35: $T_{\mathrm{cycle}}=5.9\ $\mathrm{\SIUnitSymbolMicro s}$$. This includes
the steady-flow scheme ($T_{\mathrm{cycle}}^{(1)}=5.9\
$\mathrm{\SIUnitSymbolMicro s}$$).

The requirement on the number of measurement devices is given by Eq. 30:

$\displaystyle m$
$\displaystyle\geq\left\lceil\frac{\tau_{\mathrm{meas}}^{m=1}K_{\mathrm{loop}}}{3\tau_{\mathrm{sh}}+4\tau_{\textsc{cz}}+3\tau_{\textsc{h}}}\right\rceil=\left\lceil\frac{K_{\mathrm{loop}}}{5}\right\rceil$

Hence, with $m=\lceil 5/5\rceil\approx 1$ measurement devices per loop, we can
fit in $K_{\mathrm{loop}}=5$ qubits to carry out the $5$-layer distillation
scheme and maintain the same code cycle time ($T_{\mathrm{cycle}}^{(5)}=5.9\
$\mathrm{\SIUnitSymbolMicro s}$$).

####  F.1.2 Ancilla pipeline is rate-limiting

We need $m<K_{\mathrm{loop}}/5$. If we want to fit in $K_{\mathrm{loop}}=10$
qubits, this implies $m=1$. The corresponding code cycle time is just the
ancilla circuit time given by Eq. 29:

$\displaystyle
T_{\mathrm{cycle}}=T_{\mathrm{eff}}^{\mathrm{anc}}=4\tau_{\textsc{cz}}-\tau_{\textsc{h}}+K_{\mathrm{loop}}\frac{\tau_{\mathrm{meas}}^{m=1}}{m}\approx
10.95\ $\mathrm{\SIUnitSymbolMicro s}$.$

###  F.2 Adding buffering to the data pipeline

Substituting the time needed for the different operations into Eq. 33 and
focusing on the case of $K_{\mathrm{loop}}=10$, the minimum code cycle time is
given by

$\displaystyle
T_{\mathrm{loop}}^{\mathrm{min}}=T_{\mathrm{loop}}^{\mathrm{min}*}=\frac{\tau_{\mathrm{sh}}+4\tau_{\textsc{cz}}+K_{\mathrm{loop}}\tau_{\mathrm{meas}}}{4}=(0.55+\frac{5}{2m})\
$\mathrm{\SIUnitSymbolMicro s}$.$

This will satisfy both Eqs. 31 and 24 as long as $m=1$.

For $m=1$, the minimum code cycle time is simply:

$\displaystyle T_{\mathrm{cycle}}^{*}=3T_{\mathrm{loop}}^{\mathrm{min}*}=9.15\
$\mathrm{\SIUnitSymbolMicro s}$.$

which allow us to fit $10$ qubits into the pipeline to carry out the
$10$-layer distillation scheme ($T_{\mathrm{cycle}}^{(10)}=9.15\
$\mathrm{\SIUnitSymbolMicro s}$$). The corresponding time gap is
$\tau_{\mathrm{gap}}=T_{\mathrm{loop}}^{\mathrm{min}^{*}}/9=0.34\
$\mathrm{\SIUnitSymbolMicro s}$$.

###  F.3 Space-time overhead saving for magic state distillation

From the arguments above, with $m=1$ measurement devices per loop, the time
required for no 1-layer, 5-layer and 10-layer schemes for the magic state
distillation circuit are:

$\displaystyle T_{\mathrm{cycle}}^{(1)}$ $\displaystyle=5.9\
$\mathrm{\SIUnitSymbolMicro
s}$\quad\text{(\lx@cref{creftype~refnum}{sec:two_pipe_time_new})}$
$\displaystyle T_{\mathrm{cycle}}^{(5)}$ $\displaystyle=5.9\
$\mathrm{\SIUnitSymbolMicro
s}$\quad\text{(\lx@cref{creftype~refnum}{sec:two_pipe_time_new})}$
$\displaystyle T_{\mathrm{cycle}}^{(10)}$ $\displaystyle=9.15\
$\mathrm{\SIUnitSymbolMicro
s}$\quad\text{(\lx@cref{creftype~refnum}{sec:one_pipe_time_new})}$

Substituting into Table 2, we get $24$ times space-time saving by using the
$5$-layer scheme and $140$ times space-time saving by using the $10$-layer
scheme.

Note that with these new operation times using ESR, we are not able to simply
increase the number of measurement devices $m$ in each loop to improve the
efficiency of the pipeline. This is because the measurement time is much
closer to the second slowest step: CZ gates. Therefore the measurement step
will not be the rate-limiting step anymore when there are more than two
measurement devices in each loop and improving its efficiency will not the
pipelining efficiency. However, it is possible to add more gate devices at the
same time to further parallelise the other steps as well, which can then
improve the efficiency of the whole pipeline.

##  Appendix G Time Costs for Qubit Pipelines

Here we will consider different ways $k$ qubits can flow through a pipeline
with $M$ steps. The time required for the rate-limiting step is given as
$\tau_{\mathrm{max}}$. For every step in the pipeline, we can put buffering
regions before and after the step, and we will call them _the entry buffer_
and _the exit buffer_ of the step, respectively. These buffers can temporarily
hold the qubits if necessary and otherwise cost zero time. When we say we have
a qubit stream with a qubit time gap $\tau_{\mathrm{gap},m}$ in between the
$m^{\text{th}}$ and $(m+1)^{\text{th}}$ steps, it means that after any given
qubit in the qubit stream exits the $m^{\text{th}}$ exit buffer and enters the
$m+1^{\text{th}}$ exit buffer, the next qubit will do the same after time
$\tau_{\mathrm{gap},m}$. The time gap for the input qubit stream is denoted as
$\tau_{\mathrm{gap},0}$ and the time gap of the qubit stream at the output is
simply $\tau_{\mathrm{gap},M}$. We always have to make sure the time gap
between the qubits entering the $m^{\text{th}}$ step is larger than the
processing time of the $m^{\text{th}}$ step $\tau_{m}$. Hence, sometime there
is a need for changing the time gap from $\tau_{\mathrm{gap},m-1}$ to
$\tau_{m}$ before entering the $m^{\text{th}}$ step if
$\tau_{\mathrm{gap},m-1}<\tau_{m}$.

