# Incommensurate and multiple-q magnetic misfit order in the frustrated
quantum-spin-ladder material antlerite, Cu3SO4(OH)4

Anton A. Kulbakov  Elaheh Sadrollahi  Florian Rasch  Institut für Festkörper-
und Materialphysik, Technische Universität Dresden, 01069 Dresden, Germany
Maxim Avdeev  Australian Nuclear Science and Technology Organisation, Lucas
Heights, NSW 2234, Australia  School of Chemistry, The University of Sydney,
Sydney 2006, Australia  Sebastian Gaß  Laura Teresa Corredor Bohorquez
Institute for Solid State Research, Leibniz IFW Dresden, 01069 Dresden,
Germany  Anja U. B. Wolter  Institute for Solid State Research, Leibniz IFW
Dresden, 01069 Dresden, Germany  Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter – ct.qmat, Technische Universität
Dresden, 01069 Dresden, Germany  Manuel Feig  Roman Gumeniuk  Institut für
Experimentelle Physik, TU Bergakademie Freiberg, 09596 Freiberg, Germany
Hagen Poddig  Anorganische Chemie II, Technische Universität Dresden, 01069
Dresden, Germany  Markus Stötzer  Anorganische Chemie I, Technische
Universität Dresden, 01069 Dresden, Germany  F. Jochen Litterst  Institut für
Physik der Kondensierten Materie, Technische Universität Braunschweig, 38106
Braunschweig, Germany  Inés Puente-Orench  Instituto de Nanociencia y
Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza, Zaragoza 50009,
Spain  Institut Laue-Langevin, 71 Avenue des Martyrs, CS 20156, CEDEX 9, 38042
Grenoble, France  Andrew Wildes  Institut Laue-Langevin, 71 Avenue des
Martyrs, CS 20156, CEDEX 9, 38042 Grenoble, France  Eugen Weschke  Helmholtz-
Zentrum Berlin für Materialien und Energie, BESSY II, 12489 Berlin, Germany
Jochen Geck  Dmytro S. Inosov<EMAIL_ADDRESS>Institut für
Festkörper- und Materialphysik, Technische Universität Dresden, 01069 Dresden,
Germany  Würzburg-Dresden Cluster of Excellence on Complexity and Topology in
Quantum Matter – ct.qmat, Technische Universität Dresden, 01069 Dresden,
Germany  Darren C. Peets<EMAIL_ADDRESS>Institut für Festkörper-
und Materialphysik, Technische Universität Dresden, 01069 Dresden, Germany

###### Abstract

In frustrated magnetic systems, the competition amongst interactions can
introduce extremely high degeneracy and prevent the system from readily
selecting a unique ground state. In such cases, the magnetic order is often
exquisitely sensitive to the balance among the interactions, allowing tuning
among novel magnetically ordered phases. In antlerite, Cu3SO4(OH)4, Cu2+
($S=1/2$) quantum spins populate three-leg zigzag ladders in a highly
frustrated quasi-one-dimensional structural motif. We demonstrate that at zero
applied field, in addition to its recently reported low-temperature phase of
coupled ferromagnetic and antiferromagnetic spin chains, this mineral hosts an
incommensurate helical+cycloidal state, an idle-spin state, and a multiple-$q$
phase which is the magnetic analog of misfit crystal structures. The
antiferromagnetic order on the central leg is reentrant. The high tunability
of the magnetism in antlerite makes it a particularly promising platform for
pursuing exotic magnetic order.

##  I Introduction

When attempting to expand our understanding of magnetism and reveal exotic new
magnetic phases or the influence of less-studied magnetic interactions, it is
necessary to prevent the exchange interactions from producing a more-
conventional magnetic ground state. This can be accomplished by introducing
strong frustration — the competition among interactions — most commonly by
arranging the spins in a geometry that pits different interactions against
each other [1, 2, 3]. This impedes the spin system selecting a unique global
ground state, leading to a wide variety of physical phenomena in which
fluctuations, quantum mechanical effects, and fine details of the spin-spin
interactions can be crucial [4]. Another approach is to investigate low-
dimensional structures, where the reduced number of magnetic interactions
makes it more difficult to stabilize magnetic order.

To observe quantum mechanical effects, the spin must be small, preferably
$S=1/2$. The stability of the $3d^{9}$ electronic configuration of Cu2+ makes
it probably the most accessible magnetic ion for quantum magnetism, while its
low spin-orbit coupling makes it a nearly pure-spin moment. Cu2+ is best known
in complex oxides, typically in octahedral coordination. These materials
predominantly form square magnetic sublattices, for instance in the cuprate
superconductors [5]. Frustration can arise in such lattices from a competition
between nearest- and next-nearest-neighbor interactions, but not through
geometric constraints. Low-dimensional copper ladder compounds are well-
established and are known to exhibit incommensurate magnetic order, among
other interesting physics [6], but geometric frustration does not play a
significant role in these materials since their copper sublattices are
typically bipartite.

Figure 1: (a) Copper sublattice in antlerite. (b) Low-temperature magnetically
ordered phase at 2.2 K, based on Ref. [7].

Copper-based minerals are nevertheless a rich source of novel frustrated
lattices, often built up of distorted Cu2+ triangles [8], and have proven a
rich platform for novel physics. Examples include the candidate quantum spin-
liquid state in herbertsmithite [9, 10]; enormous effective moments in
atacamite Cu2Cl(OH)3 [11]; and spinon-magnon interaction in botallackite
Cu2(OH)3Br [12]. The natural mineral antlerite considered in this work is a
three-leg-ladder compound with a magnetic sublattice comprising Cu2+ ions
arranged in distorted triangles [13], realizing a zigzag ladder — its copper
sublattice is depicted in Fig. 1(a). This structure has been studied far less
than the conventional ladder cuprates, and may be expected to reveal new
physics.

Neutron diffraction on antlerite initially indicated that only the outer legs
of the ladder possess an ordered moment, whereas the central leg exhibited
“idle-spin” behavior [14, 15]. While such a picture would be very interesting,
follow-up studies seriously questioned this result. First, density-functional-
theory (DFT) calculations showed strong antiferromagnetic coupling along the
central leg of the ladder, which would be expected to lead to order [16]. In
this work, most relevant exchange interactions were considered, including
Dzyaloshinskii-Moriya terms, and a number of possible spin configurations were
suggested. Then, Fujii et al. [17] performed specific-heat characterization
and proton NMR measurements on antlerite single crystals in magnetic field.
They found a vastly more complex magnetic phase diagram than previously
imagined, which hosts four distinct magnetic phases in zero field and at least
five other field-induced phases for $B\parallel c$ alone. Their NMR results
could only be explained if the system had at least four distinct magnetic
sites, at odds with the idle-spin picture. Finally, the low-temperature
magnetic state was determined by neutron powder diffraction and DFT
calculations to comprise mutually antialigned ferromagnetic outer legs and an
antiferromagnetic inner leg, with significant canting [7], as depicted in Fig.
1(b). The ordered moment on the outer legs of the ladder was nearly the full
moment expected for Cu2+, while the moment on the central leg was $\sim$20%
lower at 2.2 K. This suggests that quantum fluctuations play a relatively
minor role in the ferromagnetic legs of the ladder. DFT calculations were only
able to converge on this state once updated atomic positions were available,
and even then indicated other states at similar energy, with both the inner
and outer legs of the ladder independently on the verge of a phase transition.
This suggests a rich and tunable phase diagram with parameters such as field,
strain, pressure, or chemical substitution, as the published $H$–$T$ phase
diagrams already hint at [15, 17].

Here, we verify and identify the remaining low-field magnetic phases,
revealing incommensurate and multiple-$q$ order, as well as an idle-spin state
at intermediate temperatures which closely resembles the previously proposed
ground state [14].

##  II Experimental Details

### Sample preparation.

Synthetic antlerite was prepared by hydrothermal synthesis under autogenous
pressure at 180 ∘C in a Teflon-lined stainless steel autoclave.
CuSO${}_{4}\cdot 5$H2O (Alfa Aesar, 99%) and Cu(OH)2 (Alfa Aesar, 94%) in a
typical molar ratio of 1:2 were well mixed in distilled water inside the
liner, which was then sealed inside the autoclave. Deuterated samples for
neutron scattering were prepared using D2O (Acros Organics, 99.8% D) in place
of water. The mixtures were heated at 90 ∘C/h to 180 ∘C in a convection drying
oven, held at that temperature to react, cooled at 50–65 ∘C/h to 50 ∘C, then
cooled freely to room temperature. Reaction times of 1–3 days were used for
powder, 5 days for typical (100–200 $\mu$m) single crystals, or up to several
months when larger crystals were desired. Examples of several larger crystals
are shown on millimetre-ruled graph paper in the inset to Fig. 4(b).

### Physical properties.

Magnetization was measured using a Quantum Design Magnetic Property
Measurement System MPMS3, through a SQUID-detected vibrating-sample
magnetometer (VSM), in both zero-field-cooled-warming (ZFC) and field-cooled-
warming (FC) conditions. The 0.215-mg single crystal was mounted to a quartz
bar sample holder using Duosan glue, the contribution from which was measured
separately and subtracted. The extremely fine point spacing chosen to clearly
resolve the transitions led to increased noise in the derivative
d$(M/H)$/d$T$, so the derivatives shown in this paper incorporate a 6-point
moving average, which leads to slight broadening.

Low-temperature specific heat measurements were performed on a 4.75-mg mosaic
of $\sim$20 crystals using a Quantum Design Physical Property Measurement
System (PPMS DynaCool) equipped with a 3He refrigerator. Contributions from
the sample holder and grease were subtracted. Multiple data points were
collected at each temperature and averaged; it was necessary to discard the
first data point at most temperatures due to incomplete thermal stabilization.
The ac specific heat was measured on a small single crystal in a Cryogenic
Limited Cryogen-Free Measurement System using its ac calorimetry probe.
Measurements were performed at 16 Hz, using an ac power of 9.5 $\mu$W.
Decreasing the power to 2.3 $\mu$W made the peaks sharper at the cost of an
increased noise level; since we use these data primarily for higher
temperatures, the higher-power data are shown. In both experiments, the sample
was mounted to the measurement puck using a thin layer of Apiezon N grease.

### Diffraction.

To investigate the thermal evolution of the magnetic structure, high-intensity
neutron powder diffraction was performed on a $\sim$5 g sample at the WOMBAT
diffractometer at ANSTO [18], in Sydney, Australia, from 14 to 135.5∘ in steps
of 0.125∘, using 2.41-Å neutrons and count times of 30 min, in temperature
steps of 0.1 K through the transitions. Further high-intensity diffraction
data were collected on a $\sim$7 g powder sample at the D1B [19] and D20 [20]
diffraction beamlines at the Institut Laue-Langevin (ILL), in Grenoble,
France. Count times were 30–60 min to visualize the transitions, and $\sim$20
h for data required for refinement of the highest-temperature phase. A scan
through the transitions with 0.1-K temperature steps was performed before all
higher-resolution measurements to validate the thermometry. At D1B, data were
collected from 1 to 129∘ in steps of 0.1∘, and at D20 from 0.17 to 151.37∘ in
steps of 0.05∘. Neutrons with calibrated wavelengths $\lambda=2.529$ Å (D1B)
or 2.412 Å (D20) were selected with highly oriented pyrolytic graphite (HOPG)
[002] monochromators. On both beamlines, the contribution of the instrument to
the peak broadening was determined from the refinement of a Na2Ca3Al2F14
standard sample, while the wavelength was refined using a Si standard.
Parasitic diffraction peaks arising from the sample environment were
eliminated using radial oscillating collimators. Additional powder neutron
diffraction with polarization analysis was performed on the D7 beamline [21]
at the ILL, using a wavelength of 4.8707 Å. Powder diffraction data were
Rietveld-refined in FullProf by the full-matrix least-squares method [22],
using the scattering factors from Ref. 23.

