#  Geometric frustration and Dzyaloshinskii-Moriya interactions
in a quantum star lattice hybrid copper sulfate

Hajime Ishikawa<EMAIL_ADDRESS>Yuto Ishii  Takeshi Yajima
Yasuhiro H. Matsuda  Koichi Kindo  Institute for Solid State Physics,
University of Tokyo, Kashiwa, Chiba, 277-8581, Japan  Yusei Shimizu  Institute
for Materials Research, Tohoku University, Oarai, Ibaraki 311-1313, Japan
Ioannis Rousochatzakis  Department of Physics, Loughborough University,
Loughborough LE11 3TU, UK  Ulrich K. Rößler  Oleg Janson  o.janson@ifw-
dresden.de Institute for Theoretical Solid State Physics, Leibniz IFW Dresden,
01069 Dresden, Germany

###### Abstract

We study the magnetism of a layered, spin-$\frac{1}{2}$ organic-inorganic
copper sulfate, which is a close realization of the star lattice
antiferromagnet, one of the playgrounds of geometric frustration and
resonating valence bond physics in two spatial dimensions. Our thermodynamic
measurements show no ordering down to 0.1 K and a characteristic field-induced
entropic shift, revealing the presence of an infinite number of competing
states down to very low energy scales. The response to external magnetic
fields shows, in addition, a peculiar anisotropy, reflected in the formation
of a 1/3 magnetization plateau (stable up to full saturation around 105 T) and
a paramagnetic, Curie-like susceptibility for one direction of the field
(${\bf H}\parallel{\bf c}$), and a completely different response in other
field directions. Our first-principles density functional theory calculations
and exact diagonalizations show that these experimental puzzles are
distinctive signatures of a strong interplay between geometric frustration and
sizable Dzyaloshinskii-Moriya interactions, and the emergence of a continuous
U(1) symmetry at low energy scales.

#### Introduction.

The problem of tiling a plane with regular convex polygons fascinated
architects since ancient times, engendering a rich cross-cultural heritage. In
modern condensed matter physics, the very same patterns appear in a different
context: antiferromagnetically coupled localized spins arranged on the
vertices of a tiling are at heart of frustrated magnetism [1, 2] and high-
temperature superconductivity [3]. The emergent spin lattices split into two
classes. Bipartite lattices, such as the square or the honeycomb lattice,
feature the classical Néel ground state (GS), which is globally compatible
with all antiferromagnetic (AFM) bonds. Such compatibility is fundamentally
impossible in geometrically frustrated lattices, where only local constraints
can be satisfied. For instance, in the kagome lattice (Fig. 1 a), the local
constraint imposes a 120∘-spin structure within a each triangle. Since
infinitely many _global_ configurations satisfy this _local_ constraint, the
classical GS manifold is infinitely degenerate. In the case of
spin-$\frac{1}{2}$, quantum fluctuations may stabilize exotic GSs such as
quantum spin liquids [4, 5, 6] and valence bond phases [7, 8, 9].

The spin-$\frac{1}{2}$ star lattice antiferromagnet (Fig. 1 a) is one of the
paradigmatic models of geometrically frustrated magnetism in two spatial
dimensions [2, 1, 10]. As in the kagome, the triangle-based structure leads to
an infinite ground state degeneracy at the classical level [11], and an array
of unconventional ground states (GSs) in the spin $S=1/2$ limit, including
valence bond solids [11, 12, 13], resonating valence bond states [14], chiral
spin liquids [15], as well as magnetic field-induced phase transitions [11,
13]. Unlike the kagome, the star lattice features two inequivalent nearest-
neighbour (NN) bonds. The respective Heisenberg exchange interactions are
often called $J_{\rm{T}}$ and $J_{\rm{D}}$, as the spins on these bonds are
part of a triangle and a dimer, respectively. The lattice can also be viewed
as a (decorated) honeycomb lattice made of triangles.

Layered sulfates often provide peculiar spin lattices such as kagome lattice
in natural minerals [16, 17] and organic-inorganic materials [18, 19]. In
2020, Sorolla _et al._ reported [(CH3)2(NH2)]3Cu3(OH)(SO4)4$\cdot$0.24H2O,
(Dimmethylammonium Copper Sulfate, called DiMACuS hereafter), which is the
first realization of the spin-$\frac{1}{2}$ star lattice made of Cu^2+
(3$\textit{d}^{9}$) ions [20]. The crystal structure features inorganic
[Cu3(OH)(SO4)4]^3- layers separated by dimethylammonium cations and crystal
water molecules. In addition to $J_{\rm{T}}$ and $J_{\rm{D}}$, the next
nearest neighbor interaction called $J_{\rm{H}}$ here (Fig. 1 b), which form
the hexagon, may be present due to the rotation of the triangles. Similarly
modified next nearest neighbor interactions are found in a classical kagome
AFM and their effect on the GS has been examined previously [21]. Sorolla _et
al._ observed paramagnetic behavior down to 1.8 K in spite of AFM Weiss
temperature of 41 K and proposed DiMACuS as a quantum spin liquid candidate
[20].

In this Letter, we report the successful synthesis of millimeter-size single
crystals of DiMACuS as well as pure powder suitable for detailed magnetic
characterizations. We performed magnetization measurements on single crystals
down to 0.1 K, and measured the entire magnetization process of the powder
sample in pulsed magnetic fields up to 120 T. We also measured the specific
heat with and without magnetic fields. Two remarkable experimental results are
the absence of magnetic ordering down to 0.1 K and a peculiar magnetization
anisotropy, with a 1/3 plateau visible only in ${\bf H}\\!\parallel\\!{\bf
c}$. By combining first-principles calculations with analytical and numerical
model simulations, we show that DiMACuS is a quantum star lattice magnet with
$J_{\rm{T}}\gg J_{\rm{D}}$ and substantial chiral Dzyaloshinskii-Moriya (DM)
interactions within the spin triangles.

Figure 1:  (a) Kagome and star lattices. Close to the isolated-triangle limit
($J_{\rm{T}}\gg{}J_{\rm{D}}$), the star lattice can be described as honeycomb
lattice made of spin triangles. (b) Isotropic exchange interactions in DiMACuS
(top): the $({\bf a},{\bf b},{\bf c})$ frame shown refers to the hexagonal
unit cell. Local crystal structure (bottom) described by CuO5 pyramid and SO4
tetrahedra, as plotted using VESTA [22]. A picture of a single crystal is also
shown. (c) Observed and calculated powder XRD patterns.

#### Synthesis.

The sample was prepared by reacting 0.4 g of CuSO4$\cdot$5H2O, 0.5 ml of
sulfuric acid and 5 ml of N,N-dimethylformamide in a glass vial at 80°C for a
few days. Aggregation of bright green crystals including hexagonal plates of a
few millimeter size is formed (Fig. 1 b). The aggregation is recovered by
decantation and washed by N,N-dimethylformamide. The sample is immediately
vacuum dried and stored inside an argon-filled glove box as the crystal is
hygroscopic. The crystal is covered by Apiezon-N grease or sealed inside a
plastic tube when handling in air for measurements. Powder x-ray diffraction
measurement is performed on the crushed crystals by a diffractometer with
Cu-K$\alpha_{1}$ radiation (Smart Lab, Rigaku). The observed pattern matches
well with the calculated pattern of DiMACuS with the space group
$\textit{R}\bar{3}$ and lattice constants a = 12.7222(2) and c = 26.0836(4) Å,
indicating the successful synthesis: the pattern calculation is performed by
Fullprof software [23] including the effect of preferred orientation along
c-axis.

#### Low-field magnetization measurements.

The magnetization of a single crystal was measured by a SQUID magnetometer
(MPMS-XL, Quantum Design) in the $T$-range 1.8–300 K and magnetic fields of up
to 5 T. We made a Curie-Weiss fit to the magnetic susceptibility $\chi$ using
the expression
$\chi(\textit{T})\\!=\\!\chi_{0}\\!+\\!\textit{C}/(\textit{T}\\!+\\!\Theta)$,
where C and $\Theta$ are the Curie constant and Weiss temperature,
respectively, and $\chi_{0}$ is the $T$-independent term. Above 150 K, we
obtain $C$=0.469(8) emu/mol-Cu$\cdot$K ($\mu_{\rm{eff}}$=1.94 $\mu_{\rm{B}}$),
$\Theta$=43(2) K, and $\chi_{0}=-3.0(1)\times 10^{-4}$ emu/mol-Cu for ${\bf
H}\\!\parallel\\!{\bf c}$. The same fit for the data in ${\bf
H}\\!\parallel\\!{[120]}$, parallel to the edge of the hexagonal crystal,
yields a similar result with slightly larger Curie constant: $C$=0.498(8)
emu/mol-Cu$\cdot$K ($\mu_{\rm{eff}}$=1.99 $\mu_{\rm{B}}$), $\Theta$=43(2) K
and $\chi_{0}=-3.0(1)\times 10^{-4}$ emu/mol-Cu. Note that no clear anisotropy
was observed in $\chi(\textit{T})$ in ${\bf H}\\!\parallel\\!{[120]}$ and
${\bf H}\\!\parallel\\!{\bf a}$ above 1.8 K. The linear behavior of the
inverse susceptibility $(\chi-\chi_{0})^{-1}$ justifies the fits (Fig. 2 a).
The enhancement of $\mu_{\rm{eff}}$ with respect to the spin-only value (1.73
$\mu_{\rm{B}}$) is typical for Cu^2+ compounds and shows a slight anisotropy
in the $g$ factor ($g_{\parallel}\\!\simeq\\!2.24$ vs
$g_{\perp}\\!\simeq\\!2.30$).

Figure 2:  (a) $T$-dependence of inverse susceptibility at 1 T and the Curie-
Weiss fits. (b) Magnetic susceptibility at 0.1 and 1 T below 2 K. (c)
Magnetization curves of the single crystal and powder sample at 4.2 K and up
to 65 T. The magnetic field derivative for powder data is also shown. (d)
Magnetization curve of the single crystal at 0.13 K. (e) Magnetization curve
of the powder sample up to 120 T measured at $T_{\text{initial}}$ = 4.2 K and
its field derivative. (f) Specific heat divided by $T$ at 0 and 9 T below 20 K
and magnetic entropy estimated as in the main text.