There are two ways to change the time gap:

  * •

Increasing the time gap from $\tau_{\mathrm{gap}}$ to
$\tau_{\mathrm{gap}}^{\prime}$, we need to buffer the $n^{\text{th}}$ qubit by
the amount of $(n-1)(\tau_{\mathrm{gap}}^{\prime}-\tau_{\mathrm{gap}})$. Note
that all qubits are buffered other than the first qubit.

  * •

Decreasing the time gap from $\tau_{\mathrm{gap}}$ to
$\tau_{\mathrm{gap}}^{\prime}$, we need to buffer the $n^{\text{th}}$ qubit by
the amount of $(k-n)(\tau_{\mathrm{gap}}-\tau_{\mathrm{gap}}^{\prime})$. Note
that all qubits are buffered other than the last qubit.

###  G.1 Constant time gap

Let us suppose we want to maintain a time gap of $\tau_{\mathrm{gap}}$ for the
qubit stream throughout the pipeline:
$\tau_{\mathrm{gap},m}=\tau_{\mathrm{gap}}\quad\forall m$. Such a qubit stream
can pass through all the steps that have $\tau_{m}\leq\tau_{\mathrm{gap}}$
without any buffering. However, whenever we need to pass through a step with
$\tau_{m}>\tau_{\mathrm{gap}}$, we need to temporary increase the time gap
from $\tau_{\mathrm{gap}}$ to $\tau_{m}$, passing through the step, and the
reduce the time gap from $\tau_{m}$ to $\tau_{\mathrm{gap}}$ again. This will
buffer the _first qubit_ by the amount of
$(k-1)(\tau_{m}-\tau_{\mathrm{gap}})$. Hence, after passing through all steps
in the pipeline, the first qubit will be buffered by an amount of

$\displaystyle\Delta
T_{\mathrm{gap}}=(k-1)(\sum_{m=1}^{M}\max(\mathrm{\tau_{m}-\tau_{\mathrm{gap}}},0))$
(36)

Hence, the effective time needed to process the first qubit is now

$\displaystyle T_{\mathrm{eff}}=T_{\mathrm{circ}}+\Delta T_{\mathrm{gap}}.$
(37)

Time needed to process any additional qubit is simply given by the time gap
$\tau_{\mathrm{gap}}$ and thus the total time needed to process $k$ qubits
using this pipeline is

$\displaystyle T_{\mathrm{pipe}}(k)$
$\displaystyle=T_{\mathrm{eff}}+(k-1)\tau_{\mathrm{gap}}.$ (38)

If we have $\tau_{\mathrm{gap}}=\tau_{\mathrm{max}}$, we then have
$\max(\mathrm{\tau_{m}-\tau_{\mathrm{gap}}},0)=0$ for all $m$ and hence the
pipelining time is simply given by Eq. 3 as expected, in which the first qubit
will need $T_{\mathrm{circ}}$ to complete the pipeline while and any
additional qubit will need $\tau_{\mathrm{max}}$, regardless of the depth of
the pipeline $M$.

Using Eq. 36, we can rewrite Eq. 38 as

$\displaystyle T_{\mathrm{pipe}}(k)$ $\displaystyle=T_{\mathrm{circ}}$ (39)
$\displaystyle\quad+(k-1)(\tau_{\mathrm{gap}}+\sum_{m=1}^{M}\max(\tau_{m}-\tau_{\mathrm{gap}},0)).$

which can also be viewed as Eq. 38 with $\Delta T_{\mathrm{gap}}=0$ and
$\tau_{\mathrm{gap}}^{\prime}=\tau_{\mathrm{gap}}+\sum_{m=1}^{M}\max(\tau_{m}-\tau_{\mathrm{gap}},0)$.
Hence, if we have $\tau_{\mathrm{gap}}<\tau_{\mathrm{max}}$, then the time
needed for processing an additional qubit through the pipeline is now
$\tau_{\mathrm{gap}}+\sum_{m=1}^{M}\max(\mathrm{\tau_{m}-\tau_{\mathrm{gap}}},0)$,
which is dependent on the depth of the pipeline $M$.

###  G.2 Steady flow around the loop

Due to the repeated use of the loop in the loop pipeline, an interesting
pipelining scheme besides trying to achieve a steady flow throughout the whole
pipeline will be achieving a steady flow only on the loop instead (excluding
measurement and initialisation), which we will simply call the _steady-loop-
flow_ scheme. This is essentially the same scheme as the steady-flow scheme in
Sec. II.1, but only taking into account the steps on the loop. Hence, the
smallest time gap we can have for the _steady-loop-flow_ scheme is simply:

$\displaystyle\tau_{\mathrm{max}}^{\mathrm{loop}}=\max_{\text{on
loop}}\tau_{i}.$ (40)

which is the time needed for the slowest step on the loop.