### Spectroscopy.

X-ray absorption spectroscopy was measured at room temperature at beamline
UE46 PGM-1 at BESSY-II, in Berlin [24]. A counter voltage was applied to
prevent photoelectrons from entering the detector. Data were collected in
total-fluorescence-yield mode to ensure bulk sensitivity, and were normalized
to the incoming beam current. Further normalization was based on the average
signals in the 909.9–917.9 and 952.5–953.1 eV energy ranges, well below and
above the L2 and L3 edges.

Optical absorption spectra in the ultraviolet (UV), visible (vis) and near-
infrared (NIR) range from 200 to 900 nm were collected at room temperature in
a Varian Cary 4000 spectrophotometer with a scan rate of 300 nm/min, a step of
0.5 nm, a spectral bandwidth of 2 nm, and averaging time of 0.1 s. 15 mg of
antlerite crystal was ground together with 200 mg of BaSO4 to avoid excessive
absorption.

### Muon Spin Rotation.

Muon spin rotation and relaxation ($\mu$SR) experiments were performed on
Cu3SO4(OD)4 powder at the Swiss Muon Source (S$\mu$S) at the Paul Scherrer
Institut (PSI), Switzerland, using the nearly 100% spin-polarized positive-
muon beam at the GPS instrument [25]. Spectra were measured at temperatures
down to 1.5 K in zero field (ZF), weak longitudinal field (wLF, 2 mT field
applied along the initial muon spin direction) and weak transverse field modes
(wTF, 5 mT field applied perpendicular to the initial muon spin direction).
The sample was wrapped in an aluminized mylar foil and mounted within the beam
spot between the tines of an ultrapure copper fork. Thermal contact was
achieved via helium exchange gas. Analysis of the data was performed in the
time domain by the least-squares method using the program Musrfit [26].

##  III X-ray Absorption Spectroscopy

Figure 2: (a) X-ray absorption spectra of antlerite at the Cu L2/L3 edges at
room temperature, taken in total-fluorescence-yield mode for
$\vec{E}\parallel\hat{b}$ and $\hat{c}$. (b,c) Views of the first coordination
sphere for the Cu sites on one ladder.

We begin by verifying the oxidation state of Cu in Cu3SO4(OH)4. Cu L-edge
x-ray absorption spectra collected in total-fluorescence-yield mode for beam
polarizations along $\hat{b}$ and $\hat{c}$ are shown in Fig. 2(a). The strong
peak at 931.2 eV is the characteristic L3 line of $3d^{9}$ Cu2+, while the
weak hump at 934.3 eV suggests a small $3d^{10}$ contribution; these energies
and the Cu2+ peak shape are in good agreement with those found in Ref. 27 for
CuO and Cu2O. This indicates that copper atoms in antlerite exist
predominantly in the expected localized $3d^{9}$ (Cu2+) configuration, while
the $3d^{10}$ contribution may indicate a small excess population of protons,
perhaps due to some OH- ligands actually being H2O. This was not detectable in
our structure refinements. A $3d^{10}$ contribution is also observed in
materials with strongly-correlated electrons, where some hole density is
transferred to the ligands to reduce on-site repulsion at the Cu site, and in
some CuO samples [28]. The peak heights are slightly lower than expected
compared to the jump across the absorption edge, an indication of strong self
absorption which we have not corrected for. The splitting between the L3 and
L2 lines is 19.8 eV, consistent with reports on other Cu2+ materials [29].
This is a measure of the spin-orbit coupling in the $2p$ orbitals [30].

No significant difference was observed between polarizations parallel and
perpendicular to the ladders. As can be seen from the CuOx polyhedra for a
single ladder in Figs. 2(b) and 2(c), the local coordinate system has very
different orientations on the two copper sites as well as on adjacent sites on
the outer legs of the ladder, so x-ray absorption is not able to determine
exactly in which orbitals the holes in antlerite reside.

##  IV Phase Transitions

Figure 3: (a) Normalized magnetization $M/H$ data for antlerite, in fields
$H\parallel a$ and $b$ of 1000 Oe. The inset plots the difference between ZFC
and FC data for each axis, showing a sharp onset at the 5.0-K transition for
$H\parallel b$. Vertical lines represent the transition temperatures extracted
from the specific heat data in Fig. 4. (b) Derivatives of the magnetization
curves in (a), identifying the transitions. (c) Curie-Weiss fits to all four
inverse normalized magnetization datasets above 80 K.

We now turn to the phase transitions at low field. Magnetization data
collected under zero-field-cooled- and field-cooled-warming conditions are
shown in Fig. 3(a) for 1000 Oe fields parallel to the $a$ and $b$ axes. Two
transitions are visible for fields along $a$, at 5.00 and 5.55 K, while for
fields along $b$ transitions appear at 4.97 and 5.30 K and are accompanied by
a broad transition centered near 2.9 K. The transitions around 5.0–5.5 K are
clearer in the derivative plot shown in Fig. 3(b). For fields along $b$ the
transition at 4.97 K is a ferromagnetic-like increase in magnetization on
cooling, while all other observed transitions manifest as an
antiferromagnetic-like decrease. A slight splitting can be observed between
the field-cooled and zero-field-cooled data, with a clear onset at 5.0 K for
$b$-axis fields as seen in the difference between ZFC and FC data, plotted in
the inset to Fig. 3(a), indicating either the polarization or freezing of some
spin degree of freedom.

The unambiguous detection of three distinct, sharp magnetic ordering
transitions around 5 K with an additional transition at lower temperature
confirms the result of Fujii et al. [17] based on comprehensive 1H-NMR and
specific heat data, and indicates that their rather complex $H$–$T$ phase
diagram is most likely correct. The visibility of the two highest-temperature
transitions only for fields along $a$ and $b$, respectively, but not for both
orientations, is unusual, and may explain why other groups did not observe all
transitions.

The first transition encountered upon cooling, at 5.55 K, evidently concerns
spin components perpendicular to $b$, the direction along which the three-leg
ladders run, and is antiferromagnetic. The second transition, at 5.30 K, is
more surprising since it evidently concerns antiferromagnetic moments
perpendicular to $a$, but neither the plane of any three-leg spin ladder nor
its normal lies along $a$, so $a$ might not be expected to be a special
direction for spin order. The third transition around 5.0 K occurs at
temperatures that are within the uncertainty for the two field orientations,
and is assumed to be a single transition. This is antiferromagnetic along $a$,
but the small increase in magnetization and slight difference between ZFC and
FC data below this temperature for fields along $b$ would appear to suggest a
transition involving a canting angle toward the $b$ axis. We will show that
this is actually the result of a partial loss of long-range order as we enter
the gap between two phases in a reentrant system. Finally, around 2.9 K there
is a loss of susceptibility for $H\parallel b$, which we will show is the same
reentrant magnetism.

The paramagnetic state was characterized by an extended Curie-Weiss law which
considers a small temperature-independent paramagnetic contribution $\chi_{0}$
as in Ref. 15: $\chi(T)=\chi_{0}+C/(T-\Theta_{\text{CW}})$. This constant
offset reproduces the curvature seen in the inverse normalized magnetization
in Fig. 3(c). Due to the more pronounced anisotropy in the paramagnetic state
reported below 50 K, the temperature region from 80 to 300 K was fit as in
Ref. 15. Our extended Curie-Weiss fits yield an effective moment
$\mu_{\text{eff}}=1.96\pm 0.04$ $\mu_{\text{B}}$ and
$\Theta_{\text{CW}}=0.3\pm 0.2$ K for fields along $a$, and
$\mu_{\text{eff}}=1.87\pm 0.04$ $\mu_{\text{B}}$ and $\Theta_{\text{CW}}=4\pm
3$ K for fields along $b$.

The effective moments $\mu_{\text{eff}}$ are similar to those in Ref. 15 and
significantly above the spin-only value of 1.73 $\mu_{\text{B}}$ expected for
$S=1/2$ Cu2+, indicating an orbital contribution. This results in a $g$ factor
slightly larger than 2, as also reported based on ESR results [31], and
supports the scenario of non-negligible spin-orbit coupling proposed
previously [15, 14]. The small and slightly positive Curie-Weiss temperatures
indicate a balance between antiferromagnetic and ferromagnetic exchange
pathways, with a slight edge for ferromagnetism. DFT calculations and the low-
temperature magnetic structure indeed suggest ferromagnetic interactions on
the two outer legs of the ladder and antiferromagnetic exchange on the central
leg [7].

Figure 4: (a) Zero-field low-temperature specific heat data on antlerite,
showing four clear transitions. Red points were measured on warming and
turquoise on cooling. The inset plots $c_{P}/T$ vs $T^{2}$. (b) ac Calorimetry
of antlerite to higher temperatures. The inset shows several single crystals
on mm-ruled graph paper.

Our low-temperature specific heat data, shown in Fig. 4(a), indicate sharp
phase transitions at 2.80, 5.01, 5.30, and 5.57 K, defined using entropy-
conserving constructions around each jump; measurements through the higher-
temperature peaks on warming and cooling detected no hysteresis. The
transition temperatures are consistent with Ref. 17 and our magnetization data
above — this agreement is emphasized by the vertical lines in Fig. 3, which
mark the transition temperatures extracted from the specific heat data. The
upper three transitions were also clearly distinguished in our ac specific
heat data, shown to higher temperatures in Fig. 4(b), which did not extend to
low enough temperature to clearly resolve the 2.8-K transition. The inset to
Fig. 4(a), which replots the low-temperature data as $c_{P}/T$ vs $T^{2}$,
shows that the 2.8-K transition is associated with a collapse in entropy.
Below 1.6 K the specific heat data vary as $T^{2.5}$, with a slight upward
deviation below 0.5 K that may be associated with nuclear magnetism. As
previously reported [7], no further phase transitions are observed down to
0.35 K.

Figure 4(b) shows ac specific heat data to higher temperatures; it was not
possible to put these data in absolute molar units. A weak, broad hump around
230 K is associated with the glass transition of Apiezon N grease. The
magnetic entropy persists up to at least 20 K and conceivably as high as 40 K,
several times higher than any magnetic transition. This likely arises from the
low dimensionality, as well as frustration among the spins in each ladder.

##  V Magnetic Phases

Figure 5: Temperature-dependent magnetic peak intensity, after subtraction of
a high-temperature background. (a,b) Magnetic intensity color maps (after
subtraction of background measured above $T_{\text{N}}$) from (a) WOMBAT,
ANSTO and (b) D1B, ILL, highlighting the coupling between the incommensurate
and commensurate magnetism. The strong magnetic peak between 16∘ and 18∘ is
(100). The white line in (b) is a guide to the eye. (c) Peak intensity of
(100), the strongest commensurate magnetic reflection in (b), as a function of
temperature. (d) Temperature dependence of the peak intensity in the stronger
incommensurate reflection in (b). (e) Integrated intensity in the background
at low angles (11–12∘) of the background-unsubtracted data presented in (b),
as a function of temperature.