Below $\sim$50 K, the susceptibility starts to deviate from the Curie-Weiss
behavior (Fig. 2 a). In accord with a previous study [20], neither anomalies
indicative of magnetic ordering nor signatures of a spin gap formation were
observed down to 1.8 K. To obtain more information on the low-$T$ regime, we
measured magnetization of the single crystal samples down to 0.1 K by the
capacitive Faraday method [24]. Again, the results do not show any sign of
magnetic ordering, despite the relatively large $\Theta$ value (Fig. 2 b).
This is one of the key experimental results. Furthermore, the low-$T$
susceptibility is highly anisotropic and is significantly suppressed in ${\bf
H}\\!\parallel\\!{[120]}$ than in ${\bf H}\\!\parallel\\!{\bf c}$. More
importantly, at the lowest field measured (0.1 T), $\chi_{\parallel}$ exhibits
paramagnetic behaviour ($\chi_{\parallel}\\!\propto\\!1/T$) down to 0.1 K,
whereas $\chi_{\perp}$ appears to saturate at low $T$. So the longitudinal and
transverse responses are qualitatively different. This is another key
experimental result, which suggests the presence of anisotropic interactions.

#### High-field magnetization measurements.

In a magnetic field, the star lattice models can feature exotic behavior with
a cascade of phases [11, 13]. To study the emergence of field-induced phases
in DiMACuS, we performed magnetization measurements in a pulsed high magnetic
field. Measurements at 4.2 K up to 65 T were performed by the induction method
in a magnetic field with the pulse length of 4 milliseconds (Fig. 2 c). As in
the magnetic susceptibility, M is much larger in ${\bf H}\\!\parallel\\!{\bf
c}$ than in ${\bf H}\\!\parallel\\!{[120]}$, with the powder data taking
intermediate values. While the magnetization increases rapidly below 20 T, its
slope $\frac{\partial{}M}{\partial{}H}$ first decreases as the field
increases, but then saturates around 30 T, which is reminiscent of a
magnetization plateau.

This feature is examined in more detail by measuring the magnetization process
at 0.13 K (Fig. 2 d, inset). In ${\bf H}\\!\parallel\\!{\bf c}$, the
magnetization increases steeply and becomes essentially flat above 2.5 T (with
$\frac{\partial{}M}{\partial{}H}$ exhibiting a saturating behaviour already at
1 T), at around 0.37 $\mu_{\rm{B}}$/Cu. With $g_{\parallel}\\!\simeq\\!2.24$
(estimated from the Curie-Weiss fit) this is close to $\frac{1}{3}$ of the
full moment. Therefore, we conclude that DiMACuS exhibits a $\frac{1}{3}$
magnetization plateau in ${\bf H}\\!\parallel\\!{\bf c}$. On the other hand,
the magnetization in ${\bf H}\\!\parallel\\!{[120]}$ gradually increases
without a clear flat region. The presence of a clear 1/3 plateau for ${\bf
H}\\!\parallel\\!{\bf c}$ despite the presence of anisotropic interactions is
the third key experimental result.

The end of the plateau-like regime is signalled by the increase of
$\frac{\partial{}M}{\partial{}H}$ which becomes noticeable around 40 T at 4.2
K. A more evident increase of magnetization is observed in ${\bf
H}\\!\parallel\\![120]$, where magnetization is suppressed at low magnetic
fields. The extrapolated (dashed) lines (Fig. 2 c) cross at 53 T, which may be
considered as the onset of the jump to saturation for ${\bf H}\\!\perp\\!{\bf
c}$. To study the behavior in higher fields, we performed magnetization
measurements by the induction method up to around 120 T (Fig. 2 e) generated
by the destructive single turn coil method with the pulse length of 7
microseconds [25]. Powder sample was used due to the small sample space.
$\frac{\partial{}M}{\partial{}H}$ exhibits a minimum at around 30 T and a
maximum at around 65 T, which are consistent with the data obtained in the
measurements up to 65 T, albeit with large noise caused by the magnetic field
generation. The field derivative becomes small and almost constant above 105
T, indicating that the fully polarized state is reached. Indeed, the
magnetization at 105 T is approximately three times larger than the value at
30 T in the $\frac{1}{3}$ plateau-like region.

#### Specific heat measurements.

The specific heat $C_{p}$ of the single crystal was measured by the relaxation
method using a commercial apparatus (PPMS, Quantum Design). At zero field,
$C_{p}/T$ shows a low-$T$ upturn (Fig. 2 f), which may indicate long-range
ordering below 0.1 K (the lowest $T$ in $\chi$ measurements). Measurements in
9 T applied along the c-axis reveal a considerable enhancement of $C_{p}/T$
above 2 K. While we do not have data below 2 K at 9 T, the system is already
in the $\frac{1}{3}$ plateau phase (Fig. 2 d), and the remaining magnetic
entropy should be small. Hence, the field-induced enhancement of $C_{p}/T$
above 2 K gives strong evidence for the existence of low-lying magnetic
excitations residing below 2 K in zero field and which are propelled to higher
energies by the field. This behaviour is a hallmark of isolated or weakly-
interacting spin-$S$ degrees of freedom emerging at low-energy scales. To shed
further light on this, we estimate the entropy content of the low-lying
excitations by integrating the difference of $C_{p}/T$ between 9 and 0 T from
2 K up to 20 K. The obtained entropy is approximately 1.8 J/K mol-Cu-1 at 20
K, very close to $\frac{1}{3}\textit{R}\ln 2$. This value is associated with
one doublet degree of freedom per Cu triangle. This is the fourth key
experimental finding.

#### Microscopic modeling.

We now set out to develop a microscopic description that accounts for all
experimental findings. Based on the crystal structure of DiMACuS, we can
identify three inequivalent exchange paths between neighbouring spins,
$J_{\rm{T}}$, $J_{\rm{D}}$, and $J_{\rm{H}}$ (Fig. 1 b), which can be
estimated by first-principles density-functional-theory (DFT) calculations. To
this end, we employed the generalized gradient approximation (GGA) [26] as
implemented in the full-potential code FPLO version 21 [27]. Following [20],
we keep only Cu3(OH)(SO4)4 magnetic layers and discard the organic cations. A
uniform background charge was used to retain electroneutrality. Since the
experimental structure features the unusually short hydroxyl bond length of
about 0.82 Å, we calculated the optimal hydrogen position ($z/c$ = 0.370053)
with respect to the GGA total energy. Next, we performed magnetic supercell
calculations using the GGA+$U$ functional with the Coulomb repulsion $U_{d}$
of $8.5\pm 1$ eV and Hund’s exchange $J_{d}$ of 1 eV, respectively, and the
fully localized limit as the double counting correction. Magnetic exchange
integrals were estimated by mapping the GGA+$U$ energies of eight different
magnetic configurations onto a classical Heisenberg model; the resulting
redundant linear problem was solved by the least-squares method. For
$U_{d}\\!=\\!9.5$ eV, we found $J_{\rm{T}}\\!=\\!81.5$ K,
$J_{\rm{D}}\\!=\\!5.4$ K, and $J_{\rm{H}}\\!=\\!0.3$ K (practically
negligible) [28]. This choice of $U_{d}$ is justified by the excellent
agreement between the calculated Weiss temperature
$\Theta\\!=\\!\frac{1}{2}J_{\rm{T}}\\!+\\!\frac{1}{4}J_{\rm{D}}\\!+\\!\frac{1}{2}J_{\rm{H}}\\!=\\!42.2$
K and its experimental value.

The $J_{\rm{T}}$–$J_{\rm{D}}$ model in the strong-coupling limit $J_{\rm{T}}$
$\gg$ $J_{\rm{D}}$ suggests an effective description in terms of isolated
$S=1/2$ Heisenberg triangles. Indeed, the isolated triangle model with
$J_{\rm{T}}$ = 58.5 K and $g$ = 2.205 accounts for the experimental $\chi(T)$
measured in ${\bf H}\\!\parallel\\!{\bf c}$ [28] and reproduces the
$\frac{1}{3}$ plateau, which corresponds to each triangle being in the
$S_{z}\\!=\\!\frac{1}{2}$ member of the Zeeman-split doublet (a similar
situation is observed in the frustrated cuprate volborthite featuring magnetic
trimers [29, 30]).

However, the isotropic model fails to describe the observed anisotropy.
Moreover, an isolated $S\\!=\\!1/2$ AFM Heisenberg triangle has two doublet
GSs and not one [28], and therefore does not capture the
$\frac{1}{3}\textit{R}\ln 2$ entropy content of the low-lying excitations
deduced from the $C_{p}$ data. To determine the relevant anisotropies, we
performed full relativistic noncollinear DFT+$U$ total energy calculations and
compute the bilinear exchange matrix $J^{\alpha\beta}_{\rm{T}}$ using the
energy mapping method [31]. Total energies were calculated with the projector-
augmented wave code VASP version 5.4.4 [32, *VASP_2] using standard
pseudopotentials [34] and the energy cutoff of 400 eV. These calculations were
done on a 2$\times$2$\times$2 $k$-mesh; for the interaction parameters, we
used $U_{d}$ and $J_{d}$ of 9.5 and 1 eV, respectively. In this way, we found
that [28]: i) the symmetric and traceless part of the exchange anisotropy is
extremely weak and can be safely disregarded, and ii) the dominant Heisenberg
exchange in DiMACuS is accompanied by a sizable DM anisotropy on the
$J_{\rm{T}}$ bonds (the DM anisotropy on the $J_{\rm{D}}$ bonds vanishes due
to inversion symmetry). The ${\bf D}_{\rm{T}}$ vectors are perpendicular to
the respective bonds and form acute angles of $\sim$50∘ to each other, with
$D_{\rm{T}}^{\parallel}/J_{\rm{T}}\\!=\\!0.274$,
$D_{\rm{T}}^{\perp}/J_{\rm{T}}\\!=\\!0.493$, and $|{\bf
D}_{\rm{T}}|/J_{\rm{T}}\\!=\\!0.56$ [28]. The latter ratio is remarkably
large, yet not unprecedented for cuprates [35, 36]. The sizable DM is
supported by the simulated susceptibility showing excellent agreement for both
field directions down to $\sim\\!10$ K (Fig. 3 a). It allows us to further
refine the magnetic exchanges: $J_{\rm{T}}\\!=\\!57$ K and $|{\bf
D}_{\rm{T}}|/J_{\rm{T}}\\!=\\!0.42$.

The DM interactions have a drastic impact on the physics of weakly coupled
triangles. The leading contribution stems from $D_{\rm{T}}^{\parallel}$, while
the influence of in-plane DM components is zero to first order in
$D_{\rm{T}}^{\perp}/J_{\rm{T}}$ [28]. Our numerical simulations (Fig. 3 and
[28]), which include all DM components, confirm this explicitly. So, to a good
approximation, $D_{\rm{T}}^{\perp}$ can be safely disregarded, and the system
effectively features two emergent (i.e., approximate) symmetries at low
energies: U(1) spin rotation and a threefold spatial rotation, both around the
${\bf c}$ axis.