The _steady-loop-flow_ scheme will be different from the the _steady-flow_
scheme _only when the rate-limiting step is initialisation or measurement_ ,
so that we have
$\tau_{\mathrm{max}}=\tau_{\mathrm{init/meas}}>\tau_{\mathrm{max}}^{\mathrm{loop}}$.
By definition, the steady-loop-flow scheme does not require any buffers on the
loop. Hence, we have the same cycling period $T_{\mathrm{loop}}$ as the
steady-flow scheme. Due to the smaller time gap
$\tau_{\mathrm{max}}^{\mathrm{loop}}<\tau_{\mathrm{max}}$, we will be able to
use the steady-loop-flow scheme to fit more qubits in the pipeline compared to
the steady-flow scheme according to Eq. 4.

However, using the steady-loop-flow scheme comes with additional time costs.
While the steady-loop-flow scheme does not need any buffers on the loop, it
will require buffers before and after it passes through the rate-limiting step
(initialisation or measurement) to maintain the qubit time gap
$\tau_{\mathrm{max}}^{\mathrm{loop}}$. The amount of such buffering put on the
first qubit is denoted as $\Delta T_{\mathrm{gap}}^{\mathrm{loop}}$. As shown
in Appendix G, to process $k$ qubits using the steady-loop-flow scheme, the
amount of buffering we need to apply is:

$\displaystyle\Delta
T_{\mathrm{gap}}^{\mathrm{loop}}=(k-1)(\sum_{m}\max(\tau_{m}-\tau_{\mathrm{max}}^{\mathrm{loop}},0)).$
(41)

Here we sum over all steps in the pipeline and only the steps are slower than
the rate-limiting elements on the loop (i.e. measurements and/or
initialisations) are effectively included.

Hence, the effective time needed to process the first qubit is given by Eq. 37
with $\Delta T_{\mathrm{gap}}=\Delta T_{\mathrm{gap}}^{\mathrm{loop}}$:

$\displaystyle T_{\mathrm{eff}}^{\mathrm{loop}}=T_{\mathrm{circ}}+\Delta
T_{\mathrm{gap}}^{\mathrm{loop}}.$ (42)

The total time needed to process $k$ qubits using this pipeline is given by
Eq. 38 with $T_{\mathrm{eff}}=T_{\mathrm{eff}}^{\mathrm{loop}}$ and
$\tau_{\mathrm{gap}}=\tau_{\mathrm{max}}^{\mathrm{loop}}$:

$\displaystyle T_{\mathrm{pipe}}(k)$
$\displaystyle=T_{\mathrm{eff}}^{\mathrm{loop}}+(k-1)\tau_{\mathrm{max}}^{\mathrm{loop}}.$
(43)

###  G.3 Varying time gap

If we allow the time gap to vary, we will have a series of peaks (local
maxima) and troughs (local minima). Each transition from a peak to a trough
will be called a _fall_. Suppose there are $L$ falls throughout with the
$\ell^{\text{th}}$ fall being the transition from $t_{\mathrm{pk},\ell}$ to
$t_{\mathrm{tr},\ell}$, then the buffering we need to apply to the first qubit
to carry out these $L$ falls is simply

$\displaystyle\Delta
T_{\mathrm{gap}}=(k-1)(\sum_{\ell=1}^{L}\tau_{\mathrm{pk},\ell}-\tau_{\mathrm{tr},\ell}).$

Hence, using Eq. 38, the total time needed for pipelining all $k$ qubits (i.e.
the time taken for the last qubit to exit the pipeline) is:

$\displaystyle
T_{\mathrm{pipe}}(k)=T_{\mathrm{circ}}+(k-1)(\tau_{\mathrm{gap},M}+\sum_{\ell=1}^{L}\tau_{\mathrm{pk},\ell}-\tau_{\mathrm{tr},\ell}).$

Figure 26: A diagram showing the change of the time gap of the qubit stream as
it pass through different steps in the pipeline.

If $\tau_{\mathrm{gap},M}$ is a trough, then
$\tau_{\mathrm{tr},L}=\tau_{\mathrm{gap},M}$ and we have

$\displaystyle
T_{\mathrm{pipe}}(k)=T_{\mathrm{circ}}+(k-1)(\tau_{\mathrm{pk},L}+\sum_{\ell=1}^{L-1}\tau_{\mathrm{pk},\ell}-\tau_{\mathrm{tr},\ell}).$
(44)

Otherwise, if $\tau_{\mathrm{gap},M}$ is a peak, then
$\tau_{\mathrm{tr},L+1}=\tau_{\mathrm{gap},M}$ and we have:

$\displaystyle
T_{\mathrm{pipe}}(k)=T_{\mathrm{circ}}+(k-1)(\tau_{\mathrm{pk},L+1}+\sum_{\ell=1}^{L}\tau_{\mathrm{pk},\ell}-\tau_{\mathrm{tr},\ell}).$
(45)

In either case, if there are $Q$ peaks throughout the whole process with their
time gaps denoting as $\\{\tau_{\mathrm{pk},q}\ |\ 1\leq q\leq Q\\}$. In
between these $Q$ peaks, there are $Q-1$ troughs (i.e. excluding any troughs
at the beginning and end of the pipeline) with their corresponding time gaps
denoting as $\\{\tau_{\mathrm{tr},q}\ |\ 1\leq q\leq Q-1\\}$. A example is
shown in Fig. 26. Then by defining $\tau_{\mathrm{tr},0}=0$, the additional
time cost in Eqs. 44 and 45 can be combined into:

$\displaystyle
T_{\mathrm{pipe}}(k)=T_{\mathrm{circ}}+(k-1)(\sum_{q=1}^{Q}\tau_{\mathrm{pk},q}-\tau_{\mathrm{tr},q-1}).$
(46)

We will restrict our scheme such that at all peaks, we have
$\tau_{\mathrm{pk},q}=\tau_{\mathrm{gap},m}=\tau_{m}$ in which we have the
$q^{\text{th}}$ peak occurs at the $m^{\text{th}}$ step. Hence,
$\sum_{q=1}^{Q}\tau_{\mathrm{pk},q}$ is simply the sum of time required for
some subset of steps where the time peak happens, and thus it will be smaller
than the overall time require for the all steps
$\sum_{q=1}^{Q}\tau_{\mathrm{pk},q}<T_{\mathrm{circ}}$, which implies that

$\displaystyle T_{\mathrm{pipe}}(k)$
$\displaystyle=T_{\mathrm{circ}}+(k-1)(\sum_{q=1}^{Q}\tau_{\mathrm{pk},q}-\tau_{\mathrm{tr},q-1})$
$\displaystyle<kT_{\mathrm{circ}}.$

i.e. the pipelining scheme will always have time saving over the sequential
scheme.