The magnetic order in the lowest-temperature phase has already been reported
[7] [see also Fig. 1(b)], so we concentrate here on the higher-temperature
phases. A temperature-dependent map of magnetic intensity through the sharp
transitions at 5.0–5.6 K near the strongest magnetic reflection was obtained
from WOMBAT (ANSTO) data by subtracting the intensity in the paramagnetic
phase at 7.9 K. The result is shown in Fig. 5(a). It is immediately clear that
the highest-temperature magnetically ordered phase is incommensurate — the (1
0 0) reflection splits into (1$\pm\delta$ 0 0), where $\delta$ varies between
$\sim 0.07$ and 0.12. The noninteger $\delta$ appears to vary continuously
without locking into a rational value, indicating that this phase is not
commensurate. The $\sim$0.5 K temperature shift in these data is most likely
associated with a difference in calibration of temperature sensors used in the
measurements.

Similar data collected on the D1B diffractometer at the ILL are shown in Fig.
5(b), where the intensity in the paramagnetic phase at 5.9 K has been
subtracted. Of particular note, the incommensurate and commensurate peaks
coexist for $\sim$0.5 K. This is not phase separation or coincidental
coexistence, for instance from different regions of the sample or separate Cu
ladders. The incommensurate peaks form an hourglass, pinching toward the point
at which the commensurate peak is extinguished, which implies that these
wavevectors are strongly coupled. A similar pinching is observed in structural
Bragg peaks in the incommensurate charge-density-wave phase of 2H-TaSe2 [32],
and is a fingerprint of multiple-$q$ order. The phase diagram of Fujii et al.
[17] implies that this multiple-$q$ phase exists in only a very narrow window
for fields along $b$, but it may be their “$\beta$1” phase for fields along
$c$, in which case it would occupy a far broader swath of the phase diagram.
In an earlier $H$–$T$ phase diagram based on magnetization measurements [15],
a similar bubble was also observed for fields along $a$, extending down to
quite low temperatures.

The temperature dependence of the strongest magnetic reflection is shown in
Fig. 5(c). The integrated peak intensity extracted from a Gaussian fit
decreases toward the transition in an order-parameter-like fashion, as might
be expected. This peak vanishes at the middle transition, which according to
magnetization and specific heat data is at 5.30 K and represents
antiferromagnetic order largely perpendicular to $a$. The most obvious
candidate from the low-temperature state would be the outer (Cu2) legs of the
ladder — the spins on these legs lie perpendicular to $a$ at low temperature,
with opposite legs antialigned [7]. We may thus expect to find commensurate
ordering of the Cu2 sites below 5.30 K.

The integrated intensity in the incommensurate peaks, shown in Fig. 5(d) based
on a three-Gaussian fit, is peaked immediately above the temperature at which
the commensurate peak is extinguished. The magnetization data suggest a spin
orientation perpendicular to $b$ in the incommensurate phase, consistent with
a ripple along $a$ as suggested by the peaks at (1$\pm\delta$ 0 0) and
suggesting a cycloidal state. Since all spins in the lowest-temperature state
are perpendicular or nearly perpendicular to $b$, it is difficult to speculate
based on the magnetization data alone which spins participate in the
incommensurate phase, although it may be natural to expect the Cu1 site to
play a key role if we associate the commensurate peak primarily with Cu2.

Figure 6: Refinement of the incommensurate phase. Data from D20 are shown at
(a) 6.0 K in the paramagnetic state and (b) 5.3 K where the incommensurate
phase begins to yield to multiple-$q$ order; a refinement of their difference
is plotted in (c), and the new magnetic peaks are highlighted in an inset in
(b). Artifacts due to imperfect subtraction of strong nuclear peaks are marked
with arrows. The refined incommensurate order is shown in (d), and (e) shows a
view along $a$ of the elliptical helical order on the corner (upper and lower)
and body-centered (middle) Cu2 ladders. The order pattern on selected Cu1 and
Cu2 sites is shown in (f), using the same axes as in (d).

Another feature in Fig. 5(b) may not be immediately obvious — below roughly 3
K, the background magnetic intensity values in this colormap are strongly
negative. The presence of a diffuse component in the subtracted 5.9 K data
would be expected, since we see indications in the specific heat of short-
range order up to $\sim$40 K. However, such a diffuse component should
condense into the magnetic Bragg peaks upon entering the long-range-ordered
state, presumably around 5.0-5.6 K. The temperature dependence of the diffuse
background does indeed show a decrease in that temperature range, but it
exhibits a much stronger collapse below 3 K, as seen in Fig. 5(e). The order-
parameter-like behavior seen in Fig. 5(c) shows no evidence of this intensity
condensing into the (100) peak.

##  VI Magnetic Structures

Neutron diffraction was also able to fully identify the zero-field magnetic
phases. To obtain the statistics required to refine the higher-temperature
incommensurate structure based on its much-weaker peaks, we collected
additional diffraction data on the D20 neutron diffractometer at the ILL. Data
were first collected as a function of temperature to validate the thermometry,
then for longer times at 6.0 K in the paramagnetic phase and at 5.3 K where
the (1$\pm\delta$ 0 0) peaks reach their maximum intensity. Figure 6(a) shows
the baseline data from D20 in the paramagnetic phase at 6.0 K, in Fig. 6(b) we
present data at 5.3 K, and the difference between these two scans is shown in
Fig. 6(c). Arrows mark artifacts from the imperfect subtraction of a few
strong nuclear peaks, which is likely due to the peaks becoming sharper as
phonons freeze out. This difference was refined to determine the magnetic
structure. Since 5.3 K is on the cusp of the multiple-$q$ phase, weak traces
of the latter phase are visible, but the incommensurate peaks dominate. A
contribution from the multiple-$q$ phase was found to be present at the 1.3%
level and was refined as discussed below.

Of the four possible magnetic irreducible representations, only $\Gamma_{2}$
gave an acceptable description of the magnetic intensity, but the
unconstrained application of $\Gamma_{2}$ leads to physically unreasonable
spin alignments on some ladders — otherwise-equivalent ladders would have very
different spin orientations, which would only be possible if the extremely
weak interladder couplings overpowered the strong intraladder interactions. If
we constrain the Cu2 ordered moments to have equal magnitude on any given
ladder, with spins on opposite legs exactly antialigned as in the lowest-
temperature phase, and allow only one angle between consecutive spins on the
central legs, a physically plausible magnetic model is obtained which does not
lead to any reduction in the quality of the refinement. As shown in Fig. 6(d),
the incommensurate phase corresponds to elliptical-helical order of the
antialigned ferromagnetic spins on the outer Cu2 legs of the ladder paired
with cycloidal order on the inner Cu1 leg. The helical and cycloidal order on
selected Cu sites are also shown separately in Fig. 6(f), and a view of the
elliptical order along the $a$ axis is shown in Fig. 6(e). The propagation
vector in this phase is ($\delta$ 0 0), where $\delta\approx 0.1$ at 5.3 K.
The elliptical helix on the outer legs has major and minor axes of 0.41(4) and
0.15(8) $\mu_{\text{B}}$, while the cycloidal ordered moment on the central
leg is 0.09(7) $\mu_{\text{B}}$. If we exclude magnetic peaks which overlap
with nuclear peaks, the same order is obtained, including ellipticity, but
with fewer peaks it is more difficult to quantify the ellipticity. These are
small fractions of the full 1 $\mu_{\text{B}}$ moment, most likely because
these refinements are performed on data collected at $\sim$95% of
$T_{\text{N}}$, where one would not ordinarily expect fully ordered moments.
There may also be a reduction in the ordered moments due to a competition
between the inner and outer legs, for instance if these would prefer
incompatible ordered states but are forced to compromise. The angle between
adjacent spins along the central leg refines to 108(40)∘, likely due to a
competition between nearest-neighbor and next-nearest-neighbor interactions
along the leg, combined with the exchanges to the outer legs. A magnetic
crystallographic information file (mCIF) describing this refinement is
available as an arXiv ancillary file as described in Appendix B. We note that
several other forms of order were considered, for instance SDW+cycloidal, but
these refinements produced visibly poorer fits, and in particular failed to
adequately describe the peaks between 20 and 24∘.

Elliptical helical order has been reported previously in the Cu2+ chain
compounds NaCu2O2 [33], LiCu2O2 [34], and linarite PbCuSO4(OH)2 [35], but
these had smaller ellipticity. The high ellipticity in antlerite suggests a
strong proclivity of the Cu2 spins to point along $c$, as found in the
antialigned ferromagnetic spin arrangement on this site at lower temperatures.

The phase between 2.8 and 5.0 K, and in particular how it differs from the
lowest-temperature magnetically ordered phase depicted in Fig. 1(b), is more
subtle. As seen in Fig. 5(a-c), the strongest magnetic peak does not appear to
be affected by the transition at 2.8 K. Since most other magnetic Bragg peaks
sit atop far stronger structural Bragg peaks, small changes there are more
difficult to detect in conventional neutron diffraction. We performed
polarized-neutron diffuse scattering at the D7 diffractometer at the ILL in
order to cleanly separate the magnetic intensity (in the spin-flip channel)
from the structural contribution, using a six-point ($XYZ$) method [36].
Results are shown as a function of scattering vector $|\bm{Q}|$ in Fig. 7(a)
for several temperatures. The behavior in the (100) and (1$\pm\delta$ 0 0)
magnetic reflections is consistent with that shown in Fig. 5, but we observe
nonmonotonic temperature dependence in the (010) peak at $|\bm{Q}|=1.0413$
Å-1. This peak is present in the low-temperature state at 1.5 K, absent in the
intermediate phase at 3.0 and 4.0 K, and reappears in the multiple-$q$ and
incommensurate phases above 5.0 K.

Figure 7: Identifying the magnetic structure between 2.8 and 5.0 K. (a) Spin-
flip (magnetic) scattering intensity as a function of temperature, collected
using polarized neutrons on D7; data at higher temperatures have been shifted
for clarity. Solid curves are refinements according to the magnetic structures
we identify. Dashed lines mark the strong (100) magnetic peak and the weaker
(010) peak at $|\bm{Q}|=1.0413$ Å-1 which is inconsistent with an idle-spin
state, while purple dash-dotted lines indicate the incommensurate positions.
(b) Refined magnetic structure at 4.0 K, representative of the phase between
2.8 and 5.0 K. (c) Refinement of the magnetic intensity in the D1B data in the
“idle-spin” state at 4.0 K. Intensity at 6.0 K has been subtracted; arrows
mark the positions of strong nuclear peaks.

Our previous work on the lowest-temperature magnetically ordered state in
antlerite identified magnetic intensity in a number of peaks that would not be
present in the previously-proposed idle-spin state, including the (010) peak
[7]. These peaks are associated both with antiferromagnetic order on the inner
leg and strong canting on the outer legs, presumably due to strong coupling
with the inner leg. This contribution to the order is evidently reentrant. The
absence of these peaks between 2.8 and 5.0 K indicates that the intermediate-
temperature phase is essentially the idle-spin phase previously proposed to be
the ground state [14, 15]. The magnetic structure we refine for this phase
based on data from D1B is shown in Fig. 7(b), and the refinement is shown in
Fig. 7(c); an mCIF file is supplied as an ancillary file as described in
Appendix B. This state features no long-range order on the central leg and no
canting on the outer legs; the refined magnetic moment on the outer legs is
0.802(8) $\mu_{\text{B}}$ at 4.0 K, where the enhancement relative to the
incommensurate phase is likely due to the lower temperature. Refinements of D7
data at 3.0 and 4.0 K found a similar state. This is the same magnetic space
group as at lower temperature, $Pn^{\prime}m^{\prime}a^{\prime}$ (number
62.449). We will refer to this phase as “idle-spin” as in the previous papers,
and return to the question of what the Cu1 spins actually do in this phase.

Figure 8: Multiple-$q$ phase, present in Cu3SO4(OD)4 between 5.0 and 5.3 K,
based on refinements at 5.3 K.