Hence, for ${\bf H}\parallel{\bf c}$, the states of the system have well
defined total moment $S_{z}$ along ${\bf c}$ and chirality $\ell$ (irreps of
threefold spatial rotation). Now, in the absence of $D_{\rm{T}}^{\parallel}$,
each triangle has two doublet GSs, separated by the $S\\!=\\!3/2$ quartet by a
gap of $3J_{T}/2\\!\simeq\\!85$ K (Fig. 3 b, left). At lower temperatures, the
$J_{T}$ scale disappears from the problem, and we are left with two doublets
of opposite chirality $\ell$ (irreps of threefold rotation symmetry).
$D_{\rm{T}}^{\parallel}$ lifts the degeneracy of the two doublets and
introduces a new energy scale, the gap
$\Delta\\!=\\!\sqrt{3}|D_{\rm{T}}^{\parallel}|\\!\simeq\\!35$ K, (Fig. 3 b,
right). At lower $T$, this scale also disappears from the problem, and we are
left with one doublet per triangle, in agreement with the $C_{p}$ data (see
Fig. 3 d, which also demonstrates the characteristic field-induced entropic
shift seen experimentally, at the level of two coupled triangles).

Figure 3:  (a) Magnetic susceptibility at 1 T for different field directions
(symbols) and fits (lines) with the anisotropic triangle Heisenberg model with
$|{\bf D}|/J\\!=\\!0.42$ ($J\\!=\\!56.5$ K, $g\\!=\\!2.249$, and
$\chi_{0}\\!=\\!-2.82\times 10^{-4}$ emu / mol Cu for ${\bf H}\\!\perp\\!{\bf
c}$ and $J\\!=\\!58.5$ K, $g\\!=\\!2.205$, and $\chi_{0}\\!=\\!-3.18\times
10^{-4}$ emu / mol Cu for ${\bf H}\\!\parallel\\!{\bf c}$). (b) Spectra of a
Heisenberg triangle and a triangle with $D_{\rm{T}}^{\parallel}$. (c)
Magnetization curve simulated for the parameters from (a). (d) Simulated
specific heat for an isolated triangle ($J_{D}\\!=\\!0$) at 0 and 9 T, and for
two triangles coupled with $J_{D}\\!=\\!4$ K. (e) Powder-averaged
magnetization isotherm of an isolated triangle with $|{\bf D}|/J\\!=\\!0.42$
(solid line, obtained by averaging over a Fibonacci sphere of 500 points), in
comparison with the isotropic triangle (dashed lines) and the experiment
(circles). Matplotlib [37] was used for plotting (a) and (c)-(e). For plotting
scripts and numerical data, see Ref. [38].

The peculiar anisotropy of DiMACuS is governed by the emergent U(1) symmetry:
unaffected by a field along ${\bf c}$, but broken for fields perpendicular to
${\bf c}$. This is manifest as follows [28]. $D_{\rm{T}}^{\parallel}$ gives
rise to a characteristic locking between $S_{z}$ and $\ell$: the
$S_{z}\\!=\\!\pm 1/2$ members of one doublet have $\ell\\!=\\!\pm 1$, whereas
for the other doublet $\ell\\!=\\!\mp 1$. Hence, a field along ${\bf c}$ does
not couple the two doublets; instead, it gives rise to a Zeeman splitting for
each doublet, the Curie-like susceptibility
$\chi_{\parallel}\\!\propto\\!1/(4T)$ at temperatures sufficiently below
$\Delta$, and the flat 1/3 magnetization plateau (Fig. 3 c), independent of
$D_{\rm{T}}^{\parallel}$. By contrast, an in-plane field breaks the U(1)
symmetry explicitly and thus couples the two doublets, leading to a standard
level repulsion, with each doublet retaining its twofold degeneracy. At
low-$T$, $\chi_{\perp}$ approaches a finite value (Fig. 3 a), inversely
proportional to $D_{\rm{T}}^{\parallel}$, and the magnetization behaves as
$h/\sqrt{h^{2}+3(D_{\rm{T}}^{\parallel})^{2}}$ (where
$h\\!=\\!g_{\perp}\mu_{B}H$): linear at low fields (Fig. 3 c), and
asymptotically approaching the 1/3 value, without developing a flat plateau.
Precisely this behavior is observed experimentally in Fig. 2 (c,d).

Next, we consider the high-field magnetization of the isolated-triangle model.
On exiting the 1/3 plateau, the magnetization shows an almost abrupt jump to
full saturation at $H^{\ast}$ equal to $3/2J_{\rm{T}}$ in the isotropic case.
$D_{\rm{T}}$ propels the saturation to higher fields, but only for ${\bf
H}\\!\parallel\\!{\bf c}$. By using the above refined values of $J_{\rm{T}}$
and $D_{\rm{T}}$ in analytical expressions for $H^{\ast}$ [28], we obtain
$H_{\parallel}^{\ast}\\!\simeq\\!64$ T and $H_{\perp}^{\ast}\\!\simeq\\!56$ T.
The latter is in satisfactory agreement with the endpoint of the plateau-like
region for ${\bf H}\\!\perp\\!{\bf c}$ (Fig. 2 c). The difference between
$H_{\parallel}^{\ast}$ and $H_{\perp}^{\ast}$ is further corroborated by
numerical simulations of the powder-averaged magnetization (Fig. 3, e). We
note finally that, in the powder measurements, the value of $H^{\ast}$ is
distributed between $H_{\parallel}^{\ast}$ and $H_{\perp}^{\ast}$, effectively
destroying the magnetization jumps predicted for both field directions.

The isolated triangle model does not fully account for the specific heat data:
the expected gapped zero-field spectrum (Fig. 3 d) is in sharp contrast with
the divergence observed below 0.5 K (Fig. 2 f). The root cause of this
discrepancy are interactions between the triangles: for a minimal model of two
triangles coupled with $J_{\rm{D}}$, such a peak readily appears at a
temperature very close to the experimentally observed (Fig. 3 d). Obviously,
the actual connectivity of triangles in DiMACuS follows a honeycomb lattice,
and its realistic simulation requires a much larger number of spins. The
interactions between the triangles can be described by an effective model, in
which each spin triangle is treated as a rigid entity. While a full analysis
of this model is beyond our scope, we note that the rescaling of effective
spin lengths also reduces the dominant exchange scale from $J_{\rm{D}}$ to
$J_{\rm{D}}/9$, which may explain the lack of ordering down to 0.1 K.

Finally, we comment on the disorder in the dimethylammonium molecule and
defect in the crystal water intercalated between the inorganic layers. A
slight orientational disorder of the dimethylammonium molecule is reported
[20]. On the organic molecular Mott insulator, there is an argument that
random freezing of the electric polarization of the molecules causes magnetic
bond randomness, that can result in a gapless spin liquid behavior [39]. In
contrast to the organic Mott insulator where the molecule itself carries the
spin, the dimethylammonium and H2O molecules in DiMACuS are non-magnetic and
not involved in the magnetism in the Cu3(OH)(SO4)4 layer. Indeed, no
structural disorder is reported in the Cu3(OH)(SO4)4 layer [20]. While we
anticipate the effects of disorder on the magnetism to be weak, this may still
hinder the ordering at very low $T$.

#### Summary and outlook.

We demonstrate DiMACuS as a realization of a spin-$\frac{1}{2}$ star lattice
antiferromagnet, one of the paradigms for geometric frustration and resonating
valence bond physics. The main experimental puzzles, including the absence of
magnetic ordering down to very low temperatures, the characteristic field-
induced entropic shift seen in the specific heat data, and the peculiar
anisotropy in the magnetic response, can all be accounted for by the strong
frustration in the Cu triangles and the sizable Dzyaloshinskii-Moriya
anisotropy. Further studies are needed to clarify if DiMACuS orders
magnetically at a very low temperature and investigate the possible role of
disorder.

Star lattice magnets are known to exhibit chiral spin liquid state in the
presence of Kitaev-type anisotropic interaction [15]. Substituting Cu with
divalent Co may substantially enhance the exchange anisotropy and give rise to
bond-dependent interactions that underlie the Kitaev physics. As demonstrated
in a kagome system, cobaltate analogs of cuprates exist in nature [40] and can
be synthesized [41]. Synthesis of relative materials with different magnetic
cations will pave the way to explore the star lattice magnetism from the
extreme quantum case to the classical large-$S$ limit.

###### Acknowledgements.

We thank Dr. D. Nishio-Hamane for taking the optical image of the crystal. We
thank Prof. Z. Hiroi for fruitful discussions and U. Nitzsche for technical
assistance. HI was supported by JSPS KAKENHI Grants No. JP22H04467 and No. JP
22K13996. OJ was supported by the Leibniz Association through the Leibniz
Competition and the German Forschungsgemeinschaft (DFG, German Research
Foundation) through SFB 1143 (project-id 247310070). IR acknowledges the
support by the Engineering and Physical Sciences Research Council, Grant No.
EP/V038281/1.

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Supplemental Material:
Geometric frustration and Dzyaloshinskii-Moriya interactions
in a quantum star lattice hybrid copper sulfate

Hajime Ishikawa, Yuto Ishii, Takeshi Yajima, Yasuhiro H. Matsuda, Koichi
Kindo, Yusei Shimizu, Ioannis Rousochatzakis, Ulrich K. Rößler, and Oleg
Janson

In this supplementary material we provide technical derivations and auxiliary
information for: i) the first-principles calculations of the isotropic
(Heisenberg) and anisotropic, in particular Dzyaloshinskii–Moriya (DM),
interactions in DiMACuS (Sec. I), ii) the quantum-mechanical problem of a
single spin-1/2 triangle with antiferromagnetic (AF) exchange $J$ and DM
interactions (Sec. II), and iii) simulations of the magnetic susceptibility of
this model and fits to the experimental magnetic susceptibility of DiMACuS
that facilitate an accurate estimation of the exchange parameters (Sec. III).

##  I Estimation of the exchange integrals

###  I.1 Isotropic (Heisenberg) exchanges

In DFT calculations, we consider the experimental crystal structure of DiMACuS
[1], remove the organic cation, add a uniform background charge to ensure the
electroneutrality, and use the GGA-optimized position for H atoms. All
calculations are done using the rhombohedral unit cell, whose volume is three
times smaller than that of the hexagonal cell depicted in Fig. S1 (left).
Estimates for the three leading exchanges as a function of the Coulomb
repulsion $U_{d}$ are provided in Table S1. The value of $U_{d}=9.5$ eV gives
an excellent agreement with the experimental Weiss temperature.