For the sum $\sum_{q=1}^{Q}\tau_{\mathrm{pk},q}-\tau_{\mathrm{tr},q-1}$, we
know that one of the peak would be $\tau_{\mathrm{max}}$. Hence, we can
extract out $\tau_{\mathrm{max}}$ and pair up the peaks and troughs before the
$\tau_{\mathrm{max}}$ peak using “falls” and pair up the peaks and troughs
after the $\tau_{\mathrm{max}}$ peak using “rises”. In this way, we can show
that
$\sum_{q=1}^{Q}\tau_{\mathrm{pk},q}-\tau_{\mathrm{tr},q-1}\geq\tau_{\mathrm{max}}$.
i.e. any time gap variation will never outperform the scheme with a _constant_
time gap of $\tau_{\mathrm{max}}$.

When there is only $1$ peak, i.e. $Q=1$, which would occur at the rate-
limiting element with $\tau_{\mathrm{pk},1}=\tau_{\mathrm{max}}$, we would
have:

$\displaystyle
T_{\mathrm{pipe}}(k)=T_{\mathrm{circ}}+(k-1)\tau_{\mathrm{max}}$

which is simply Eqs. 3 and 2. i.e. there is no additional time cost if there
is just one peak in the time gap variation.

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# From spin liquid to magnetic ordering in the anisotropic kagome
Y-Kapellasite Y3Cu9(OH)19Cl8: a single crystal study

Dipranjan Chatterjee  Université Paris-Saclay, CNRS, Laboratoire de Physique
des Solides, 91405, Orsay, France  Pascal Puphal  Max-Planck-Institute for
Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany  Quentin
Barthélemy  Université Paris-Saclay, CNRS, Laboratoire de Physique des
Solides, 91405, Orsay, France  Jannis Willwater  Institut für Physik der
Kondensierten Materie, Technische Universität Braunschweig, Braunschweig,
Germany  Stefan Süllow  Institut für Physik der Kondensierten Materie,
Technische Universität Braunschweig, Braunschweig, Germany  Christopher Baines
Laboratory for Muon-Spin Spectroscopy, Paul Scherrer Institut, 5232 Villigen,
Switzerland  Sylvain Petit  LLB, CEA, CNRS, Université Paris-Saclay, CEA
Saclay, Gif-sur-Yvette, France  Eric Ressouche  Univ. Grenoble Alpes, CEA,
IRIG, MEM, MDN, 38000 Grenoble, France  Jacques Ollivier  Institut Laue-
Langevin, 38042 Grenoble, France  Katharina M. Zoch  Physikalisches Institut,
Goethe-Universität Frankfurt, Frankfurt am Main, Germany  Cornelius Krellner
Physikalisches Institut, Goethe-Universität Frankfurt, Frankfurt am Main,
Germany  Michael Parzer  Institute of Solid State Physics, TU Wien, 1040
Vienna, Austria  Alexander Riss  Institute of Solid State Physics, TU Wien,
1040 Vienna, Austria  Fabian Garmroudi  Institute of Solid State Physics, TU
Wien, 1040 Vienna, Austria  Andrej Pustogow  Institute of Solid State Physics,
TU Wien, 1040 Vienna, Austria  Philippe Mendels  Université Paris-Saclay,
CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France  Edwin
Kermarrec  Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides,
91405, Orsay, France  Fabrice Bert  Université Paris-Saclay, CNRS, Laboratoire
de Physique des Solides, 91405, Orsay, France

###### Abstract

Y3Cu9(OH)19Cl8 realizes an original anisotropic kagome model hosting a rich
magnetic phase diagram [M. Hering _et al_ , npj Computational Materials 8, 1
(2022)]. We present an improved synthesis of large phase-pure single crystals
via an external gradient method. These crystals were investigated in details
by susceptibility, specific heat, thermal expansion, neutron scattering and
local $\mu$SR and NMR techniques. At variance with polycristalline samples,
the study of single crystals gives evidence for subtle structural
instabilities at 33 K and 13 K which preserve the global symmetry of the
system and thus the magnetic model. At 2.1 K the compound shows a magnetic
transition to a coplanar (1/3,1/3) long range order as predicted
theoretically. However our analysis of the spin wave excitations yields
magnetic interactions which locate the compound closer to the phase boundary
to a classical _jammed_ spin liquid phase. Enhanced quantum fluctuations at
this boundary may be responsible for the strongly reduced ordered moment of
the Cu2+, estimated to be $\sim$ 0.075 $\mu_{B}$ from $\mu$SR.

##  I Introduction

Low-dimensional materials with strong magnetic frustration, such as compounds
with decoupled antiferromagnetic kagome layers, are prototypical systems to
search for an experimental realization of the quantum spin-liquid (QSL) state
[1, 2]. This long sought state features no static magnetic order, despite
sizeable magnetic interactions, but rather macroscopic entanglement and
fractional excitations. Experimental signatures of a QSL state in kagome
materials were found first in herbertsmithite, ZnCu3(OH)6Cl2 [3, 4] and in the
closely related Zn-doped barlowite [5, 6]. Thus Cu-based systems with a
perfect kagome layer have been strongly investigated.