Since the previously proposed idle-spin phase is indeed present in the system
but at a higher temperature, one may wonder whether the previous reports of an
idle-spin state were due to a thermometry issue. The neutron diffraction data
that led to the original idle-spin conclusion were collected at 1.4 K [37,
14], which is half the temperature of the phase transition in question. A
thermometry issue of that magnitude is very unlikely. The (010) peak is
structurally forbidden, so interference from structural Bragg reflections is
also not a satisfying explanation. This peak may simply have been too weak to
be clearly observed at the short wavelengths used. We also note that antlerite
is expected to be exquisitely sensitive to parameters such as strain,
pressure, and chemical doping, so it is conceivable, although considerably
less likely, that inadvertent doping or overly assertive packing of the powder
sample could also have suppressed the 2.8-K transition.

The collapse in diffuse magnetic background intensity below $\sim$3 K in Fig.
5(b) agrees with the significant reduction on cooling of the diffuse
scattering hump centered around 0.7 Å-1 in Fig. 7. The differences between the
lowest-temperature phase and the “idle-spin” state above 2.8 K are the loss of
antiferromagnetic order on the central Cu1 leg of the ladder and canting on
the outer legs, so this large change in diffuse intensity is primarily
associated with the Cu1 site.

Having identified the phases below 5.0 K and above 5.3 K, we can now piece
together the multiple-$q$ phase between 5.0 and 5.3 K. The outer legs of the
ladder are presumably ordered ferromagnetically and antialigned, as they are
in the lower-temperature phases, since the (100) magnetic peak is not
sensitive to the transition at 5.0 K. Spins on the central leg evidently
retain a strong tendency to stay in the plane of the cycloid, but fail to
order below 5.0 K. This indicates that the multiple-$q$ phase is a combination
of ferromagnetic outer legs with a cycloidal inner leg. The elliptical helix
on the outer legs of the ladder becomes constant-amplitude antialigned
ferromagnetic order, which to some extent decouples the outer legs of the
ladder from the central leg and allows the incommensurate wavevector to return
to its higher-temperature value, again without any sign of locking into a
commensurate value. The multiple-$q$ order was refined as a minor component at
5.3 K according to this model, producing an ordered moment on the outer legs
of 0.87(28) $\mu_{\text{B}}$. The resulting magnetic structure is depicted in
Fig. 8, and an mCIF file is provided as an ancillary file as described in
Appendix B. The ordered moment and orientation on the central leg were
constrained to be identical to that in the incommensurate phase.

Within a ladder the spin order is commensurate in all four phases, although
not collinear. The relative spin orientation within every ladder is also the
same, as would be expected because the interladder interactions are far weaker
than the exchange coupling within a ladder, which we quantified in Ref. 7. The
incommensurate modulation along $a$ is perpendicular to the ladders, implying
that the interladder interactions play a key role in stabilizing the
incommensurate order. This indicates strong limitations to the picture of an
isolated ladder, as has been used in calculating the phase diagrams for
Cu3SO4(OH)4 as a function of the exchange parameters and its ground state [7].
The incommensurate order we have identified in antlerite at elevated
temperatures arises from a relative rotation of the spins from one ladder to
the next, and is not captured by the calculations, since they neglected
interladder interactions. We also note that although having an incommensurate
modulation along $a$ is broadly reminiscent of the cycloidal state proposed
for Se-substituted antlerite, Cu3SeO4(OH)4 [38], the ordered state is quite
different. In the Se analog similar order with a different pitch was reported
— a commensurate $(\frac{1}{7}\,0\,0)$ rather than a ($\sim 0.1$ 0 0)
propagation vector; however, the spins on the outer legs were cycloidal rather
than helical, while the spins on the inner leg did not participate. This was
reported by the same group that found an idle-spin ground state in antlerite,
so it is possible that nonzero ordered moments on the central leg in the Se
analog are present but were missed for similar reasons, and it may be worth
revisiting this material to look for signatures of ordering on the central
leg.

The spins on the outer legs in the incommensurate phase in antlerite prefer to
lie parallel to ±c, as in the lowest-temperature state. Their spatial
oscillation is presumably due to the interactions with the central leg
destabilizing the preferred spin alignment. The circular cycloid on the
central leg suggests that this site is driving the incommensurate order.
Indeed, our DFT calculations [7] found that the strongest exchange interaction
in antlerite is the antiferromagnetic $J_{5}$ along the central leg of the
ladder.

While strong antiferromagnetic interactions on the central leg evidently drive
the transition to magnetic order at 5.55 K, frustration results in
noncollinear order along each central leg, and we additionally find a
cycloidal state along $a$. The noncollinear order along the central leg of the
ladder may arise from bond frustration — DFT indicated that the second-
neighbor exchange along this leg, $J_{6}$, is roughly half of its nearest-
neighbor $J_{5}$ and is the third-strongest interaction in the system.
Alternating interactions with the outer legs may also play a role.

The second- and fourth-strongest interactions in antlerite are ferromagnetic
exchanges on the outer legs. With twice as many spins on the outer legs as on
the central leg, these interactions may begin to overpower those on the
central leg as they strengthen on cooling. It appears that first the outer
legs partially decouple and become constant-amplitude ferromagnetic, rather
than helical, then their ferromagnetism becomes strong enough to destroy the
cycloidal order on the central leg. Finally, the central leg becomes
antiferromagnetic, forming the lowest-temperature state.

While incommensurate order as seen here is already relatively uncommon,
multiple-$q$ phases are rarer still. Such phases are especially uncommon in
centrosymmetric materials, with the few prominent examples including USb [39]
and phase III in CeB6 [40, 41]. However, these examples are rare-earth
compounds. Multiple-$q$ magnetic order is extremely uncommon in compounds
based on $3d$ transition metals, with the best-known examples being
$\gamma$-Mn alloys [42]; the recently-reported hedgehog- and skyrmion-lattice
phases in SrFeO3 [43]; and the suggested triple-$q$ order in Na2Co2TeO6 [44].
Multiple-$q$ order most commonly arises from Dzyaloshinskii-Moriya
interactions, which sum to zero by symmetry over a unit cell in
centrosymmetric crystal structures like that of antlerite and result from
spin-orbit coupling which is generally weak in the $3d$ block; from an
interplay between Kondo physics and exchange interactions, which is not
expected in transition metal compounds; from bond-dependent Kitaev
interactions as proposed in Na2Co2TeO6 [44]; or from degeneracy among possible
ordering directions in highly symmetric lattices as found in face-centered-
cubic $\gamma$-Mn, which is unlikely to be the case in orthorhombic antlerite.
Here, the fine balance required for multiple-$q$ order evidently comes from
competing exchange interactions.

We also note that multiple-$q$ phases ordinarily exist only in very small
regions of a material’s magnetic phase diagram, usually forming as a bubble
within other magnetically ordered phases. This also seems to be the case in
antlerite. However, while these multiple-$q$ bubbles seldom extend over broad
temperature ranges, the published magnetic phase diagrams of antlerite show an
intriguing tail to zero temperature for fields around 1–1.8 T along $c$ and
around 3–4 T along $a$ [15, 17], apparently connected to the multiple-$q$
phase, suggesting that this phase may occupy a significant swath of the phase
diagram. This should be investigated further once sufficiently large single
crystals are available to enable neutron diffraction in magnetic field.

An even rarer aspect of the multiple-$q$ phase in antlerite is that it
combines commensurate antiferromagnetic order on one magnetic site with
incommensurate order on another. The only other reported example of such a
situation is Co3TeO6 [45, 46, 47], whose magnetic sublattice includes five
magnetic Co sites (including both octahedral and tetrahedral coordination) and
has an additional antiferroelectric transition within one of its magnetically
ordered phases. This structural complexity has greatly complicated the
identification and investigation of its magnetic order, and led to the papers
cited here to very different determinations of several of its phases.

##  VII Diffuse Scattering in the Paramagnetic Phase

Figure 9: Diffuse scattering in Cu3SO4(OD)4 in the paramagnetic phase at 6.0
K. (a) Fit (red) of the magnetic diffuse scattering intensity (gray), and its
residual (blue). (b-d) Reverse-Monte-Carlo reconstructed scattering intensity
for $(HK0)$, $(0KL)$ and $(H0L)$ cuts through reciprocal space, respectively.
(e) Extracted spin-spin correlation functions as a function of distance. The
ferromagnetic nearest neighbor correlations along the outer legs are visible
at the lowest distances.

We now return to the diffuse scattering presented in Fig. 7. We used the
Spinvert software package [48] to perform a reverse Monte Carlo refinement of
the spin-flip neutron powder diffraction data, to recover the three-
dimensional magnetic diffuse scattering pattern and the spin-pair correlation
function. Reconstructed diffuse scattering intensity for antlerite in the
paramagnetic phase at 6.0 K based on a 5$\times$5$\times$5 supercell is shown
in Fig. 9(b-d) for different cuts through reciprocal space. This is what we
predict would be observed on a single crystal, if a sufficiently large crystal
were available. Figure 9(a) shows the resulting fit to the powder data, and
Fig. 9(e) shows the extracted spin-spin correlation function. The shortest
Cu–Cu distance in antlerite is a ferromagnetic exchange on the outer leg,
followed by the antiferromagnetic link on the central leg and the second
ferromagnetic exchange on the outer leg. Accordingly, the correlations in Fig.
9(e) are ferromagnetic at the shortest distance, then become mainly
antiferromagnetic on average. Beyond $\sim$8 Å the correlations average out to
zero due to powder averaging — the sphere corresponding to this radius begins
to include a large number of spins.

Spin-flip scattering was also performed to higher temperatures, as shown in
Fig. 13 in the Appendix, where the diffuse peak is seen to shift to lower
$|\bm{Q}|$ on warming but does not vanish up to 200 K, indicating persistent
short-range correlations. We note that this measurement method integrates over
the inelastic scattering from the sample, and hence probes the time-equal
correlations rather than time-averaged correlations which give no energy
change to the neutrons. Correlations in paramagnetic fluctuations, which are
all dynamic, will give a $|\bm{Q}|$ dependence in the time-equal correlations.

##  VIII Muon Spin Rotation and Relaxation

We performed muon spin rotation and relaxation ($\mu$SR) experiments to obtain
additional information about the magnetic phases in antlerite, and to
distinguish whether the diffuse contributions are dynamic or static on the
timescale of $\mu$SR. $\mu$SR probes local fields and their fluctuations
through their effect on implanted positive muons $\mu^{+}$ (hereafter referred
to as “muons” for simplicity), which stop at specific sites in the crystal
structure, ordinarily near anions. Muons have a lifetime of 2.2 $\mu$s and
decay into a positron and neutrinos. The decay positrons are emitted
preferentially along the muon spin direction, leading to a decay asymmetry
which rotates as the muon spins precess about local magnetic fields. Positron
detectors positioned along and opposite the direction of the incoming
polarized muon beam yield a time-dependent asymmetry $A(t)$, the primary
signal in $\mu$SR.

In a wTF experiment with a weak external magnetic field $B_{\text{ext}}$
applied perpendicular to the initial muon polarization, muon spins in non-
magnetic environments will precess about this field, leading to oscillations
in the asymmetry with a frequency $\nu_{\mu}=\gamma_{\mu}B_{\text{ext}}/2\pi$,
where the muon gyromagnetic ratio $\gamma_{\mu}$ = 135.54 $\mu$s-1T-1. The
asymmetry signal will be an oscillation within a decaying envelope:

$A(t)=A(0)e^{-\lambda_{\text{para}}t}\cos(2\pi\nu_{\mu}t),$ (1)

with $\lambda_{\text{para}}$ being the paramagnetic relaxation rate.