Figure S1:  Left: Crystal structure of the magnetic Cu3(OH)(SO4)4 layers in DiMACuS, featuring distorted CuO4 plaquettes (blue) and SO4 tetrahedra (orange). The unit cell parameters are $a=b=12.7222$ Å, $c=26.0836$ Å, $\alpha=\beta=90^{\circ}$, $\gamma=120^{\circ}$ (hexagonal axes) or $a=b=c=11.31263$ Å, $\alpha=\beta=\gamma=68.435^{\circ}$ (rhombohedral axes). All calculations are done for the rhombohedral cell. The hexagonal unit cell is plotted with gray lines. Right: Crystalline environments facilitating the three leading in-plane exchange couplings $J_{\rm{T}}$, $J_{\rm{D}}$, and $J_{\rm{H}}$. The crystal structure was plotted using vesta [2]. Table S1:  Exchange integrals $J_{\rm{T}}$, $J_{\rm{D}}$, $J_{\rm{H}}$ and the Weiss temperature $\theta$ (all in K) estimated from GGA+$U$ ($U_{d}$ = 9.5 eV, $J_{d}$ = 1 eV) total energies for 8 magnetic configurations. exchange | bonds per spin | $d_{\text{Cu..Cu}}$ (Å) |  $J$ (K)
---|---|---|---
|  | FPLO | vasp
|  | $U_{d}=7.5$ | $U_{d}=8.5$ | $U_{d}=9.5$ | $U_{d}=9.5$
$J_{\rm{T}}$ | $1$ | $3.2797$ | $124.5$ | $101.3$ | $\bf{81}.\bf{5}$ | $77.1$
$J_{\rm{D}}$ | $\nicefrac{{1}}{{2}}$ | $4.4387$ | $8.5$ | $6.8$ | $\bf{5}.\bf{4}$ | $4.1$
$J_{\rm{H}}$ | $1$ | $6.0010$ | $0.2$ | $0.3$ | $\bf{0}.\bf{3}$ | $0.4$
$\theta$ |  |  | $64.5$ | $52.5$ | $\bf{42}.\bf{2}$ | $39.8$

###  I.2 Anisotropic components of $J_{\rm{T}}$

The experimentally observed anisotropy of the magnetic susceptibility despite
the similarity of the Weiss temperature for both field directions hints at
possible relevance of the antisymmetric DM exchange $\mathbf{D}_{\rm{T}}$. To
estimate this exchange numerically, we determine all elements of the
respective bilinear exchange matrix $\mathbf{M}_{\rm{T}}$ by using the energy
mapping method [3], and then decomposing this matrix as

${\bf M}_{\rm{T}}=J_{\rm{T}}~{}{\bf 1}+\bm{\Sigma}_{\rm{T}}+{\bf
A}_{\rm{T}}\,,$ (1)

where ${\bf 1}$ is the 3$\times$3 identity matrix and

$J_{\rm{T}}=\frac{1}{3}~{}\text{Tr}[\mathbf{M}_{\rm{T}}]$ (2)

is the Heisenberg coupling,

$\bm{\Sigma}_{\rm{T}}=\frac{1}{2}\left({\bf M}_{\rm{T}}+{\bf
M}_{\rm{T}}^{\top}\right)-\frac{1}{3}~{}\text{Tr}[{\bf M}_{\rm{T}}]~{}{\bf
1}=\bm{\Sigma}_{\rm{T}}^{\top}$ (3)

is the symmetric and traceless part of ${\bf M}_{\rm{T}}$ (${\bf
M}_{\rm{T}}^{\top}$ is the transpose matrix of ${\bf M}_{\rm{T}}$), and

${\bf A}_{\rm{T}}=\frac{1}{2}\left({\bf M}_{\rm{T}}-{\bf
M}_{\rm{T}}^{\top}\right)=-{\bf A}_{\rm{T}}^{\top}$ (4)

is the anti-symmetric part of ${\bf M}_{\rm{T}}$, which is related to the
components of ${\bf D}_{\rm{T}}$ as
$A^{\alpha\beta}_{\rm{T}}=\epsilon^{\alpha\beta\gamma}D_{\rm{T}}^{\gamma}$,
or, more explicitly,

${\bf A}_{\rm{T}}=\begin{pmatrix}0&D_{\rm{T}}^{Z}&-D_{\rm{T}}^{Y}\\\
-D_{\rm{T}}^{Z}&0&D_{\rm{T}}^{X}\\\
D_{\rm{T}}^{Y}&-D_{\rm{T}}^{X}&0\end{pmatrix}\,,$ (5)

where $X$, $Y$, $Z$ refer to the crystallographic (Cartesian) axes.

Table S2: Positions of selected atoms in the unit cell in the crystallographic (Cartesian) frame (Fig. S2). site |  $X$ (Å) |  $Y$ (Å) |  $Z$ (Å) |  | site |  $X$ (Å) |  $Y$ (Å) |  $Z$ (Å)
---|---|---|---|---|---|---|---|---
Cu(1) | $1.571222$ | $3.828742$ | $7.097030$ |  | Cu(3) | $4.579598$ | $2.909116$ | $6.169293$
Cu(2) | $3.265568$ | $5.665752$ | $4.973043$ |  | S(7) | $2.196772$ | $2.893669$ | $4.255106$
Figure S2:  Local structure of the Cu(1)-Cu(2)-Cu(3) triangles, with the
S(7)O4 tetrahedron (shaded orange) above the plane of the spins. The red
arrows show the orientations of the DM vectors on the nearest-neighbour bonds
around the triangle. We also show three different frames: i) The
crystallographic $(a,b,c)$ frame corresponding to the rhombohedral unit cell.
This frame differs from the $({\bf a},{\bf b},{\bf c})$ frame of the main
text, which corresponds to the hexagonal unit cell. ii) The Cartesian
$(X,Y,Z)$ frame pertains to the crystallographic coordinate system (same as in
Table S2). iii) The coordinate system $(x,y,z)$ of a single triangle, with the
S(7)O4 tetrahedron sitting above the plane of the spins. The transformation
$(X,Y,Z)\rightarrow(x,y,z)$ is described by $R_{\Delta}$ in Eq. (10). The
structural elements were plotted using vesta [2].

The crystal structure of DiMACuS imposes no symmetry constraints on the DM
vector $\mathbf{D}_{\rm{T}}$, but all vectors on different bonds in the cell
are connected by symmetry elements. In particular, the three vectors belonging
to the same triangle are connected by $C_{3}$ symmetry. In the triangle formed
by the atoms Cu(1), Cu(2), and Cu(3) (Table S2), we consider the first
interatomic vector $\mathbf{r}_{\text{Cu(1)-\text{Cu(3)}}}$. After performing
36 full-relativistic DFT+$U$ total energy calculations, we obtain the
following bilinear exchange matrix:

$\mathbf{M}_{\rm{T}}^{\text{Cu(1)-\text{Cu(3)}}}=\begin{pmatrix}66.0&-35.9&\phantom{+0}3.1\\\
35.9&\phantom{+}65.5&-11.3\\\
-1.6&\phantom{+}11.6&\phantom{+}69.1\end{pmatrix}\text{K}\,,$ (6)

from which we get

$J_{\rm{T}}=66.9~{}\text{K}\,,$ (7)

$\mathbf{D}_{\rm{T}}^{\text{Cu(1)-\text{Cu(3)}}}=\left(-11.5,-2.3,-35.9\right)\text{K}\,,$
(8)

and

$\bm{\Sigma}_{\rm{T}}^{\text{Cu(1)-\text{Cu(3)}}}=\begin{pmatrix}-0.9&0&0.8\\\
0&-1.3&0.2\\\ 0.8&0.2&2.2\end{pmatrix}\text{K}\,,$ (9)

in the $(X,Y,Z)$ frame. So the elements of the symmetric and traceless part of
the exchange anisotropy are extremely weak compared to the antisymmetric part,
and can therefore be disregarded.

In the following, it is more convenient to resort to a different coordinate
system, denoted by $(x,y,z)$ in Fig. S2, with the midpoint of
Cu(1)-Cu(2)-Cu(3) triangle at the origin and the $z$ axis coinciding with the
$C_{3}$ axis which crosses the midpoint and the neighboring S(7) atom.
Importantly, there are two such frames, one for the Cu triangles with the S(7)
atom above the Cu plane, and another for the Cu triangles with S(7) atoms
below the Cu plane. The two frames are related by the inversion center in the
middle of the $J_{D}$ bond that connects the two types of triangles. In the
following, we shall use the $(x,y,z)$ frame of the Cu triangles with S(7)
above the Cu plane as our global frame. The directions of the DM vectors on
the Cu triangles with S(7) atoms below the Cu plane can be obtained by
applying inversion symmetry (see site-labelling convention indicated by red
arrows in Fig. S3).

Figure S3: Lattice structure of Cu atoms in DiMACus, showing the most relevant
Heisenberg couplings. Red arrows indicate our site-labelling convention
$i\\!\to\\!j$ associated with the DM coupling term ${\bf D}_{ij}\cdot{\bf
S}_{i}\times{\bf S}_{j}$ between spin sites $i$ and $j$. For the actual
directions of these vectors see Fig. S2 and Eqs. (12)-(14).

The crystallographic (Cartesian) coordinates are transformed into this new
coordinate system by the following matrix:

$\mathbf{R}_{\Delta}=\begin{pmatrix}\phantom{+}0.35633105&-0.84786434&\phantom{+}0.39262481\\\
-0.84786434&-0.11683799&\phantom{+}0.51717980\\\
-0.39262481&-0.51717980&-0.76050695\end{pmatrix}\,.$ (10)

By shifting the origin and applying this rotation matrix to the atomic
coordinates from Table S2, we obtain the coordinates in the $(x,y,z)$ frame:

$\begin{split}\mathbf{r}_{1}&=\left(0.1001,1.8909,0\right)\text{\AA},\\\
\mathbf{r}_{2}&=\mathbf{R}_{z}(\nicefrac{{2\pi}}{{3}})\cdot\mathbf{r}_{1}=\left(-1.6876,-0.8588,0\right)\text{\AA},\\\
\mathbf{r}_{3}&=\mathbf{R}_{z}(-\nicefrac{{2\pi}}{{3}})\cdot\mathbf{r}_{1}=\left(1.5875,-1.0321,0\right)\text{\AA},\end{split}\quad\hfill\quad\begin{split}\mathbf{r}_{13}\equiv\mathbf{r}_{3}-\mathbf{r}_{1}&=\left(1.4874,-2.9231,0\right)\text{\AA},\\\
\mathbf{r}_{32}\equiv\mathbf{r}_{2}-\mathbf{r}_{3}&=\left(1.7877,2.7497,0\right)\text{\AA},\\\
\mathbf{r}_{21}\equiv\mathbf{r}_{1}-\mathbf{r}_{2}&=\left(-3.2752,0.1734,0\right)\text{\AA},\end{split}$
(11)

where $\mathbf{R}_{z}(\theta)$ describes the rotation around the new $z$ axis
by the angle $\theta$. Here, we replaced Cu(1), Cu(2), and Cu(3) superscripts
with 1, 2, and 3 to emphasize that we work in the new coordinate system
associated with a single triangle. By applying $\mathbf{R}_{\Delta}$ from Eq.
(10) to the DM vector in Eq. (8), we obtain the DM vectors in the $(x,y,z)$
frame:

$\displaystyle\mathbf{R}_{\Delta}\cdot\mathbf{D}_{\rm{T}}^{\text{Cu(1)-\text{Cu(3)}}}=\mathbf{D}_{\rm{T}}^{13}=-\mathbf{D}_{\rm{T}}^{31}$
$\displaystyle=$ $\displaystyle\left(-16.2,-8.6,33.0\right)\text{K}\,.$ (12)

The symmetrically equivalent DM vectors on other bonds of the triangle are:

$\displaystyle\mathbf{D}_{\rm{T}}^{21}=-\mathbf{D}_{\rm{T}}^{12}=$
$\displaystyle\mathbf{R}_{z}(\frac{2}{3}\pi)\cdot\mathbf{D}_{\rm{T}}^{13}$
$\displaystyle=\left(15.5,-9.7,33.0\right)\text{K}\,,$ (13)
$\displaystyle\mathbf{D}_{\rm{T}}^{32}=-\mathbf{D}_{\rm{T}}^{23}=$
$\displaystyle\mathbf{R}_{z}(-\frac{2}{3}\pi)\cdot\mathbf{D}_{\rm{T}}^{13}$
$\displaystyle=\left(0.7,18.3,33.0\right)\text{K}\,.$ (14)

The orientation of $\mathbf{D}_{\rm{T}}$ vectors with respect to the
constituents of the crystal structure is shown in Fig. S2. The vectors are
perpendicular to the respective bonds and form $60^{\circ}$ angle with the
line connecting the midpoint of the respective bond and the third Cu atom. For
instance, for $\mathbf{D}_{\rm{T}}^{13}$, this line connects the midpoint of
$\mathbf{r}_{13}$ and Cu(2). The angle between each two of the three DM
vectors is close to $50^{\circ}$.

Finally, it is convenient to express $\mathbf{D}_{\rm{T}}$ in units of
$J_{\rm{T}}\equiv\frac{1}{3}\operatorname{Tr}\left[\mathbf{M}_{\rm{T}}\right]$
and decompose them into components perpendicular and parallel to the $z$ axis
of a triangle, which coincides with the $c$ axis of the hexagonal unit cell in
Fig. S1. We obtain:

$D_{\rm{T}}^{\parallel}/J_{\rm{T}}=0.493\,,~{}~{}{D_{\rm{T}}^{\perp}}/{J_{\rm{T}}}=0.274\,,~{}~{}{D_{\rm{T}}^{\perp}}/{D_{\rm{T}}^{\parallel}}\simeq
1/\sqrt{3}\,.$ (15)

##  II The problem of a single antiferromagnetic spin-1/2 triangle with DM
interactions

We consider the problem of a single spin-1/2 triangle in the presence of AF
exchange $J$, DM interactions and an external field ${\bf H}$. As in DiMACuS,
the triangle features a threefold rotational axis along $z$ (perpendicular to
the plane of the triangle), going through the center of the triangle. This
transformation maps the individual spin operators as follows

$\begin{array}[]{l}(S_{1}^{x},S_{1}^{y},S_{1}^{z})\mapsto(-\frac{1}{2}S_{2}^{x}+\frac{\sqrt{3}}{2}S_{2}^{y},-\frac{\sqrt{3}}{2}S_{2}^{x}-\frac{1}{2}S_{2}^{y},S_{2}^{z})\\\
(S_{2}^{x},S_{2}^{y},S_{2}^{z})\mapsto(-\frac{1}{2}S_{3}^{x}+\frac{\sqrt{3}}{2}S_{3}^{y},-\frac{\sqrt{3}}{2}S_{3}^{x}-\frac{1}{2}S_{3}^{y},S_{3}^{z})\\\
(S_{3}^{x},S_{3}^{y},S_{3}^{z})\mapsto(-\frac{1}{2}S_{1}^{x}+\frac{\sqrt{3}}{2}S_{1}^{y},-\frac{\sqrt{3}}{2}S_{1}^{x}-\frac{1}{2}S_{1}^{y},S_{1}^{z})\end{array}$
(16)

###  II.1 Hamiltonian

This problem is described by the Hamiltonian

$\mathcal{H}=\mathcal{H}_{J}+\mathcal{H}_{\text{DM}}+\mathcal{H}_{\text{Z}},$
(17)

where $\mathcal{H}_{J}$ comprises the Heisenberg interactions,
$\mathcal{H}_{\text{DM}}$ the DM interactions, and $\mathcal{H}_{\text{Z}}$
the Zeeman coupling to the external field. The latter takes the usual form

$\mathcal{H}_{\text{Z}}=g\mu_{B}{\bf S}\cdot{\bf H}\,,$ (18)

where we have assumed an isotropic $g$ factor. The Heisenberg terms take the
form

$\mathcal{H}_{J}=J\left(\mathbf{S}_{1}\cdot\mathbf{S}_{2}+\mathbf{S}_{2}\cdot\mathbf{S}_{3}+\mathbf{S}_{3}\cdot\mathbf{S}_{1}\right)=\frac{J}{2}\mathbf{S}^{2}-\frac{9}{8}J\,,$
(19)

where ${\bf S}={\bf S}_{1}+{\bf S}_{2}+{\bf S}_{3}$ is the total spin. The DM
terms take the form

$\mathcal{H}_{\text{DM}}={\bf D}_{12}\cdot({\bf S}_{1}\times{\bf S}_{2})+{\bf
D}_{23}\cdot({\bf S}_{2}\times{\bf S}_{3})+{\bf D}_{31}\cdot({\bf
S}_{3}\times{\bf S}_{1})\,,$ (20)

where the DM vectors ${\bf D}_{12}$, ${\bf D}_{23}$ and ${\bf D}_{31}$ are
related to each other via the threefold rotational symmetry, namely

${\bf D}_{23}={\bf R}\cdot{\bf D}_{12},~{}~{}{\bf D}_{31}={\bf R}\cdot{\bf
D}_{23}\,,$ (21)

where

${\bf R}=\left(\begin{array}[]{ccc}-\frac{1}{2}&-\frac{\sqrt{3}}{2}&0\\\
\frac{\sqrt{3}}{2}&-\frac{1}{2}&0\\\ 0&0&1\end{array}\right)\,.$ (22)

More explicitly, denoting the three components of ${\bf D}_{12}$ by $a$, $b$
and $c$, we have

${\bf D}_{12}=\left(a,b,c\right),~{}~{}~{}{\bf
D}_{23}=\left(-\frac{1}{2}a-\frac{\sqrt{3}}{2}b,\frac{\sqrt{3}}{2}a-\frac{1}{2}b,c\right),~{}~{}~{}{\bf
D}_{31}=\left(-\frac{1}{2}a+\frac{\sqrt{3}}{2}b,-\frac{\sqrt{3}}{2}a-\frac{1}{2}b,c\right)\,.$
(23)

For later purposes, let us denote the terms arising from the out-of-plane
components of the DM vectors by $\mathcal{H}_{\text{DM},z}$ and the terms
arising from the in-plane components by $\mathcal{H}_{\text{DM},xy}$, namely

$\mathcal{H}_{\text{DM}}=\mathcal{H}_{\text{DM},z}+\mathcal{H}_{\text{DM},xy}\,.$
(24)

The first term, which takes the form

$\displaystyle\mathcal{H}_{\text{DM},z}$ $\displaystyle=$ $\displaystyle
c\left(S_{1}^{x}S_{2}^{y}-S_{1}^{y}S_{2}^{x}+S_{2}^{x}S_{3}^{y}-S_{2}^{y}S_{3}^{x}+S_{3}^{x}S_{1}^{y}-S_{3}^{y}S_{1}^{x}\right)$
(25) $\displaystyle=$
$\displaystyle-\frac{c}{2i}\left(S_{1}^{+}S_{2}^{-}-S_{1}^{-}S_{2}^{+}+S_{2}^{+}S_{3}^{-}-S_{2}^{-}S_{3}^{+}+S_{3}^{+}S_{1}^{-}-S_{3}^{-}S_{1}^{+}\right),$

plays a dominant role in DiMACuS, as we discuss in the main text.

###  II.2 Convenient set of basis states and matrix representation of
$\mathcal{H}$

A convenient set $\mathcal{B}_{1}$ of basis states which diagonalizes
$\mathcal{H}_{J}$. This set is labeled as $\\{|S_{23},S,M\rangle\\}$, where
$S_{23}$ is the spin quantum number associated with ${\bf S}_{23}={\bf
S}_{2}+{\bf S}_{3}$, $S$ is the total spin quantum number (associated with
${\bf S}={\bf S}_{1}+{\bf S}_{23}$), and $M$ is its projection along the $z$
axis. The addition of three spins-1/2 gives two doublets and one quartet:

$\frac{1}{2}\otimes\frac{1}{2}\otimes\frac{1}{2}=\frac{1}{2}\otimes(0\oplus
1)=\frac{1}{2}\oplus\frac{1}{2}\oplus\frac{3}{2}\,.$ (26)

More explicitly, the quantum number $S_{23}$ can be either 0 or 1. For
$S_{23}=0$, we get the $S=1/2$ doublet

$\displaystyle|0,1/2,1/2\rangle=\frac{1}{2}\left(|\\!\uparrow\uparrow\downarrow\rangle-|\\!\uparrow\downarrow\uparrow\rangle\right),~{}~{}~{}|0,1/2,-1/2\rangle=\frac{1}{2}\left(|\\!\downarrow\uparrow\downarrow\rangle-|\\!\downarrow\downarrow\uparrow\rangle\right)\,.$
(27)

For $S_{23}=1$, the total spin can be either $1/2$ or $3/2$. The former
consists of the states

$|1,1/2,1/2\rangle=\frac{1}{\sqrt{6}}\left(2|\downarrow\uparrow\uparrow\rangle-|\uparrow\downarrow\uparrow\rangle-|\uparrow\uparrow\downarrow\rangle\right)\,,~{}~{}~{}|1,1/2,-1/2\rangle=\frac{1}{\sqrt{6}}\left(2|\uparrow\downarrow\downarrow\rangle-|\downarrow\uparrow\downarrow\rangle-|\downarrow\downarrow\uparrow\rangle\right)\,,$
(28)

whereas the $S=3/2$ quartet consists of

$\begin{array}[]{l}|1,3/2,3/2\rangle=|\\!\uparrow\uparrow\uparrow\rangle\\\
|1,3/2,1/2\rangle=\frac{1}{\sqrt{3}}\left(|\\!\downarrow\uparrow\uparrow\rangle+|\\!\uparrow\downarrow\uparrow\rangle+|\\!\uparrow\uparrow\downarrow\rangle\right)\\\
|1,3/2,-1/2\rangle=\frac{1}{\sqrt{3}}\left(|\\!\uparrow\downarrow\downarrow\rangle+|\\!\downarrow\uparrow\downarrow\rangle+|\\!\downarrow\downarrow\uparrow\rangle\right)\,,\\\
|1,3/2,-3/2\rangle=|\\!\downarrow\downarrow\downarrow\rangle\,.\end{array}$
(29)

According to Eq. (19), the $\\{|S_{23},S,M\rangle\\}$ basis is an eigenbasis
of $\mathcal{H}_{J}$, with

$\mathcal{H}_{J}|S_{23},S,M\rangle=\left(\frac{J}{2}S(S+1)-\frac{9}{8}J\right)|S_{23},S,M\rangle\,.$
(30)

The two doublets are degenerate with energy $-3J/4$ and the quartet has energy
$3J/4$.