Around the A2+Cu3(OH)6Cl2 family [7] a new similar series of compounds has
been found with A3+Cu3(OH)6Cl3 [7, 8, 9], which realizes perfect kagome layers
in a kapellasite-type structure. However, for Y3+ the compound is not stable
in water solutions, leading to a Cl-OH substitution YCu3(OH)6+xCl3-x with
$x=1/3$ (or Y3Cu9(OH)19Cl8) [10, 11]. This substitution induces a distortion
of the magnetic lattice as the Y atoms are moved out of the kagome plane [10].
The distorted kagome lattice of Cu2+ ions is scarcely studied with the
exception of the family around (Cs,Rb)2Cu3(Ti,Sn,Zr,Hf)F12 [12, 13, 14, 15].
However recent structural investigations of low temperature diffraction on
various kagome systems show that a structural instability induces distortions
in barlowite, claringbullite [16], volborthite [17] and vesigneite [18] making
it a typical structural motif at low temperatures.

In Y-kapellasite ($x=1/3$) that we investigate in this article, the distortion
yields two nonequivalent Cu sites and a unique magnetic model with three
different nearest neighbor interactions, while still retaining rotational
symmetry around the hexagons of the anisotropic kagome lattice (see inset Fig.
11 and Ref. [11]). The discovery of this compound has triggered a detailed
theoretical study [19] which disclosed the full classical phase diagram of the
model and its richness. Interestingly, besides two long range ordered phases
with propagation vectors $Q=(1/3,1/3)$ and $Q=(0,0)$, a large area in the
phase diagram, coined ”classical spin liquid” phase, which encompasses the
isotropic kagome model, opens up. It remains so far largely unexplored,
although it could realize an unprecedented ”jammed” spin liquid phase,
characterized by a discrete ground state degeneracy, in a disorder free model.
Besides, first principle calculations confirm the relevance of this
anisotropic nearest-neighbor model for Y-kapellasite and locates it in the
$(1/3,1/3)$ long range ordered phase [19]. Here, we report a comprehensive
study of the physical properties of large phase pure single crystals that can
be compared to the theoretical prediction and which show marked deviations
with respect to former studies of polycristalline samples [11] or other
synthesis route for crystals [20].

The paper is organized as follows. In sections II and III the synthesis of
phase-pure large single crystals is described and macroscopic measurements of
the susceptibility, heat capacity and thermal expansion are presented,
providing evidence for a series of structural and magnetic transitions below
33 K. In sections IV and V, the transitions are investigated in detail by
neutron diffraction and by Cl NMR, which shows in particular good agreement
with the theoretically predicted spin order in the ground state. In section
VI, muon spin spectroscopy is used to prove the bulk nature of the 2.1 K
magnetic transition. In section VII, the spin wave excitations are studied by
inelastic neutron scattering and give a novel insight into the determination
of the magnetic interactions beyond first principle calculations. Finally in
section VIII, we summarize and discuss our results in comparison with previous
ones on polycristalline samples and in light of theoretical studies.

##  II Synthesis and susceptibility

The crystal growth of Y-Kapellasite was originally reported in Ref. [10],
where 0.59 g Y2O3, 0.82 g CuO and 0.89 g CuCl${}_{2}\cdot$2(H2O) in 10 ml H2O
were heated up to the dissolution point of Y2O3, followed by a slow cooling to
crystallize. However, these crystals of an average size of 1x1x1 mm3 suffered
small CuO inclusions since the growth takes place on the surface of the
polycrystalline CuO starting material, as the dissolution point of Y2O3 and
the crystallization point of the compound lie above the maximum in solubility
of CuO. Reference [20] describes a synthesis of inclusion free crystals using
LiOH, Y(NO3)${}_{3}\cdot 6$H2O and CuCl${}_{2}\cdot 2$H2O. However the
reported crystals are somewhat disordered as a partial occupation of the Y
site is observed, similarly to what was found in the related compound
YCu3(OH)6Cl3 [8]. This likely hints at a phase mixture in both cases due to
contact with water. Indeed, a low percentage of occupation of Y for
YCu3(OH)6Cl3 [8] lies out of the kagome plane, which represents structural
parts of Y-Kapellasite Y3Cu9(OH)19Cl8. For the reported inclusion free
Y3Cu9(OH)19Cl8 [20], the partial occupation of Y can be explained similarly,
since the specific heat shows a mixture of the signal observed for $x=0$ and
$x=1/3$ as shown in Fig. 2.

We thus improved the synthesis for inclusion-free large bulk single crystals
suitable for neutron studies via a horizontal external gradient growth method
in a thick-walled quartz ampule as described in detail for herbertsmithite in
Ref. [21]. The growth is realized by slowly dissolving CuO in a YCl3-H2O (or
YCl3-D2O for deuterated crystals) solution and transporting it to the cold
end. The growth is executed in a three zone furnace with a gradient of
25$\degree$C and a temperature of 240$\degree$C at the hot end (note that the
elevated pressure at this temperature requires especially thick quartz
ampules). The gradient was optimized as too low temperatures yielded a mixture
of Y-Kapellasite and Clinoatacamite. Afterwards, the inclusion-free hexagonal
single crystals have an average size of 3x3x1 mm3 up to 3x3x3 mm3, if grown
over several weeks. The pictures of single crystalline Y-kapellasite are
displayed in Fig. 1 a) and b), which show a transparent specimen without any
visible impurity inclusions.