In Cu3SO4(OD)4 at higher temperatures $A(t)$ is consistent with a paramagnetic
phase, but we find an initial asymmetry $A(0)$ of 0.23 in the paramagnetic
regime. This is significantly lower than the 0.28 expected for a fully
polarized muon beam in the setup used (as also directly verified with a
different sample). The asymmetries recorded in our ZF and wLF experiments are
similarly reduced to $\sim$0.22, as seen in the paramagnetic state at 6.5 K in
Fig. 14, and this reduction is seen at all temperatures. A reduced initial
asymmetry in a weak transverse field is an indication that some of the
implanted muons form muonium, as is often found in insulating and
semiconducting materials. Muonium signals are determined by large hyperfine
and dipolar interactions and are beyond our time resolution [49], but details
of the mechanism for muonium formation in antlerite may be of interest for
future investigation.

ZF and wLF experiments provide information about the temperature-dependent
development, distributions, and spin dynamics of local magnetic fields
$B_{\text{loc}}$ in the long-range ordered regime and on the approach to it.
The depolarization seen in zero field in the paramagnetic regime is dominated
by strong damping with a shape that is well described by a static Gaussian
Kubo-Toyabe function (Eq. 2) [50, 51]. The application of a weak longitudinal
field of 2 mT suppresses this damping (see Fig. 14). This means that the field
distribution in the paramagnetic state in Cu3SO4(OD)4 is due to small,
randomly directed fields from static spins that can be overcome by even a
small applied field. The origin of this field distribution can be attributed
to the nuclear spins, most likely of 17O when the muon is attached closely to
oxygen.

Analysis of the 2-mT LF spectra reveals a much weaker damping which is now
exponential due to fields from rapidly fluctuating electronic moments as
expected in the paramagnetic state. The damping increases upon cooling toward
the ordering temperature as the fluctuations slow down. Below 8 K, however, we
observe that an increasing part of the signal becomes much more rapidly
damped. As addressed below, we find a reduction of wTF asymmetries (see Fig.
16) in the same temperature range. Both observations likely indicate an onset
of short-range order above $T_{\text{N}}$, as is often found in low-
dimensional magnetic systems.

When $\mu$SR is performed in a magnetically ordered state, additional local
fields are present which are typically much larger than the applied field and
are randomly oriented with respect to it. Precession about these randomly
oriented fields leads to depolarization of the muon spins. This is reflected
in a rapid loss of the asymmetry signal after implantation. This clear
difference in behavior enables the detection of ordered phases, including
short-range order and fluctuations slower than the $\sim$10 $\mu$s timescale
to which muons are sensitive. For antlerite in a transverse field of 5 mT, the
asymmetry starts to decrease below 8 K, followed by a rapid drop below 6 K,
and saturates by 5.3 K (see Fig. 16), demonstrating that the entire sample
volume hosts long-range magnetic order. Within our temperature uncertainty of
$\pm$0.3 K, this finding is consistent with the first ordering temperature of
$T_{\text{N}}$ = 5.55 K derived from specific heat and magnetization data
(Figs. 3, 4). In the ordered state we still find some minor residual asymmetry
oscillating due to only the applied field, presumably from muons which stop in
the sample holder; this temperature-independent background asymmetry
$a_{\text{BG}}$ = 0.03 is taken into account in the analysis of the ZF and wLF
experiments.

Figure 10: Early-time $\mu^{+}$ spin-polarization asymmetry in Cu3SO4(OD)4 at
temperatures below $T_{\text{N}}$ from wLF measurements. Solid curves are fits
to Eq. 3. The spectra at higher temperatures are shifted vertically for
clarity; the common asymmetry scale is given only for the 1.5-K spectrum.
Spectra up to 6 $\mu$s are presented in Fig. 15 in Appendix D.

Spectra below $T_{\text{N}}$ in weak longitudinal field are shown in Fig. 10
and to longer decay times in Fig. 15. Up to 4 K they reveal at least two
spontaneous rotation signals in addition to a broad, damped pattern that can
be best reproduced again with a static Gaussian Kubo-Toyabe function. The
different signals are associated with different muon stopping sites near
oxygen sensing different local fields due to neighboring Cu ions. Unlike the
Gaussian-damped signal found above $T_{\text{N}}$, the signal at low
temperatures is not affected by an applied longitudinal field of 2 mT, i.e.,
the field distribution comes from static fields much larger than 2 mT and must
be related to static electronic moments. For parametrizing these complex
spectra we use a phenomenological approach described in Appendix D. For each
site $i$ = 1,2 we have a contribution $a_{\text{rot},i}$ to the asymmetry, a
precession frequency $\nu_{\mu,i}$ about the local field $B_{i}$, a transverse
relaxation rate $\lambda_{\text{t},i}$, and a longitudinal relaxation rate
$\lambda_{\text{l}}$ which is directly proportional to the spin fluctuation
rate. In addition we introduce a static Gaussian Kubo-Toyabe component with
asymmetry contribution $a_{\text{GKT}}$ and damping parameter $\sigma$.

The longitudinal damping $\lambda_{\text{l}}$ (extracted from the behavior at
long times once the oscillating signal is already damped out) is nearly zero
up to 3.1 K . This indicates that the electronic spin system is static on the
timescale of $\mu$SR in all magnetically ordered phases, notably including the
idle-spin state. Starting at 4.2 K we see an increase of longitudinal damping
that, as will be discussed below, can be related with the increased Kubo-
Toyabe contribution.

Figure 11: Temperature dependence of weak-LF-$\mu$SR parameters in
Cu3SO4(OD)4: (a) normalized asymmetry, (b) spontaneous $\mu^{+}$ spin
precession frequency $\nu_{\mu,i}$, (c) transverse relaxation rate
$\lambda_{\text{t},i}$, and (d) damping parameter $\sigma$ of the Gaussian
component. Blue circles indicate the regime in which only one frequency is
observed. Dashed lines indicate the phase transitions identified in
magnetization and specific heat — phase I is incommensurate, II is
multiple-$q$, III is idle-spin, and IV is the fully-ordered low-temperature
state.

Figure 11(a) plots the temperature dependence of the fractional contributions
$a_{i}$ to the background-corrected asymmetry. The asymmetries of the rotating
signals stay unchanged below 4.2 K within the uncertainty. Starting from 4.2 K
the asymmetries of the Gaussian signal and one of the rotating signals
increase at the expense of the other rotating signal. This means that at some
muon sites there is a gradual change in spin order combined with an increase
of static disorder. Finally, only a single frequency is observable at 4.8 and
5.3 K. The latter two temperatures correspond approximately to the temperature
range where incommensurate order was observed, and will be discussed
separately.

The rotation frequencies $\nu_{\mu,i}$ vary only weakly up to 3 K and then
decrease with increasing temperature, as seen in Fig. 11(b). This represents
the strength of the magnetic field on the muon site due to the ordered
moments, so such a temperature dependence is expected. Due to the limited
number of temperatures, it is not possible to resolve the shape clearly enough
to determine at which transition each frequency extrapolates to zero.

We present the transverse relaxation rates $\lambda_{\text{t},i}$ in Fig.
11(c). The apparent strong damping of the rotation signals indicates a
distribution of frequencies due to inhomogeneous broadening. For the first
rotation signal we find a strong increase in $\lambda_{\text{t}}$ when warming
from 2 to 3 K. This means that the local field distribution at this muon site
broadens around the lowest temperature transition and suggests that Cu1 is
more strongly coupled to the first site than the second site. The second
frequency has a much higher damping at low temperature, which is reduced
somewhat on warming toward the second transition at 5.0 K. There is clearly a
much greater variation in local fields at this site than at site 1.

The damping $\sigma$ of the Gaussian signal decreases significantly on warming
from 1.5 to 3 K, i.e., the static local field distribution narrows and the
fields become more homogeneous. This may be related to a spin rearrangement in
the neighborhood of the muon site. For higher temperatures the damping stays
essentially unchanged within the uncertainty.

We now return to the data at 4.8 and 5.3 K. The depolarization patterns at
these temperatures are different in shape from all the others. Two distinct
rotating signals can no longer be resolved, and at the earliest times (0–0.1
$\mu$s) a finite phase shift $\varphi$ of $-$50(5)∘ is visible (see Fig. 10).
Fits to the spectra at 3.1 and 4.2 K give a reduced phase shift of $-$25(5)∘
for the rotating signal ($i=2$) with greater asymmetry. At 1.5 and 2 K this
phase shift is consistent with zero. Phase shifts of the order observed at 4.8
and 5.3 K are fingerprints of the complex field distributions which arise in
incommensurately modulated spin structures [52, 53], in agreement with the
incommensurate or multiple-$q$ phases refined from neutron diffraction data in
a similar temperature range.

Starting at 4.2 K the asymmetry contribution from the Gaussian signal has
increased at the expense of the rotating signal [see Fig. 11(a)]. In contrast
to lower temperatures, there is a finite depolarization for times $t>1$ $\mu$s
(see Fig. 15) showing that the local fields are now fluctuating. For the data
analysis we thus use a so-called dynamic Gaussian Kubo-Toyabe function [54,
55] instead of the static Kubo-Toyabe profile used at lower temperatures. We
extract fluctuation rates that are relatively slow: 1.3(2) and 2.3(5) $\mu$s-1
at 4.8 and 5.3 K, respectively. A dynamic fit of the Gaussian signal at 4.2 K
results in a considerably lower fluctuation rate of only 0.15(5) $\mu$s-1. At
both temperatures the dynamic fluctuations are visible in the changes of the
Gaussian damped signal, while the rotating signal indicates incommensurate
order of the Cu spins. Note that at lower temperatures where we see two
rotating signals, the fluctuations are slower, and we see a weaker, static
Gaussian damped signal. Since the most probable muon sites lie close to oxygen
atoms, which bridge the two Cu sites, the muons may be expected to sense
fields originating from both Cu1 and Cu2. Without knowing the locations of the
muon stopping sites it is unfortunately not possible to separate the roles of
these sites.

In summary, our $\mu$SR data indicate an onset of short-range order below 8 K
followed by full long-range order at 5.3 K. Signals at 5.3 and 4.8 K support
the presence of incommensurate order and indicate slowly fluctuating local
fields. At the lowest temperatures the two oscillating signals with their
temperature-dependent frequencies are clearly associated with static,
commensurate long-range order. A reduced phase shift in the idle-spin phase at
3.1 and 4.2 K indicates that vestiges of the incommensurate modulation persist
in this phase. The spins are static on muon timescales here, which may
indicate that the idle-spin phase is either glassy or short-range. The
splitting of ZFC and FC magnetization below 5.0 K argues against slow
fluctuations, and there cannot be long-range order within a ladder and
disorder from one ladder to the next, as this would produce clear signatures
in the spin-flip neutron scattering data. The splitting between ZFC and FC
data is small, possibly because the idle spins remain constrained in the plane
of the cycloid, or perhaps because the spins are trained into particular
orientations when the system passes through the phases at higher and lower
temperature. The persistence of a minor Gaussian-damped signal to low
temperatures indicates a random component in the local spin arrangement which
remains disordered to the lowest temperatures, perhaps an additional canting
angle. This is most likely associated with the Cu1 site, where we refine a
somewhat reduced ordered moment.

##  IX Summary and Outlook

We have confirmed that Cu3SO4(OH)4 undergoes three magnetic ordering
transitions between 5.00 and 5.55 K in zero field, followed by a fourth
transition at 2.8 K, as was suggested by the more-complicated magnetic phase
diagram for antlerite proposed in Ref. 17. Our magnetization, $\mu$SR, and
neutron scattering data have allowed us to identify the spin structures of
these low-field phases.

