<1$, white regions are unimportant.