The dc-susceptibility of a 15.7 mg single crystal of Y-Kapellasite measured in
a Quantum Design MPMS XL7 squid system with a 1 T field applied along the $c$
axis is shown in Fig. 1 c). It reproduces published results [10, 22], with an
antiferromagnetic Curie-Weiss constant $\theta_{\rm{cw}}=102(1)$ K and
$g_{\rm{c}}=2.40(1)$. Measurements in a lower 0.01 T magnetic field, shown in
the inset, reveal a bifurcation of the field cooled and zero field cooled
curves below $\sim 5$ K, and a maximum around 2.5 K. These low field
measurements point at a magnetic transition which is investigated in details
with neutron scattering and resonance techniques in the following sections. At
variance with reference [20], we observe no magnetic transition at 11 K. Such
a transition, reminiscent of the one observed at 12 K in the $x=0$
counterpart, supports a phase mixture scenario for the crystal of reference
[20].

Figure 1: a) Y3Cu9(OH)19Cl8 single crystal along the $c$ axis. b) Image in
transmission light of an as grown single crystal. c) Susceptibility (M/H)
versus temperature with a 1 T field applied along $c$. Bottom left inset:
field cooled (FC) and zero field cooled (ZFC) measurements at 0.01 T. Upper
right inset: anisotropic kagome lattice with 3 different nearest neighbor Cu-
Cu bonds and the related 3 different magnetic interactions.

##  III Specific heat and thermal expansion

Figure 2: a) Specific heat divided by temperature in the temperature range
from 1.8 to 40 K for the $x=1/3$ optimized large crystals, compared to
literature results for $x=1/3$ and $x=0$ powder samples [11] and the reported
inclusion free $x=1/3$ crystals from Ref. [20]. b) Thermal expansion along the
$s$ direction of an optimized large Y-Kapellasite single crystal. c) and d)
Insets: enlarged views of the peaks detected in main panel b). Note the
different vertical scales in the two insets.

Specific heat was measured with the standard option of a Quantum Design
Physical Properties Measurement System (PPMS), using a 5.935 mg inclusion-free
single crystal formerly described. Its temperature evolution is shown in Fig.
2 a) and compared to the published results for $x=0$ and $x=1/3$
polycrystalline samples [11] and the $x=1/3$ single crystal reported in Ref.
[20]. Unexpectedly, our $x=1/3$ single crystal presents a clear double peak at
33 K, which is absent in powder samples of the same compound. The underlying
entropy amounts to 0.1537 J/mol K, below 3% of $R$ln2, calculated by using the
powder sample as a baseline. There is no associated sharp feature in the
susceptibility in this temperature range so that the double peak is likely
signaling a structural transition. Note that this 33 K anomaly was not
reported in the single crystals of Ref. [20] and that, on the contrary, the
large hump around 15 K seen in the latter crystals and in the $x=0$ powder
sample, associated to magnetic transitions, is absent from our phase-pure
$x=1/3$ single crystal. Only at 2.1 K do we find a sharp peak in specific
heat, which was already reported for this compound both in powder and crystals
[10, 11], and stems from a magnetic long range ordering transition as shown
later in this article. For the disordered powder sample $x=1/3$, the
underlying entropy of the transition is reduced and previous muon spin
relaxation studies had shown the absence of long range order [11], possibly
arising from the slightly altered structure due to the suppression of the 33 K
transition.

To inspect the 33 K anomaly in more detail, we performed thermal expansion
experiments along the crystallographic $c$-axis, i.e. perpendicular to the
kagome layers, using a capacitive dilatometer. Figure 2 b) displays the
thermal expansion coefficient $\alpha$ on the same temperature scale as the
specific heat data in panel a). We find a similar anomaly below 33 K as in
$C/T$, consisting of two slightly broadened, and thus overlapping peaks with a
ratio between the sizes of the higher-temperature and lower-temperature peaks
comparable to a). The relative length change incorporated in these two sharp
features amounts to $\Delta L/L=10^{-4}$. In addition, we discover another,
significantly smaller anomaly around 13 K, which can be seen in more details
in the inset c). Below 5 K an anomaly in $\alpha$ forms as approaching the
transition at 2.1 K, signalling the importance of magneto-elastic coupling as
antiferromagnetism sets in. Overall, the anomalies at 13 K and 33 K reveal
clear structural effects in this compound with a distorted kagome lattice,
without noticeable changes upon applying a magnetic field of 9 T along the
$c$-axis.