Now, let us examine the symmetry of these states under the threefold rotation
which maps the spin indices (123) to (312). In general, the eigenvalues of
$C_{3}$ can be labeled as $e^{i\frac{2\pi}{3}\ell}$, where $\ell=0$, $1$ and
$-1$. The quartet belongs to the sector $\ell=0$, since
$C_{3}|1,3/2,M\rangle=|1,3/2,M\rangle$ for any $M$. The two doublets are not
eigenstates of $C_{3}$ but we can form linear combinations which are. These
combinations can be labeled as $|\ell,\sigma\rangle$ with $\ell=\pm 1$ and
$\sigma=\pm 1/2$ (or $\uparrow$ and $\downarrow$), and are given by:

$|\ell,\sigma\rangle=\frac{1}{\sqrt{3}}\left(|-\sigma,\sigma,\sigma\rangle+e^{-i\frac{2\pi}{3}\ell}|\sigma,-\sigma,\sigma\rangle+e^{i\frac{2\pi}{3}\ell}|\sigma,\sigma,-\sigma\rangle\right)\,,$
(31)

with

$\displaystyle
C_{3}~{}|\ell,\sigma\rangle=e^{i\frac{2\pi}{3}\ell}~{}|\ell,\sigma\rangle\,.$
(32)

Note that the $M=\pm 1/2$ members of the quartet [second and third line of
(29)] are also of the form of Eq. (31), but with $\ell=0$. Altogether, we have
the following new set $\mathcal{B}_{2}$ of basis states:

$\mathcal{B}_{2}=\left\\{|\ell\\!=\\!1,\uparrow\rangle,~{}|\ell\\!=\\!-1,\downarrow\rangle,~{}|\ell\\!=\\!-1,\uparrow\rangle,~{}|\ell\\!=\\!1,\downarrow\rangle,~{}|\\!\uparrow\uparrow\uparrow\rangle,~{}|\ell\\!=\\!0,\uparrow\rangle,~{}|\ell\\!=\\!0,\downarrow\rangle,~{}|\\!\downarrow\downarrow\downarrow\rangle\right\\}\,.$
(33)

In this basis, the Hamiltonian in the presence of a field

${\bf H}={\bf h}/(g\mu_{B}),~{}~{}{\bf h}=h_{\perp}(\cos\phi\hat{\bf
x}+\sin\phi\hat{\bf y})+h_{z}\hat{\bf z}\,,$ (34)

takes the form

$\mathcal{H}\\!=\\!\frac{1}{2}\\!\left(\begin{array}[]{cc|cc|cccc}-\frac{3J}{2}\\!-\\!\sqrt{3}c\\!+\\!h_{z}&0&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-h_{\perp}e^{-i\phi}}&0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\gamma^{\ast}}&0\\\
0&-\frac{3J}{2}\\!-\\!\sqrt{3}c\\!-\\!h_{z}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-h_{\perp}e^{i\phi}}&0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-\gamma}&0&0\\\
\hline\cr
0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-h_{\perp}e^{-i\phi}}&-\frac{3J}{2}\\!+\\!\sqrt{3}c\\!+\\!h_{z}&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sqrt{3}\gamma}&0&0&0\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-h_{\perp}e^{i\phi}}&0&0&-\frac{3J}{2}\\!+\\!\sqrt{3}c\\!-\\!h_{z}&0&0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-\sqrt{3}\gamma^{\ast}}\\\
\hline\cr
0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sqrt{3}\gamma^{\ast}}&0&\frac{3J}{2}\\!+\\!3h_{z}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sqrt{3}h_{\perp}e^{-i\phi}}&0&0\\\
0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-\gamma^{\ast}}&0&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sqrt{3}h_{\perp}e^{i\phi}}&\frac{3J}{2}\\!+\\!h_{z}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2h_{\perp}e^{-i\phi}}&0\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\gamma}&0&0&0&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2h_{\perp}e^{i\phi}}&\frac{3J}{2}\\!-\\!h_{z}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sqrt{3}h_{\perp}e^{-i\phi}}\\\
0&0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-\sqrt{3}\gamma}&0&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sqrt{3}h_{\perp}e^{i\phi}}&\frac{3J}{2}\\!-\\!3h_{z}\\\
\end{array}\right)_{\\!\\!\\!\\!\mathcal{B}_{2}}.~{}~{}~{}~{}~{}~{}~{}$ (35)

where $\gamma\equiv\frac{\sqrt{3}}{2}e^{i2\pi/3}(a+ib)$.

###  II.3 Energy spectrum

The field dependence of the energy spectrum is shown in Fig. S4 for ${\bf
h}\\!\parallel\\!{\bf z}$ (panel a) and ${\bf h}\\!\parallel\\!{\bf x}$-axis
(panel b), for a representative set of DM parameters
$(c,d_{\perp}\equiv\sqrt{a^{2}+b^{2}})\\!=\\!(0.2J,0)$ (solid lines) and
$(0.2J,0.2J)$ (dotted lines).

Figure S4: Evolution of the energy spectrum of
$\mathcal{H}_{J}\\!+\\!\mathcal{H}_{\text{DM}}\\!+\\!{\bf h}\cdot{\bf S}$,
where ${\bf h}\\!=\\!g\mu_{B}{\bf H}$, for ${\bf h}\\!\parallel\\!{\bf z}$ (a)
and ${\bf h}\\!\parallel\\!{\bf x}$-axis (b). The DM parameters $c$ and
$d_{\perp}\equiv\sqrt{a^{2}+b^{2}}$ are $(c,d_{\perp})=(0.2J,0)$ for the blue
solid lines, and $(0.2J,0.2J)$ for the red dotted lines. The number labels
give the degeneracy of each energy level.

The main features of Fig. S4 can be understood by examining the general
structure of the matrix of Eq. (35):

  * •

$\mathcal{B}_{2}$ is an eigenbasis of $\mathcal{H}_{\text{DM},z}$. This is
because $\mathcal{H}_{\text{DM},z}$ is invariant under: i) $C_{3}$ and thus
does not couple states with different $\ell$; ii) $U(1)$ rotations around the
$z$ axis in spin space and thus does not couple states with different $S_{z}$.

  * •

At zero field, $\mathcal{H}_{\text{DM},z}$ lifts the degeneracy of the two
doublet states of $\mathcal{H}_{J}$ and gives rise to one Kramers doublet
$\\{|\ell\\!=\\!1,\uparrow\rangle,|\ell\\!=\\!-1,\downarrow\rangle\\}$ with
$E\\!=\\!-3J/4\\!-\\!\sqrt{3}c/2$ and a second Kramers doublet
$\\{|\ell\\!=\\!-1,\uparrow\rangle,|\ell\\!=\\!1,\downarrow\rangle\\}$ with
$E\\!=\\!-3J/4\\!+\\!\sqrt{3}c/2$. Essentially, $c$ locks the quantum number
$\ell$ to the value of $\sigma$.

  * •

Unlike $\mathcal{H}_{\text{DM},z}$, $\mathcal{H}_{\text{DM},xy}$ [matrix
elements highlighted in blue color in Eq. (35)] is not diagonal in the
$\mathcal{B}_{2}$ basis. More importantly, $\mathcal{H}_{\text{DM},xy}$ does
not couple the two doublets, but only connects the doublets with the high-
energy quartet. As a result, the leading correction to the energy eigenvalues
from the transverse DM components $a$ and $b$ is second order in
$d_{\perp}/J$, where $d_{\perp}\\!\equiv\\!\sqrt{a^{2}+b^{2}}$. In other
words, the transverse DM components have little influence on the spectrum (see
difference between solid and dotted lines in Fig. S4), and are therefore not
as significant as the $z$ components. In particular,
$\mathcal{H}_{\text{DM},xy}$ does not influence the quartet, as long as we are
away from level anticrossings.

  * •

$\mathcal{H}_{\text{DM},z}$ does not affect the quartet. So, in the absence of
$\mathcal{H}_{\text{DM},xy}$, the quartet will show the usual Zeeman splitting
for any field direction, see solid lines in Fig. S4. A nonzero $d_{\perp}$
gives rise to an excited level antocrossing at high fields, see dotted lines
in Fig. S4.

  * •

For a magnetic field along the $z$-axis, the Zeeman coupling is diagonal in
the basis $\mathcal{B}_{2}$ because it conserves $S_{z}$ and $\ell$ (the
Zeeman term is invariant under $C_{3}$). Hence, a field along $z$ gives rise
to the usual Zeeman splitting of each separate multiplet, see Fig. S4 (a).

  * •

By contrast, for in-plane fields, the Zeeman coupling [matrix elements
highlighted in red color in Eq. (35)] connects states with the same $\ell$ and
$\Delta S_{z}=\pm 1$. As a result, the in-plane magnetic field gives rise to:
i) a level-repulsion of the two doublets of
$\mathcal{H}_{J}+\mathcal{H}_{\text{DM},z}$, without lifting the two-fold
degeneracy of each doublet, and, ii) the usual Zeeman splitting of the
quartet. Figure S4 (b) illustrates these aspects.

We can also deduce the following analytical results for the energy spectrum in
special cases:

  * •

For fields along the $z$-axis and $d_{\perp}\\!=\\!0$, the energies are given
by the diagonal elements of the matrix of Eq. (35), namely

$\text{spectrum
of}~{}\mathcal{H}_{J}\\!+\\!\mathcal{H}_{\text{DM},z}\\!+\\!h_{z}S^{z}:~{}\boxed{-\frac{3J}{4}-\frac{\sqrt{3}|c|}{2}\pm\frac{h_{z}}{2},~{}~{}-\frac{3J}{4}+\frac{\sqrt{3}|c|}{2}\pm\frac{h_{z}}{2},~{}~{}\frac{3J}{4}\pm\frac{3h_{z}}{2},~{}~{}\frac{3J}{4}\pm\frac{h_{z}}{2}}\,,$
(36)

  * •

For fields along the $z$-axis and $d_{\perp}\\!\neq\\!0$, the energies are
given by

$\\!\\!\\!\\!\\!\text{spectrum
of}~{}\mathcal{H}_{J}+\mathcal{H}_{\text{DM}}+h_{z}S^{z}:\boxed{-\frac{\sqrt{3}}{4}c\\!+\\!\frac{\zeta_{1}}{4}\sqrt{3d_{\perp}^{2}\\!+\\!(\sqrt{3}c\\!+\\!3J\\!+\\!\zeta_{2}2h_{z})^{2}},\frac{\sqrt{3}}{4}c\\!+\\!h_{z}\\!+\\!\frac{\zeta_{1}}{4}\sqrt{9d_{\perp}^{2}\\!+\\!(\sqrt{3}c\\!-\\!3J\\!+\\!\zeta_{2}2h_{z})^{2}}},$
(37)

where $\zeta_{1}=\pm 1$ and $\zeta_{2}=\pm 1$.