The picture that emerges here is of a magnetic subsystem which first enters an
incommensurate elliptical helical+cycloidal state, most likely driven by
interladder interactions connecting the strongly antiferromagnetic central
legs of the ladders. While all spins participate, the refined moments are not
the full moment of Cu2+, and $\mu$SR indicates that this state still has a
dynamic component. The outer legs of the ladder have a strong tendency toward
ferromagnetic order along $c$ which fights the incommensurate order, leading
to the helical order on these sites and likely also to the rather small
ordered moment on the central leg. The incommensurate wavevector is
perpendicular to the ladders, so although all sites are incommensurately
ordered, the order along a ladder is commensurate, and this is neither the
“IC-IC-IC” nor “FM-IC-FM” state proposed previously by density-matrix
renormalization group (DMRG) based on DFT parameters [7], since in these
phases the incommensurate (IC) order runs along the ladder. The DFT/DMRG
results neglected interladder coupling and would not be able to generate the
observed incommensurate state. The observed incommensurate phase is also
distinct from the cycloidal state proposed for Cu3SeO4(OH)4 [38]. The latter
state has a similar but commensurate $(\frac{1}{7}\,0\,0)$ propagation vector,
but the spins on the outer legs were cycloidal rather than helical, while the
spins on the inner leg did not participate.

As the incommensurate phase is cooled, the outer legs evidently push the
incommensurability parameter $\delta$ toward zero. They then order
ferromagnetically, antialigned, partially decoupling from the central leg and
leaving it to revert toward its original incommensurate wavevector in a
multiple-$q$ structure. As with the incommensurate phase, since the
incommensurate wavevector presumably relies upon interladder interactions, the
multiple-$q$ state is not the FM-IC-FM state proposed by DMRG to lie nearby in
parameter space. The discovery of multiple-$q$ magnetic order in a frustrated
quantum magnet is likely to be of considerable interest, as the quantum nature
of the spins may lead to entirely new and unexpected physics.

This is actually a particularly unique multiple-$q$ structure, insofar as one
magnetic site hosts commensurate order while the other orders
incommensurately. This is the magnetic analog of misfit crystal structures, in
which separate layers of a compound fail to line up due to a large lattice
mismatch [56, 57]. The misfit materials only rarely host magnetic ions in both
types of layers. In fact, to the authors’ knowledge, there are only three
misfit-layered compounds with magnetic ions in both layers on which magnetic
properties have been investigated: (CeS)1.20CrS2 exhibits two transitions in
the magnetization, which were attributed to the two separate layers ordering
independently [58], while magnetization data on (GdS)1.27CrS2 suggested a
transition at low temperature [59] which was not investigated further. In
neither of these materials has magnetic diffraction been applied to determine
the spin arrangement in any layer. Finally, in [Ca2CoO3]0.62[CoO2], spin-
density-wave order in the CoO2 layer [60] was recently found to compete with
ferromagnetic clusters in the Ca2CoO3 layer to produce a glassy state [61].
Due largely to a scarcity of suitable materials platforms, no long-range
magnetic-misfit order has been identified in the misfit-layered materials. The
magnetic-misfit phase in antlerite, on the other hand, involves
interpenetrating magnetic lattices that are structurally commensurate, and the
magnetic misfit structure emerges spontaneously because of the different
temperatures of the incommensurate-commensurate lock-in phase transitions in
the ferro- and antiferromagnetic legs of the spin ladder. As noted above, the
results in Refs. 15, 17 suggest that this novel phase occupies a large swath
of the $H$–$T$ phase diagram. The as-yet-unidentified “$\gamma$1” and
“$\gamma$2” phases may also emerge from it at higher $H\parallel c$, which
could indicate an ability to field-tune this magnetic-misfit phase.

Upon further cooling, the ferromagnetic order evidently strengthens on the
outer legs, which appears to temporarily destabilize the order on the central
leg. We detect this as an “idle-spin” state, closely similar to that proposed
previously as the ground state [14, 15]. However, the idle spins are likely
static, at least on the timescale of $\mu$SR, possibly frozen in a glassy
state with only short-range correlations, similar to the one found in the
misfit cobaltate [61]. Finally, at 2.8 K the central leg orders
antiferromagnetically and the fully-ordered low-temperature state is obtained.

The long-range order on the central Cu1 leg is evidently reentrant. The
question remains what its spins are doing in the “idle-spin” phase between 2.8
and 5.0 K. Our magnetization data find an additional contribution to the
magnetization for fields along $b$ when cooling into this phase. This suggests
that the “idle” Cu1 spins remain locked in the plane of the cycloid. This
would make the spins more readily polarized once their long-range order is
extinguished, but only for fields perpendicular to the $ac$ plane of the
failed cycloid. When the Cu1 spins order antiferromagnetically below 2.8 K,
this enhanced magnetization falls away again.

DFT and DMRG calculations proposed that the inner and outer legs in antlerite
are independently on the cusp of a phase transition, leading to a complex
phase diagram with exchange parameters and $U$ [7]. However, the complexity of
these calculations required neglecting minor contributions such as anisotropy,
Dzyaloshinskii-Moriya (DM) interactions, and interladder exchanges. The phases
we identify at zero field indicate that these neglected interactions
contribute an additional rich complexity to the system. The outer legs of the
ladder clearly exhibit strong anisotropy, with the moments directed
preferentially along $c$. Strong canting in the previously reported lowest-
temperature phase and within the central leg in the incommensurate phase may
suggest a role for DM interactions. Meanwhile, the incommensurate order
presumably arises from relative rotations of the Cu1 moments between the
ladders, which can only arise from interladder exchange interactions. That the
phases newly identified here were absent from the calculated phase diagram is
most likely a result of having to exclude weaker interactions, making these
phases inaccessible to DMRG. The high tunability suggested by DMRG is clearly
only the tip of the iceberg, and temperature is only one tuning knob of many.
More as-yet-unidentified phases have been seen in antlerite in applied
magnetic fields [15, 17], while to our knowledge tuning by pressure and strain
have not yet been reported. Se substitution should be revisited [38], and a
very different alternation of exchanges leads to a completely different ground
state in szenicsite Cu3(MoO4)(OH)4 [62], which to our knowledge has never been
synthesized, suggesting another potential substitution series. We anticipate
that antlerite and its derivatives will host a very rich collection of
magnetic phases in an accessible range of parameters.

###### Acknowledgements.

The authors are grateful for the assistance of Philipp Schlender and Ellen
Häußler in initial sample characterization and Hubert Luetkens with $\mu$SR
measurements. This project was funded by the German Research Foundation (DFG)
via the projects B01, B03, C03, and C06 of the Collaborative Research Center
SFB 1143 (project-id 247310070); Research Training Group GRK 1621 (project-id
129760637); the Würzburg-Dresden Cluster of Excellence on Complexity and
Topology in Quantum Matter – ct.qmat (EXC 2147, project-id 390858490); and
through grants No. IN 209/9-1, PE 3318/2-1, and LI 244/12-2, and Project No.
422219907. This work is based in part on experiments performed at the Swiss
Muon Source S$\mu$S, Paul Scherrer Institut, Villigen, Switzerland. The
authors acknowledge the support of the Australian Centre for Neutron
Scattering, Australian Nuclear Science and Technology Organisation, in
providing neutron research facilities used in this work. The Institut Laue-
Langevin, Grenoble (France) is acknowledged for providing neutron beam time;
data underpinning the ILL diffraction results may be accessed via Refs. 63,
64, 65.

##  Appendix A UV-vis spectroscopy

Figure 12: Optical absorption spectrum of Cu3SO4(OH)4. Data are in purple,
while the fit described in the text is in red.

The infrared spectrum of antlerite has been reported several times [14, 66,
67, 68], but we are only aware of one report of its optical response in the
ultraviolet, visible, and near-infrared regimes [69], and this has significant
gaps. We show the absorption spectrum of Cu3SO4(OH)4 in these frequency
regimes in Fig. 12. This spectrum was fit to five Gaussians in the frequency
domain, yielding peaks at 207, 303, 373, 444, and 909 nm (48236, 32996, 26845,
22500, and 11007 cm-1), which are marked on the figure. The previous work
found peaks in this frequency range at 49260, 44445, 16390, and 10990 cm-1,
corresponding to 203, 225, 610, and 910 nm [69]. These were attributed to
${}^{2}\text{A}_{\text{1g}}(d_{x^{2}-y^{2}})\rightarrow{}^{2}\text{B}_{\text{2g}}(d_{xy})$
(10990 cm-1),
${}^{2}\text{A}_{\text{1g}}(d_{x^{2}-y^{2}})\rightarrow{}^{2}\text{B}_{\text{3g}}(d_{yz})$
(16390 cm-1), and charge transfer excitations at higher wavenumber. Our data
would not allow us to clearly distinguish a shoulder at 225 nm, and we are not
able to resolve the weak hump reported at 610 nm. The previous report had
large gaps in the spectrum, most notably between $\sim$250 and 595 nm, which
would have prevented the observation of additional features at intermediate
wavelengths.

We consider it plausible that the peaks at shorter wavelength are indeed
charge transfer excitations as previously reported, while the longer-
wavelength peaks are $d$–$d$ excitations. A reliable assignment of all
observed modes would require detailed modeling and calculation of excited
states beyond the scope of this work — antlerite has two Cu sites with
different coordination environment [13, 14], neither site has a highly
symmetric coordination sphere, there are multiple types of ligands, and
electron-electron interactions are thought to be very strong in this system,
all of which add considerable complexity to such a calculation. Spectra in the
ultraviolet, visible, and infrared frequency ranges can serve as a fingerprint
of a substance, so having a broader spectrum available (when combined with the
previously available infrared results) is likely to prove useful for chemical
identification purposes.

##  Appendix B Magnetic Refinements

As arXiv ancillary files, we provide magnetic crystallographic information
files (mCIF) describing our magnetic refinements:

Phase | Temperature | Filename
---|---|---
Idle Spin | 4.0 K | IdleSpin.mcif
Multiple-$q$ | 5.3 K | MultiQ.mcif
Incommensurate | 5.3 K | Icom.mcif

##  Appendix C Diffuse Magnetic Intensity to Higher Temperatures

Figure 13: Diffuse magnetic intensity at higher temperatures. Solid lines
represent the magnetic form factor of Cu2+. The 6.0-K dataset is identical to
that in Fig. 9(a).

Figure 13 shows the diffuse magnetic intensity collected on the D7
diffractometer to higher temperatures. The data at all temperatures are fit to
$I=AF^{2}$, where $I$ is the intensity, $F$ is the magnetic form factor for
Cu2+, and $A$ is 0.2290(19) for all datasets in the units used. The diffuse
peak shifts to lower $|\bm{Q}|$, but a significant diffuse contribution
persists to at least 200 K.

##  Appendix D Additional Muon Spin Rotation and Relaxation ($\mu$SR) Results

Figure 14: ZF and wLF spectra in the paramagnetic state at 6.5 K. The damping
is suppressed significantly by longitudinal field. Figure 15:  wLF spectra in
the long-range magnetically ordered states.

The depolarization in ZF spectra in the paramagnetic regime is dominated by
strong damping described by a static Gaussian Kubo-Toyabe function

$G_{z}(t)^{\text{GKT}}=\frac{2}{3}[1-(\sigma t)^{2}]e^{-\frac{1}{2}(\sigma
t)^{2}}+\frac{1}{3}.$ (2)

Here the Gaussian damping width $\sigma$ is $\gamma_{\mu}\Delta$, with
$\Delta$ being the width of the Gaussian distribution of local field
components $B_{x,y}$ around zero. $G_{z}$ is the component of the asymmetry
along the direction of the incoming muon beam.