##  IV Diffraction

T (K) | 0.065 | 8 | 20 | 40
---|---|---|---|---
| x/a | y/b | z/c | x/a | y/b | z/c | x/a | y/b | z/c | x/a | y/b | z/c
Y1 | 0 | 0 | 0.1318(7) | 0 | 0 | 0.1314(6) | 0 | 0 | 0.1324(7) | 0 | 0 | 0.1302(5)
Y2 | 0 | 0 | 0.5 | 0 | 0 | 0.5 | 0 | 0 | 0.5 | 0 | 0 | 0.5
Cu1 | 0.6661(10) | 0.8325(11) | 0.5025(4) | 0.6658(10) | 0.8340(11) | 0.5021(4) | 0.6675(11) | 0.8328(12) | 0.5026(4) | 0.6645(7) | 0.8317(7) | 0.5021(3)
Cu2 | 0.5 | 0 | 0.5 | 0.5 | 0 | 0.5 | 0.5 | 0 | 0.5 | 0.5 | 0 | 0.5
Cl1 | 0.6745(9) | 0.0033(10) | 0.2845(2) | 0.6743(8) | 0.0037(10) | 0.28435(19) | 0.6748(9) | 0.0031(11) | 0.2845(2) | 0.6732(6) | 0.0027(6) | 0.28395(15)
Cl2 | 0 | 0 | 0.3368(5) | 0 | 0 | 0.3368(5) | 0 | 0 | 0.3365(6) | 0 | 0 | 0.3373(4)
O1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
D1 | 0.456(9) | 0.776(10) | 0.660(4) | 0.455(9) | 0.783(10) | 0.664(4) | 0.455(9) | 0.776(10) | 0.660(4) | 0.462(7) | 0.777(7) | 0.661(3)
O2 | 0.8119(14) | 0.8060(12) | 0.5435(7) | 0.8122(13) | 0.8046(12) | 0.5438(7) | 0.8129(14) | 0.8060(13) | 0.5438(8) | 0.8121(9) | 0.8035(9) | 0.5435(4)
D2 | 0.7769(19) | 0.7778(17) | 0.6011(9) | 0.7874(18) | 0.7858(17) | 0.5991(9) | 0.7723(17) | 0.7722(16) | 0.6030(8) | 0.7866(13) | 0.7827(13) | 0.6005(6)
O3 | 0.5293(12) | 0.6635(11) | 0.5540(6) | 0.5285(11) | 0.6638(10) | 0.5537(6) | 0.5283(12) | 0.6622(11) | 0.5542(6) | 0.5290(8) | 0.6634(8) | 0.5543(4)
D3 | 0.5494(17) | 0.6731(18) | 0.6065(8) | 0.5575(16) | 0.6709(17) | 0.6082(8) | 0.5452(15) | 0.6740(18) | 0.6051(8) | 0.5548(12) | 0.6692(13) | 0.6069(5)
O4 | 0.5142(12) | 0.8433(11) | 0.4631(5) | 0.5167(11) | 0.8430(10) | 0.4632(5) | 0.5154(12) | 0.8450(11) | 0.4634(6) | 0.5162(8) | 0.8441(8) | 0.4638(4)
D4 | 0.5008(16) | 0.8310(17) | 0.4096(8) | 0.5013(16) | 0.8328(16) | 0.4097(8) | 0.5018(17) | 0.8320(17) | 0.4099(8) | 0.5029(12) | 0.8337(12) | 0.4099(6)
Table 1: Rietveld refinement results of neutron diffraction data obtained at
65 mK, 8 K, 20 K and 40 K with R-3 (#148) $a=b=11.539$ Å, and $c=17.1355$ Å.
Figure 3: Temperature dependence of the neutron diffraction intensity around
the nuclear Bragg peaks a) (6 6 0) and b) (0 0 18). Structure viewed along the
$c$-axis obtained from neutron single crystal diffraction measurements at c)
40 K and d) 65 mK.

The structure of Y-Kapellasite Y3Cu9(OH)19Cl8 with R-3c was first reported in
Ref. [10], where hydrogen was placed by symmetry arguments. Later, from
neutron diffraction on deuterated powder samples, we found that the deuterium
atom next to the O1 refines to a distance above 1 Å, which is so far
unaccounted for an O-H bound. Thus in Ref. [11], we dismissed the D1, leading
to the revised chemical formula of Y3Cu9(OD)18OCl8. To tackle further this
issue we then measured the stoichiometry including the hydrogen content with a
combination of inductively coupled plasma mass spectroscopy (ICP-OES) and gas
extraction, the former with a SPECTRO CIROS CCD and the latter with an Eltra
ONH-2000 analyzer. For the determination of oxygen and hydrogen content, the
powder samples were placed in a Ni crucible and clipped and heated, where the
carrier gas takes the oxygen and hydrogen out of the sample. The oxygen reacts
with carbon, and CO2 is detected in an infrared cell, while hydrogen is
detected by a thermal conductivity cell. Each measurement is repeated three
times and compared to a standard. The quoted error bars provide the
statistical error. We thus measured three pieces of a hydrated single crystal
and a part of a powder sample of Ref. [11]. The results of:
Y3.00(3)Cu9.06(9)O19.0(2)H18.7(3)Cl8 for the crystals and
Y3.00(3)Cu9.07(9)O19.3(2)H18.8(3)Cl8 for the powder sample show a high enough
hydrogen content to assume the ideal stoichiometry of Y3Cu9(OH)19Cl8. Notably
we have also seen vibrational O-H modes that exist only for electrical field
parallel to the kagome plane solely matching to this hydrogen/deuterium
position further confirming the presence of the D1 atom. Therefore, from now
on, we use the ideal stoichiometry of Y3Cu9(OH)19Cl8.

We performed neutron diffraction on the 2 axis thermal neutron diffractometer
D23 instrument at the Institut Laue-Langevin (ILL) using an inclusion-free,
deuterated single crystal weighing around 30 mg. Measurements have been
performed in the temperature range of 40 K to 0.065 K using constant incident
neutron wavelength $\lambda$=1.27 Å. We find the R-3c structure that was
reported in Ref. [10] to be stable over the whole temperature range. The
previously mentioned D1 position can be reasonably refined in all cases to a
similar position and even a release of the occupation is possible. We find in
all cases an unusually large distance of d${}_{O-D}=1.3$ Å, which was
nevertheless considered as mentioned above. The transition around 33 K
observed in specific heat and thermal expansion is reflected here in an
increase of the Bragg peak intensity by approximately 30% as shown in Fig. 3
a) and b). Yet, it produces no change in the crystallographic cell, as the
crystallographic axes $a,b$ and $c$ remain unaltered in the investigated
temperature range. We find in Rietveld refinements a sudden jump around 33 K
for certain atomic positions, which in absolute numbers remain however
marginal. The Rietveld refinement results are summarized in table
LABEL:Rietveld and selected atomic positions are plotted normalized to the 40
K values versus temperature in Fig. 4 to visualize the sudden changes around
30 K (see also 111See Supplemental Material at [URL] for the Crystallographic
Information Files corresponding to the refined Y-Kapellasite Y3Cu9(OH)19Cl8
structure at 40 K, 20 K, 8 K and 0.065 K.). These changes lie below 1% of the
nominal value, except for deuterium and thus are hardly visible in Fig. 3 c)
and d) depicting the crystal structure along the $c$ axis at 40 K and 65 mK,
but will be relevant for the discussion of our NMR results.