  * •

For fields along the $x$-axis (the results are the same for any direction in
the $xy$-plane) and $d_{\perp}\\!=\\!0$, the energies are given by

$\text{spectrum
of}~{}\mathcal{H}_{J}\\!+\\!\mathcal{H}_{\text{DM},z}\\!+\\!h_{x}S^{x}:~{}\boxed{-\frac{3J}{4}\pm\frac{1}{2}\sqrt{3c^{2}+h_{x}^{2}}~{}(\text{each
two-
fold}),~{}~{}\frac{3J}{4}\pm\frac{3h_{x}}{2},~{}~{}\frac{3J}{4}\pm\frac{h_{x}}{2}}\,.$
(38)

  * •

For fields along the $x$-axis and $d_{\perp}\\!\neq\\!0$, we can obtain the
energies up to second order in $d_{\perp}$ using perturbation theory (valid
away from level-anticrossings). We find

$\text{spectrum
of}~{}\mathcal{H}_{J}\\!+\\!\mathcal{H}_{\text{DM}}\\!+\\!h_{x}S^{x}:~{}\boxed{-\frac{3J}{4}\pm\frac{1}{2}\sqrt{3c^{2}+h_{x}^{2}}+\delta
E^{(2)}_{\pm}~{}(\text{each two-
fold}),~{}~{}\frac{3J}{4}\pm\frac{3h_{x}}{2},~{}~{}\frac{3J}{4}\pm\frac{h_{x}}{2}}\,.$
(39)

where

$\delta
E^{(2)}_{+}=-\frac{|\gamma|^{2}}{32}\frac{(3\sqrt{3}c+h_{x}+3\sqrt{3c^{2}+h_{x}^{2}})^{2}}{(3c^{2}+h_{x}^{2}+\sqrt{3}c\sqrt{3c^{2}+h_{x}^{2}})(3J+h_{x}-\sqrt{3c^{2}+h_{x}^{2}})},~{}~{}\delta
E^{(2)}_{-}=\frac{|\gamma|^{2}}{16}\frac{4\sqrt{3}c+3h_{x}-5\sqrt{3c^{2}+h_{x}^{2}}}{\sqrt{3c^{2}+h_{x}^{2}}~{}(3J+h_{x}+\sqrt{3c^{2}+h_{x}^{2}})}\,.$
(40)

It follows that, to leading order in $d_{\perp}$: i) the zero-field energy of
the quartet is unaffected by $d_{\perp}$, and, ii) the zero-field splitting
between the two lowest doublets becomes

$\boxed{\Delta=\sqrt{3}|c|+\frac{J}{32}\left(\frac{d_{\perp}}{J}\right)^{2}\frac{12+5\sqrt{3}\frac{c}{J}}{3-\left(\frac{c}{J}\right)^{2}}}\,.$
(41)

###  II.4 Magnetization process for $d_{\perp}=0$

In the following we shall disregard the in-plane components of the DM vectors.
The magnetization can be obtained using the expression

${\bf m}=-\sum_{n}P_{n}\frac{\partial E_{n}}{\partial{\bf H}}\,,$ (42)

where $\\{E_{n},n=1-8\\}$ are the energy eigenvalues, $P_{n}=e^{-\beta
E_{n}}/Z$ are the corresponding Boltzmann probabilities,
$\beta\\!=\\!\frac{1}{k_{B}T}$ and $Z\\!=\\!\sum_{n}e^{-\beta E_{n}}$ is the
partition function. We find:

  * •

Magnetization process for ${\bf h}\\!\parallel\\!{\bf z}$: Using Eqs. (36) and
(42) leads to the analytical expression

$\boxed{\frac{m_{z}}{g\mu_{B}}=\frac{1}{2}\tanh\left(\frac{\beta
h_{z}}{2}\right)\left[1+4\frac{\cosh^{2}(\beta h_{z}/2)}{\cosh(\beta
h_{z})+e^{3\beta J/2}\cosh(\sqrt{3}\beta|c|/2)}\right]}\,.$ (43)

The evolution of $m_{z}$ with $h_{z}/J$ for $k_{B}T/J=0.005$ and $|c|/J=0$
(black), 0.1 (blue) and 0.2 (red) is shown in Fig. S5 (a). First of all, the
results show a clear 1/2 magnetization plateau. Second, the magnetization
shows very little dependence on $c$ (invisible in the graph) up to the
saturation field $h^{\ast}_{z}$, whose value seems to be the only aspect that
changes with $c$. Specifically, $h^{\ast}_{z}$ appears to increase linearly
with $c$. Analytically, we find (at $T=0$)

$\boxed{h^{\ast}_{z}=3J/2+\sqrt{3}|c|/2}\,.$ (44)

  * •

Magnetization process for ${\bf h}\\!\perp\\!{\bf z}$: Similarly, we can
obtain the magnetization $m_{x}$ for for ${\bf h}\\!\perp\\!{\bf z}$ and
$d_{\perp}\\!=\\!0$ using Eqs. (38) and (42), but the resulting analytical
expression for $m_{x}$ is too lengthy to write down explicitly. However, we
can obtain a simpler formula for the zero-temperature magnetization, where Eq.
(42) reduces to

$T=0:~{}~{}\frac{m_{x}}{g\mu_{B}}=-\frac{\partial}{\partial h_{x}}E_{1}\,,$
(45)

where $E_{1}$ is the ground state energy,

$E_{1}=\Bigg{\\{}\begin{array}[]{ll}-\frac{3J}{4}-\frac{1}{2}\sqrt{3c^{2}+h_{x}^{2}},&\text{if}~{}|h_{x}|\leq
h_{x}^{\ast}\\\ +\frac{3J}{4}-\frac{3}{2}h_{x},&\text{if}~{}|h_{x}|\geq
h_{x}^{\ast}\end{array}$ (46)

and $h_{x}^{\ast}$ is the saturation field. We find:

$\\!\\!\boxed{\frac{m_{x}}{g\mu_{B}}=\Bigg{\\{}\begin{array}[]{ll}\frac{1}{2}\frac{h_{x}}{\sqrt{3c^{2}+h_{x}^{2}}},&\text{if}~{}0\leq
h_{x}\leq h_{x}^{\ast}\\\ \frac{3}{2},&\text{if}~{}h_{x}\geq
h_{x}^{\ast}\end{array}}~{}\text{and}~{}\boxed{h_{x}^{\ast}=\frac{1}{8}\left(9J+\sqrt{9J^{2}+24c^{2}}\right)=J\big{[}\frac{3}{2}+\frac{1}{2}(c/J)^{2}+\mathcal{O}(c/J)^{4}\big{]}}\,.$
(47)

Note that the saturation field $h_{x}^{\ast}$ scales quadratically with $c$
for small $c/J$, and therefore is not influenced from $c$ as strongly as
$h_{z}^{\ast}$. This is also demonstrated in Fig. S5.

strongly on $c$ quadratically quaThe size of the magnetization jump at
$h_{x}^{\ast}$ is given by

$\frac{\delta
m_{x}}{g\mu_{B}}=\frac{3}{2}-\frac{9+\sqrt{9+244(c/J)^{2}}}{16\sqrt{3(c/J)^{2}+\frac{1}{64}(9+\sqrt{9+244(c/J)^{2}})^{2}}}=1+\frac{(c/J)^{2}}{3}+\mathcal{O}((c/J)^{4})\,.$
(48)

The low-$T$ magnetization process for a field applied along the $x$ axis is
shown in Fig. S5 (b) for $|c|/J=0$, 0.1 and 0.2. The response is qualitatively
different from Fig. S5 (a). Most importantly, the flat plateau disappears for
nonzero $c$. Second, it is the low-field regime of $m_{x}$ which is affected
by $c$, and not the high-field regime, which is opposite to what happens in
Fig. S5 (a). Third, the slope of $m_{x}$ at low fields does not diverge (as
for the case of $c\\!=\\!0$ or for fields along $z$) and decreases with
increasing $c$, see next subsection.

Figure S5: Low-$T$ magnetization process of a spin-1/2 triangle described by
$\mathcal{H}_{J}\\!+\\!\mathcal{H}_{\text{DM},z}\\!+\\!{\bf h}\cdot{\bf S}$,
for ${\bf h}\\!\parallel\\!{\bf z}$ (a) and ${\bf h}\\!\parallel\\!{\bf
x}$-axis (b). Here $k_{B}T=0.005J$, ${\bf h}=g\mu_{B}{\bf H}$ and the three
curves correspond to $|c|/J=0$ (black), 0.1 (blue) and 0.2 (red).

###  II.5 Magnetic susceptibility for $d_{\perp}=0$

We can find an analytical expression for the susceptibility using the Van
Vleck formula

$\chi_{\alpha\alpha^{\prime}}=\lim_{H\to
0}\sum_{n}P_{n}\left(\beta\frac{\partial E_{n}}{\partial
H_{\alpha}}\frac{\partial E_{n}}{\partial
H_{\alpha^{\prime}}}-\frac{\partial^{2}E_{n}}{\partial H_{\alpha}\partial
H_{\alpha^{\prime}}}\right)\,,$ (49)

where $\alpha$ and $\alpha^{\prime}$ are Cartesian components and
$\chi_{\alpha\alpha^{\prime}}=\lim_{H\to
0}\frac{M_{\alpha}}{H_{\alpha^{\prime}}}$. Due to symmetry, the susceptibility
tensor is diagonal and, furthermore, $\chi_{xx}=\chi_{yy}$. In particular, we
find (disregarding again the in-plane components of the DM vectors):

  * •

Longitudinal susceptibility: Using Eq. (49) and the expressions for the energy
eigenvalues given in Eq. (36), gives

$\boxed{\frac{\chi_{zz}}{(g\mu_{B})^{2}}=\beta
J\left(\frac{1}{4}+\frac{1}{1+e^{3\beta
J/2}\cosh(\sqrt{3}\beta|c|/2)}\right)}\,.$ (50)

The evolution of $\chi_{zz}$ with $k_{B}T/J$ for $|c|/J=0.1$ and $0.2$ is
shown in Fig. S6 (a). The curves fall almost on top of each other (they start
to deviate slightly at high $T$ for larger values of $|c|/J$). So, $\chi_{zz}$
is essentially unaffected by $c$, which is consistent with the agreement
between the three curves of Fig. S5 (a) at low fields. Moreover, at low
temperatures,

$\boxed{\text{low}~{}T:~{}~{}\frac{\chi_{zz}}{(g\mu_{B})^{2}}\to\frac{\beta
J}{4}}\,,$ (51)

which is the same expression with that of a spin-1/2 object (Curie’s law). At
high temperatures,

$\boxed{\text{high}~{}T:~{}~{}\frac{\chi_{zz}}{(g\mu_{B})^{2}}\to\frac{3}{4}(\beta
J)-\frac{3}{8}(\beta
J)^{2}-\frac{3}{32}\beta^{3}Jc^{2}+\mathcal{O}(\beta^{4})}\,,$ (52)

So, the two leading terms of the high-$T$ expansion of $\chi_{zz}$ are
unaffected by the DM interactions. These leading terms give a Curie-Weiss
temperature $k_{B}\Theta=-J/2$, the same as in the case without DM
interactions. This is a general result that is related to the fact that the DM
interactions are antisymmetric (see, e.g., appendix of Ref. [4]).