Figure 16: Normalized asymmetry from the fitted wTF spectra versus
temperature. Vertical dashed lines indicate the magnetic phase transitions
extracted from specific heat data.

For parametrizing our asymmetry spectra below $T_{\text{N}}$ we use a
phenomenological approach:

$A(t)=\sum_{i=1}^{2}a_{\text{rot,}i}G_{z,i}(t)^{\text{rot}}+a_{\text{GKT}}G_{z}(t)^{\text{GKT}}+\text{BG}$
(3)

with

$G_{z,i}(t)^{\text{rot}}=\frac{2}{3}e^{-\lambda_{\text{t},i}t}\cos(2\pi\nu_{\mu,i}t+\varphi_{i})+\frac{1}{3}e^{-\lambda_{\text{l}}t},$
(4)

where $\nu_{\mu,i}$ is the rotation frequency in local internal fields caused
by the ordered electronic moments, while $\lambda_{\text{t}}$ is the
transverse damping parameter of the oscillations, mainly caused by field
inhomogeneities, and can be seen as a direct measure of the width of the
distribution of $B_{i}$ arising from local disorder in the long-range-ordered
spin array. Here $a_{\text{rot}}$ is the contribution to the asymmetry from
the rotating signal. The two terms in $G_{z,i}(t)^{\text{rot}}$, transverse
and longitudinal, arise from the usual approach to describe the isotropic
average by assuming that $\frac{2}{3}$ of the muons experience an internal
field perpendicular to the initial orientation of their spins, resulting in
precession. For the remaining $\frac{1}{3}$ of the muons, the internal field
is oriented parallel to the muon spins and no precession takes place. However,
dynamic muon spin polarization, caused by fluctuations of the magnetic moments
surrounding the muon, introduces a nonzero longitudinal term with a relaxation
rate $\lambda_{\text{l}}$ proportional to the magnetic spin fluctuation rate.

Figure 14 shows the asymmetry in the paramagnetic state at 6.5 K, where a
longitudinal field of 2 mT suppresses the damping. Figure 15 shows wLF spectra
to longer decay times in the magnetically ordered state. Figure 16 shows the
temperature dependence of the normalized asymmetry of the wTF spectra. A
reduction is seen in points just above the highest magnetic transition,
indicating an onset of short-range order, while points below the transition
indicate ordering throughout the sample.