  * •

Transverse susceptibility: In a similar way, using Eq. (49) and the energy
expressions given in Eq. (38), gives

$\boxed{\frac{\chi_{xx}}{(g\mu_{B})^{2}}=\frac{J}{2\sqrt{3}|c|}\frac{1-e^{-\sqrt{3}\beta|c|}}{1+2e^{-3\beta
J/2}e^{-\sqrt{3}\beta|c|/2}+e^{-\sqrt{3}\beta|c|}}+\frac{5\beta
J}{4}\frac{1}{1+e^{3\beta J/2}\cosh(\beta\sqrt{3}|c|/2)}}\,.$ (53)

The evolution of $\chi_{xx}$ with $k_{B}T/J$ for $|c|/J=0$, $0.1$ and $0.2$ is
shown in Fig. S6 (b). The results show a qualitatively different behaviour
from the curves shown in Fig. S6 (a), which stems from the corresponding
difference in the low-field magnetization process shown in the two panels of
Fig. S5. Unlike $\chi_{zz}$, $\chi_{xx}$ does not diverge as $1/T$ at low $T$,
but saturates to a $T$-independent value that is inversely proportional to the
$z$ component of the DM vectors. Specifically,

$\boxed{\text{low}~{}T:~{}~{}\frac{\chi_{xx}}{(g\mu_{B})^{2}}\to\frac{J}{2\sqrt{3}|c|}}\,.$
(54)

At high temperatures,

$\boxed{\text{high}~{}T:~{}~{}\frac{\chi_{xx}}{(g\mu_{B})^{2}}\to\frac{3}{4}(\beta
J)-\frac{3}{8}(\beta
J)^{2}-\frac{1}{8}\beta^{3}Jc^{2}+\mathcal{O}(\beta^{4})}\,,$ (55)

The two leading terms are again unaffected by the DM interactions, and,
moreover, coincide with the two leading terms of the high-$T$ expansion of
$\chi_{zz}$, see Eq. (52). The third term in the expansion differs by that of
Eq. (52) by a factor of $4/3$, which reflects the general trend of the
$\chi_{xx}$ curves (corresponding to $c\neq 0$) to lie below the $\chi_{zz}$
curve, see Fig. S6.

Figure S6: $T$-dependence of the susceptibilities
$\chi_{zz}\\!=\\!\lim_{H_{z}\to 0}\frac{m_{z}}{H_{z}}$ (a) and
$\chi_{xx}\\!=\\!\lim_{H_{x}\to 0}\frac{m_{x}}{H_{x}}$ (b), for a spin-1/2
triangle described by $\mathcal{H}_{J}\\!+\\!\mathcal{H}_{\text{DM},z}+{\bf
h}\cdot{\bf S}$, where ${\bf h}\\!=\\!g\mu_{B}{\bf H}$. The curves shown
correspond to $|c|/J=0$ (black), 0.1 (blue) and 0.2 (red).

##  III Estimation of the DM parameters by fitting the experimental $\chi(T)$

Here, we search for the set of parameters – the Heisenberg exchange $J$ and
the DM exchanges $D_{\parallel}$ and $D_{\perp}$ – of the isolated triangle
model that provide an optimal description of the experimental magnetic
susceptibility for both field directions. To this end, we use two
complementary approaches. In Sec. III.1, we apply the analytical expressions
from Eqs. (50) and (53), which are valid for $D_{\parallel}=0$. In Sec. III.2,
we calculate the reduced magnetic susceptibilities numerically for different
values of $|\mathbf{D}|/J$ by keeping the $D_{\perp}/D_{\parallel}$ ratio
fixed to $\frac{\sqrt{3}}{3}$ $[$Eq. (15)$]$. In both cases, the minimal
fitting temperature $T_{\text{min}}$ is used as a free parameter.

###  III.1 Analytical expressions

Analytical expressions in Eqs. (50) and (53), supplemented with a temperature-
independent contribution $\chi_{0}$, account for the experimental
susceptibility down to approximately 15 K (Fig. S7); at lower temperatures the
discrepancies become manifest (Fig. S8, right). In the broad $T_{\text{min}}$
range between 12 and 20 K, the fitted $J$ and $D_{\parallel}$ are nearly
constant (Fig. S8, left and middle) and amount to 60 K and 22 K, respectively
($D_{\parallel}/J$ = 0.365).

Figure S7: Top: Least-square fits to the experimental (circles) susceptibility
(measured in 1 T field) as function of temperature. Fitting is done for
temperatures above $T_{\text{min}}$ = 15 K. Bottom: difference plots for the
same quantities. The plotting was made using matplotlib [5].  Figure S8: The
fitted $J$ (left) and $D_{\parallel}/J$ (middle) as a function of the minimal
fitting temperature $T_{\text{min}}$. Right: the normalized sum (over all
fitted temperature and both field directions) of squared differences between
the experimental and fitted $\chi(T)$ as a function of the minimal fitting
temperature $T_{\text{min}}$. The plotting was made using matplotlib [5].

###  III.2 Numerical evaluation

Here we include all the components of the DM vectors and calculate the
magnetization and the magnetic susceptibility of a $S=\frac{1}{2}$ spin
triangle [Eq. (17)] numerically. We work in energy units of $J=1$ and vary
$|{\bf D}|/J$ between 0.2 an 0.6, while keeping the ratio
$D_{\perp}/D_{\parallel}$ fixed to $\frac{\sqrt{3}}{3}$. The magnetic
susceptibility is obtained by choosing the direction $\alpha$ ($x$ or $z$) of
the reduced magnetic field $h^{*}=g\mu_{B}H/J$, calculating the reduced
magnetization $M^{*}_{\alpha}$ and taking its zero-field derivative w.r.t.
$h^{*}_{\alpha}$:

$M^{*}_{\alpha}(t^{*},h^{*})=-\frac{\operatorname{Tr}\left(S^{\alpha}e^{-\mathcal{H}(h^{*})/t^{*}}\right)}{\operatorname{Tr}\left(e^{-\mathcal{H}(h^{*})/t^{*}}\right)}\,,~{}~{}~{}\chi^{*}_{\alpha\alpha}(t^{*})=\Big{(}\frac{\partial}{\partial
h_{\alpha}^{*}}M^{*}_{\alpha}(t^{*},h^{*})\Big{)}_{h_{\alpha}^{*}\to 0},$ (56)

where $S^{\alpha}$ is the operator for the $\alpha$-th component of the spin,
$M_{\alpha}^{*}=\frac{M_{\alpha}}{g\mu_{B}}$,
$\chi_{\alpha\alpha}^{*}=\frac{\chi_{\alpha\alpha}}{(g\mu_{B})^{2}/J}$, and
the reduced temperature $t^{*}=\frac{k_{B}T}{J}$ is taken on a mesh of 299
points between 0.04 and 6. The reduced susceptibilities $\chi^{*}_{xx}(t^{*})$
and $\chi^{*}_{zz}(t^{*})$ calculated on this $t^{*}$ mesh are approximated by
a polynomial $\sum_{i}{c_{\alpha,i}(1/t^{*})^{i}}$, where the coefficients
$c_{\alpha,i}$ are determined numerically. The resulting polynomials are
continuous functions of $t^{*}$ and can be fitted to the experiment using the
following expression for the susceptibility of $N_{A}$ non-interacting spin
triangles:

$\chi_{\alpha\alpha}(T)=\chi_{0}+\frac{N_{\text{A}}(g\mu_{\text{B}})^{2}}{J}\sum_{i}{c_{\alpha,i}(1/t^{*})^{i}},$
(57)

where $J$, $g$, and $\chi_{0}$ are the fitting parameters that are determined
independently for $\alpha=x$ and $\alpha=z$. In this way, we find that
$\chi_{\parallel}$ is largely independent of $|\mathbf{D}|/J$ (Fig. S9,
right). In contrast, $\chi_{\perp}$ is sensitive to $|\mathbf{D}|/J$ (Fig. S9,
left), with the optimal $|\mathbf{D}|/J\simeq$ 0.42, and hence
$D_{\parallel}/J=0.364$ and $D_{\perp}/J=0.21$. These values are $25$ %
smaller than the estimates from first-principles calculations (Sec. I). At the
same time, $D_{\parallel}/J$ is in remarkable agreement with the value
obtained in the analytical fits (Sec. III.1), showing once again, that the
effect of the in-plane components of the DM vectors is weak, see Sec. II.

Figure S9: Normalized squared differences between the calculated and
experimental (measured 1 T) magnetic susceptibilities $\chi_{\perp}$ (left)
and $\chi_{\parallel}$ (right) as a function of $|\mathbf{D}|/J$ for various
minimal fitting temperatures $T_{\text{min}}$. The plotting was made using
matplotlib [5].

###  III.3 Comparison with the isotropic model

Finally, we compare the optimal solution of the anisotropic model with the
isotropic Heisenberg model. While the out-of-plane susceptibilities can be
fitted by either model equally well (Fig. S10, middle panels), only the
anisotropic model can account for the in-plane susceptibility (Fig. S10, top
panels), which becomes evident if we inspect the absolute difference (Fig.
S10, bottom panels).

Figure S10:  Fits to the experimental magnetic susceptibility with the
isolated Heisenberg triangle model (left) in comparison with the
$|\mathbf{D}|/J=0.42,D_{\perp}/D_{\parallel}=\frac{\sqrt{3}}{3}$ solution
(Sec. III.2) of the anisotropic model (right). Top and middle panels show in-
plane and out-of-plane susceptibilities, bottom panels show the absolute
difference between the fit and the experiment. The minimal fitting temperature
$T_{\text{min}}=15$ K is denoted with a dashed line. The plotting was made
using matplotlib [5].

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