## References

  * Ramirez [1994] A. P. Ramirez, Strongly geometrically frustrated magnets, Annu. Rev. Mater. Sci. 24, 453 (1994).
  * Batista  _et al._ [2016] C. D. Batista, S.-Z. Lin, S. Hayami, and Y. Kamiya, Frustration and chiral orderings in correlated electron systems, Rep. Prog. Phys. 79, 084504 (2016).
  * Schmidt and Thalmeier [2017] B. Schmidt and P. Thalmeier, Frustrated two dimensional quantum magnets, Phys. Rep. 703, 1 (2017).
  * Lacroix  _et al._ [2011] C. Lacroix, P. Mendels, and F. Mila, eds.,  _Introduction to Frustrated Magnetism_, Springer Series in Solid-State Sciences, Vol. 164 (Springer, Berlin, 2011).
  * Park and Snyder [1995] C. Park and R. L. Snyder, Structures of high-temperature cuprate superconductors, J. Am. Ceram. Soc. 78, 3171 (1995).
  * van Smaalen [1999] S. van Smaalen, Structural aspects of spin-chain and spin-ladder oxides, Z. Kristallogr. Cryst. Mater. 214, 786 (1999).
  * Kulbakov  _et al._ [2022] A. A. Kulbakov, D. Y. Kononenko, S. Nishimoto, Q. Stahl, A. Mannathanath Chakkingal, M. Feig, R. Gumeniuk, Y. Skourski, L. Bhaskaran, S. A. Zvyagin, J. P. Embs, I. Puente-Orench, A. Wildes, J. Geck, O. Janson, D. S. Inosov, and D. C. Peets, Coupled frustrated ferromagnetic and antiferromagnetic quantum spin chains in the quasi-one-dimensional mineral antlerite, Cu3SO4(OH)4, Phys. Rev. B 106, L020405 (2022).
  * Inosov [2018] D. S. Inosov, Quantum magnetism in minerals, Adv. Phys. 67, 149 (2018).
  * Shores  _et al._ [2005] M. P. Shores, E. A. Nytko, B. M. Bartlett, and D. G. Nocera, A structurally perfect ${S}=\frac{1}{2}$ kagomé antiferromagnet, J. Am. Chem. Soc. 127, 13462 (2005).
  * Norman [2016] M. R. Norman, Colloquium: Herbertsmithite and the search for the quantum spin liquid, Rev. Mod. Phys. 88, 041002 (2016).
  * Heinze  _et al._ [2021] L. Heinze, H. O. Jeschke, I. I. Mazin, A. Metavitsiadis, M. Reehuis, R. Feyerherm, J.-U. Hoffmann, M. Bartkowiak, O. Prokhnenko, A. U. B. Wolter, X. Ding, V. S. Zapf, C. Corvalán Moya, F. Weickert, M. Jaime, K. C. Rule, D. Menzel, R. Valentí, W. Brenig, and S. Süllow, Magnetization process of atacamite: A case of weakly coupled ${S}=1/2$ sawtooth chains, Phys. Rev. Lett. 126, 207201 (2021).
  * Zhang  _et al._ [2020] H. Zhang, Z. Zhao, D. Gautreau, M. Raczkowski, A. Saha, V. O. Garlea, H. Cao, T. Hong, H. O. Jeschke, S. D. Mahanti, T. Birol, F. F. Assaad, and X. Ke, Coexistence and interaction of spinons and magnons in an antiferromagnet with alternating antiferromagnetic and ferromagnetic quantum spin chains, Phys. Rev. Lett. 125, 037204 (2020).
  * Hawthorne  _et al._ [1989] F. C. Hawthorne, L. A. Groat, and R. R. Eby, Antlerite, Cu3SO4(OH)4, a heteropolyhedral wallpaper structure, Can. Mineral. 27, 205 (1989).
  * Vilminot  _et al._ [2003] S. Vilminot, M. Richard-Plouet, G. André, D. Swierczynski, M. Guillot, F. Bourée-Vigneron, and M. Drillon, Magnetic structure and properties of Cu3(OH)4SO4 made of triple chains of spins $s$=1/2, J. Solid State Chem. 170, 255 (2003).
  * Hara  _et al._ [2011] S. Hara, H. Kondo, and H. Sato, Successive magnetic transitions in candidate “idle-spin” system, Cu3(OH)4SO4, J. Phys. Soc. Jpn. 80, 043701 (2011).
  * Koo  _et al._ [2012] H.-J. Koo, R. K. Kremer, and M.-H. Whangbo, Non-idle-spin behavior and field-induced magnetic transitions of the triple chain magnet Cu3(OH)4SO4, J. Phys. Soc. Jpn. 81, 063704 (2012).
  * Fujii  _et al._ [2013] Y. Fujii, Y. Ishikawa, H. Kikuchi, Y. Narumi, H. Nojiri, S. Hara, and H. Sato, Magnetic property of a single crystal of spin-1/2 triple-chain magnet Cu3(OH)4SO4, J. Korean Phys. Soc. 62, 2054 (2013).
  * Studer  _et al._ [2006] A. J. Studer, M. E. Hagen, and T. J. Noakes, Wombat: The high-intensity powder diffractometer at the OPAL reactor, Physica B: Condens. Matter 385–386, 1013 (2006).
  * Puente Orench  _et al._ [2014] I. Puente Orench, J. F. Clergeau, S. Martínez, M. Olmos, O. Fabelo, and J. Campo, The new powder diffractometer D1B of the Institut Laue Langevin, J. Phys.: Conf. Ser. 549, 012003 (2014).
  * Hansen  _et al._ [2008] T. C. Hansen, P. F. Henry, H. E. Fischer, J. Torregrossa, and P. Convert, The D20 instrument at the ILL: a versatile high-intensity two-axis neutron diffractometer, Meas. Sci. Technol. 19, 034001 (2008).
  * Stewart  _et al._ [2009] J. R. Stewart, P. P. Deen, K. H. Andersen, H. Schober, J.-F. Barthélémy, J. M. Hillier, A. P. Murani, T. Hayes, and B. Lindenau, Disordered materials studied using neutron polarization analysis on the multi-detector spectrometer, D7, J. Appl. Crystallogr. 42, 69 (2009).
  * Rodríguez-Carvajal [1993] J. Rodríguez-Carvajal, Recent advances in magnetic structure determination by neutron powder diffraction, Physica B: Condens. Matter 192, 55 (1993).
  * Sears [1992] V. F. Sears, Neutron scattering lengths and cross sections, Neutron News 3, 26 (1992).
  * Weschke and Schierle [2018] E. Weschke and E. Schierle, The UE46 PGM-1 beamline at BESSY II, Journal of Large-Scale Research Facilities 4, A127 (2018).
  * Amato  _et al._ [2017] A. Amato, H. Luetkens, K. Sedlak, A. Stoykov, R. Scheuermann, M. Elender, A. Raselli, and D. Graf, The new versatile general purpose surface-muon instrument (GPS) based on silicon photomultipliers for $\mu$SR measurements on a continuous-wave beam, Rev. Sci. Instrum. 88, 093301 (2017).
  * Suter and Wojek [2012] A. Suter and B. M. Wojek, Musrfit: A free platform-independent framework for $\mu$SR data analysis, Phys. Procedia 30, 69 (2012).
  * Grioni  _et al._ [1992] M. Grioni, J. F. van Acker, M. T. Czyžyk, and J. C. Fuggle, Unoccupied electronic structure and core-hole effects in the x-ray-absorption spectra of Cu2O, Phys. Rev. B 45, 3309 (1992).
  * Grioni  _et al._ [1989] M. Grioni, J. B. Goedkoop, R. Schoorl, F. M. F. de Groot, J. C. Fuggle, F. Schäfers, E. E. Koch, G. Rossi, J.-M. Esteva, and R. C. Karnatak, Studies of copper valence states with Cu ${L}_{3}$ x-ray-absorption spectroscopy, Phys. Rev. B 39, 1541 (1989).
  * Jiang  _et al._ [2013] P. Jiang, D. Prendergast, F. Borondics, S. Porsgaard, L. Giovanetti, E. Pach, J. Newberg, H. Bluhm, F. Besenbacher, and M. Salmeron, Experimental and theoretical investigation of the electronic structure of Cu2O and CuO thin films on Cu(110) using x-ray photoelectron and absorption spectroscopy, J. Chem. Phys. 138, 024704 (2013).
  * Chen [1997] J. G. Chen, NEXAFS investigations of transition metal oxides, nitrides, carbides, sulfides and other interstitial compounds, Surf. Sci. Rep. 30, 1 (1997).
  * Okubo  _et al._ [2009] S. Okubo, H. Yamamoto, M. Fujisawa, H. Ohta, T. Nakamura, and H. Kikuchi, High field ESR measurements of quantum triple chain system Cu3(OH)4SO4, J. Phys.: Conf. Ser. 150, 042156 (2009).
  * Leininger  _et al._ [2011] P. Leininger, D. Chernyshov, A. Bosak, H. Berger, and D. S. Inosov, Competing charge density waves and temperature-dependent nesting in $2H$-TaSe2, Phys. Rev. B 83, 233101 (2011).
  * Capogna  _et al._ [2010] L. Capogna, M. Reehuis, A. Maljuk, R. K. Kremer, B. Ouladdiaf, M. Jansen, and B. Keimer, Magnetic structure of the edge-sharing copper oxide chain compound NaCu2O2, Phys. Rev. B 82, 014407 (2010).
  * Kobayashi  _et al._ [2009] Y. Kobayashi, K. Sato, Y. Yasui, T. Moyoshi, M. Sato, and K. Kakurai, Studies of multiferroic system of LiCu2O2: II. magnetic structures of two ordered phases with incommensurate modulations, J. Phys. Soc. Jpn. 78, 084721 (2009).
  * Willenberg  _et al._ [2012] B. Willenberg, M. Schäpers, K. C. Rule, S. Süllow, M. Reehuis, H. Ryll, B. Klemke, K. Kiefer, W. Schottenhamel, B. Büchner, B. Ouladdiaf, M. Uhlarz, R. Beyer, J. Wosnitza, and A. U. B. Wolter, Magnetic frustration in a quantum spin chain: The case of linarite PbCuSO4(OH)2, Phys. Rev. Lett. 108, 117202 (2012).
  * Ehlers  _et al._ [2013] G. Ehlers, J. R. Stewart, A. R. Wildes, P. P. Deen, and K. H. Andersen, Generalization of the classical xyz-polarization analysis technique to out-of-plane and inelastic scattering, Rev. Sci. Instrum. 84, 093901 (2013).
  * Vilminot  _et al._ [2002] S. Vilminot, M. Richard-Plouet, G. André, D. Swierczynski, F. Bourée-Vigneron, E. Marino, and M. Guillot, Synthesis, structure and magnetic properties of copper hydroxysulfates, Cryst. Eng. 5, 177 (2002).
  * Vilminot  _et al._ [2007] S. Vilminot, G. André, F. Bourée-Vigneron, M. Richard-Plouet, and M. Kurmoo, Magnetic properties and magnetic structures of Cu3(OD)${}_{4}X$O4, $X$ = Se or S: Cycloidal versus collinear antiferromagnetic structure, Inorg. Chem. 46, 10079 (2007).
  * Hälg and Furrer [1986] B. Hälg and A. Furrer, Anisotropic exchange and spin dynamics in the type-I (-IA) antiferromagnets CeAs, CeSb, and USb: A neutron study, Phys. Rev. B 34, 6258 (1986).
  * Burlet  _et al._ [1982] P. Burlet, J. Rossat-Mignod, J. M. Effantin, T. Kasuya, S. Kunii, and T. Komatsubara, Magnetic ordering in cerium hexaboride CeB6, J. Appl. Phys. 53, 2149 (1982).
  * Effantin  _et al._ [1985] J. M. Effantin, J. Rossat-Mignod, P. Burlet, H. Bartholin, S. Kunii, and T. Kasuya, Magnetic phase diagram of CeB6, J. Magn. Magn. Mater. 47–48, 145 (1985).
  * Long [1993] M. W. Long, Multiple-q structures in frustrated antiferromagnets, Int. J. Mod. Phys. B 07, 2981 (1993).
  * Ishiwata  _et al._ [2020] S. Ishiwata, T. Nakajima, J.-H. Kim, D. S. Inosov, N. Kanazawa, J. S. White, J. L. Gavilano, R. Georgii, K. M. Seemann, G. Brandl, P. Manuel, D. D. Khalyavin, S. Seki, Y. Tokunaga, M. Kinoshita, Y. W. Long, Y. Kaneko, Y. Taguchi, T. Arima, B. Keimer, and Y. Tokura, Emergent topological spin structures in the centrosymmetric cubic perovskite SrFeO3, Phys. Rev. B 101, 134406 (2020).
  * Chen  _et al._ [2021] W. Chen, X. Li, Z. Hu, Z. Hu, L. Yue, R. Sutarto, F. He, K. Iida, K. Kamazawa, W. Yu, X. Lin, and Y. Li, Spin-orbit phase behavior of Na2Co2TeO6 at low temperatures, Phys. Rev. B 103, L180404 (2021).
  * Ivanov  _et al._ [2012] S. A. Ivanov, R. Tellgren, C. Ritter, P. Nordblad, R. Mathieu, G. André, N. V. Golubko, E. D. Politova, and M. Weil, Temperature-dependent multi-k magnetic structure in multiferroic Co3TeO6, Mater. Res. Bull. 47, 63 (2012).
  * Wang  _et al._ [2013] C.-W. Wang, C.-H. Lee, C.-Y. Li, C.-M. Wu, W.-H. Li, C.-C. Chou, H.-D. Yang, J. W. Lynn, Q. Huang, A. B. Harris, and H. Berger, Complex magnetic couplings in Co3TeO6, Phys. Rev. B 88, 184427 (2013).
  * Lee  _et al._ [2017] C.-H. Lee, C.-W. Wang, Y. Zhao, W.-H. Li, J. W. Lynn, A. B. Harris, K. Rule, H.-D. Yang, and H. Berger, Complex magnetic incommensurability and electronic charge transfer through the ferroelectric transition in multiferroic Co3TeO6, Sci. Rep. 7, 6437 (2017).
  * Paddison  _et al._ [2013] J. A. M. Paddison, J. R. Stewart, and A. L. Goodwin, Spinvert: a program for refinement of paramagnetic diffuse scattering data, J. Phys.: Condens. Matter 25, 454220 (2013).
  * Cox [1987] S. F. J. Cox, Implanted muon studies in condensed matter science, J. Phys. C: Solid State Phys. 20, 3187 (1987).
  * Uemura  _et al._ [1985] Y. J. Uemura, T. Yamazaki, D. R. Harshman, M. Senba, and E. J. Ansaldo, Muon-spin relaxation in AuFe and CuMn spin glasses, Phys. Rev. B 31, 546 (1985).
  * Kubo [1981] R. Kubo, A stochastic theory of spin relaxation, Hyperfine Interact. 8, 731 (1981).
  * Yaouanc and Dalmas de Réotier [2011] A. Yaouanc and P. Dalmas de Réotier,  _Muon Spin Rotation, Relaxation, and Resonance: Applications to Condensed Matter_ (Oxford University Press, Oxford, UK, 2011).
  * Franke  _et al._ [2018] K. J. A. Franke, B. M. Huddart, T. J. Hicken, F. Xiao, S. J. Blundell, F. L. Pratt, M. Crisanti, J. A. T. Barker, S. J. Clark, A. Štefančič, M. C. Hatnean, G. Balakrishnan, and T. Lancaster, Magnetic phases of skyrmion-hosting GaV4S8-ySey ($y$=0,2,4,8) probed with muon spectroscopy, Phys. Rev. B 98, 054428 (2018).
  * Hayano  _et al._ [1979] R. S. Hayano, Y. J. Uemura, J. Imazato, N. Nishida, T. Yamazaki, and R. Kubo, Zero-and low-field spin relaxation studied by positive muons, Phys. Rev. B 20, 850 (1979).
  * de Reotier and Yaouanc [1992] P. D. de Reotier and A. Yaouanc, Quantum calculation of the muon depolarization function: effect of spin dynamics in nuclear dipole systems, J. Phys.: Condens. Matter 4, 4533 (1992).
  * Wiegers [1996] G. A. Wiegers, Misfit layer compounds: Structures and physical properties, Prog. Solid State Chem. 24, 1 (1996).
  * Ng and McQueen [2022] N. Ng and T. M. McQueen, Misfit layered compounds: Unique, tunable heterostructured materials with untapped properties, APL Mater. 10, 100901 (2022).
  * Suzuki  _et al._ [1993] K. Suzuki, T. Kondo, M. Iwasaki, and T. Enoki, Variety of magnetism in incommensurate misfit layer compounds (RES)xTS2 (RE=rare earths, T=Ta, V, Ti, Cr), Jpn. J. Appl. Phys. 32, 341 (1993).
  * Lafond and Meerschaut [1993] A. Lafond and A. Meerschaut, Préparation, détermination structurale et propriétés magnétiques d’un nouveau composé lamellaire incommensurable: Le sulfure de gadolinium et de chrome, Mater. Res. Bull. 28, 979 (1993).
  * Sugiyama  _et al._ [2003] J. Sugiyama, J. H. Brewer, E. J. Ansaldo, H. Itahara, K. Dohmae, Y. Seno, C. Xia, and T. Tani, Hidden magnetic transitions in the thermoelectric layered cobaltite [Ca2CoO3]0.62[CoO2], Phys. Rev. B 68, 134423 (2003).
  * Ahad  _et al._ [2020] A. Ahad, K. Gautam, K. Dey, S. S. Majid, F. Rahman, S. K. Sharma, J. A. H. Coaquira, I. da Silva, E. Welter, and D. K. Shukla, Magnetic correlations in subsystems of the misfit [Ca2CoO3]0.62[CoO2] cobaltate, Phys. Rev. B 102, 094428 (2020).
  * Lebernegg  _et al._ [2017] S. Lebernegg, O. Janson, I. Rousochatzakis, S. Nishimoto, H. Rosner, and A. A. Tsirlin, Frustrated spin chain physics near the Majumdar-Ghosh point in szenicsite Cu3(MoO4)(OH)4, Phys. Rev. B 95, 035145 (2017).
  * Inosov  _et al._ [2020] D. Inosov, A. A. Kulbakov, D. C. Peets, and I. Puente-Orench, Determining the zero-field magnetic structure of antlerite (2020), Institut Laue-Langevin (ILL) data, Grenoble. doi:10.5291/ILL-DATA.5-31-2734.
  * Inosov  _et al._ [2021a] D. S. Inosov, A. A. Kulbakov, A. Mannathanath Chakkingal, D. C. Peets, I. Puente-Orench, and C. Ritter, Determining the intermediate-temperature incommensurate magnetic structure of antlerite (2021a), Institut Laue-Langevin (ILL) data, Grenoble. doi:10.5291/ILL-DATA.5-31-2864.
  * Inosov  _et al._ [2021b] D. S. Inosov, A. A. Kulbakov, A. Mannathanath Chakkingal, D. C. Peets, and A. Wildes, Verifying the magnetic origin of the low-temperature diffuse signal in antlerite powder (2021b), Institut Laue-Langevin (ILL) data, Grenoble. doi:10.5291/ILL-DATA.5-32-923.
  * Bissengaliyeva  _et al._ [2012] M. R. Bissengaliyeva, L. P. Ogorodova, L. V. Mel’chakova, and M. F. Vigasina, Thermochemical investigation of natural antlerite, J. Therm. Anal. Calorim. 109, 467 (2012).
  * Bissengaliyeva  _et al._ [2013] M. R. Bissengaliyeva, N. S. Bekturganov, D. B. Gogol, S. T. Taimassova, T. A. Koketai, and M. A. Bespyatov, Heat capacities of natural antlerite and brochantite at low temperature, J. Chem. Eng. Data 58, 2904 (2013).
  * Zittlau  _et al._ [2013] A. H. Zittlau, Q. Shi, J. Boerio-Goates, B. F. Woodfield, and J. Majzlan, Thermodynamics of the basic copper sulfates antlerite, posnjakite, and brochantite, Geochemistry 73, 39 (2013).
  * Rama Subba Reddy  _et al._ [2002] R. Rama Subba Reddy, S. Lakshmi Reddy, G. Siva Reddy, and B. J. Reddy, Spectral studies of divalent copper in antlerite mineral, Cryst. Res. Technol. 37, 485 (2002).