Figure 4: Relative atomic position changes versus temperature of a) Chlorine
site 1 and 2, b) Deuterium site 1, 2, 3, 4 and c) Cu site 1.

At low temperature $T=65$ mK we have searched for magnetic ordering at
different (H,K,L) planes but we found no magnetic Bragg peaks probably due to
low magnetic moment of Cu2+ ions in the ground state. Hence extensive studies
of the low temperature regime has been done to investigate the magnetic
structure using NMR, $\mu$SR and inelastic neutron scattering, as presented in
the following sections.

##  V Nuclear Magnetic Resonance

We further investigated the structural distortions and magnetic transition
using 35Cl NMR on a $\sim 30$ mg single crystal of Y-Kapellasite
Y3Cu9(OH)19Cl8 with an external magnetic field Bext applied parallel to the
$c$-axis, i.e. perpendicular to the large facet of the sample (see Fig. 1).
The spectra are measured from room temperature down to 5 K by sweeping the
frequency in a fixed field B${}_{ext}=7.553$ T which corresponds to the
reference frequency $\nu_{0}=31.510$ MHz. At lower temperatures, the broader
spectra were recorded by sweeping the field with a fixed irradiation frequency
$\nu_{0}$.

The spectrum measured at 100 K, far above the magnetic transition and the
structural distortions, is shown in Fig. 5. We observe two distinct central
lines (see inset of Fig. 5) corresponding to the two crystallographic Cl sites
with intensity ratio close to the expected 3:1. There are indeed 18 Cl1 (resp.
6 Cl2) sites per unit cell located in between the kagome layers, close to the
center of each triangle (resp. hexagon) of the kagome structure. For clarity
in this section we denote the Cl1 and Cl2 sites as the triangular ($tri$) and
hexagonal ($hex$) sites respectively.

Figure 5: Full NMR spectrum of 35Cl at 100 K with the configuration
$B_{ext}\parallel c$ which shows 2 central lines (C) of Cl1 (triangular site)
and Cl2 (hexagonal site) and the 2 corresponding pairs of satellites (Stri and
SHex). Inset: enhanced view of the central lines (C). The dashed vertical line
indicates the reference frequency $\nu_{0}=31.510$ MHz

Since 35Cl possesses a nuclear spin $I=3/2$ with a finite quadrupolar moment,
the three allowed transitions between the Zeeman-split adjacent nuclear levels
are modified by the quadrupolar interaction with the surrounding electric
charges. We therefore observe in the full spectrum three NMR lines per Cl
site, one central line ($-1/2\leftrightarrow 1/2$ transition) and two
satellite ones ($-3/2\leftrightarrow-1/2$ and $1/2\leftrightarrow 3/2$
transitions).

At first order in perturbation, the frequency difference between two satellite
lines of a given $\alpha=tri$ or $\alpha=hex$ Cl site arising from the
quadrupolar interaction is given by [24]

$\Delta\nu_{\alpha}^{(1)}=\nu^{\alpha}_{Q}(3\cos^{2}{\theta_{\alpha}}-1-\eta_{\alpha}\sin^{2}{\theta_{\alpha}}\cos
2\phi_{\alpha})$ (1)

where $\theta_{\alpha}$ and $\phi_{\alpha}$ are the polar and azimuthal angles
defining Bext in the local frame of the principal axes of the electric field
gradient tensor (EFG), $\nu_{Q}^{\alpha}$ is proportional to the quadrupolar
moment and the largest eigenvalue of the EFG, therefore reflecting the
strength of the quadrupolar interaction, and the asymmetry parameter
$\eta_{\alpha}$ reflects the departure of the EFG from cylindrical symmetry.

Because of the high symmetry of the hexagonal Cl site, the EFG at this site is
axially symmetric ($\eta_{hex}=0$) along the $c$ axis and, under the condition
of the experiment $B_{ext}\parallel c$, $\theta_{hex}=0$. For this site, the
two satellites are maximally separated by $2\nu_{Q}^{hex}$ and we measure
directly from Fig. 5 $\nu_{Q}^{hex}=3640(5)$ kHz. The triangular Cl site is
less symmetric. We used the structure determined at 40 K in section IV by
neutron diffraction to compute the EFG at this site with a point charge
approach. From the calculated parameters $\theta_{tri}=8.3^{\circ}$,
$\phi_{tri}=123.0^{\circ}$, $\eta_{tri}=0.33$ and the measured distance
between satellites, we obtain with Eq. 1 the value $\nu_{Q}^{tri}=2180(20)$
kHz. Note that if Bext is not strictly applied along the symmetry axis $c$,
the triangular sites become non-equivalent because of a different orientation
of their EFG with respect to Bext. The minute splitting of the triangular site
satellite lines in Fig. 5 reflects such a small misalignment $\pm 1.2^{\circ}$
of our crystal in the experiment.

The quadrupolar interaction shifts the position of the central line only at
second order in perturbation [25]:

$\nu^{(2)}_{\alpha}=-\frac{(\nu^{\alpha}_{Q})^{2}}{2\nu_{0}}f(\theta_{\alpha},\phi_{\alpha},\eta_{\alpha})$
(2)

where, for the simplest case of an axially symmetric EFG, $f$ reads

$f(\theta,\phi,0)=\frac{3}{8}(1-\cos^{2}{\theta})(9\cos^{2}{\theta}-1).$ (3)

In the present case at 100 K, the second order quadrupolar shift is either
$\nu^{(2)}_{hex}=0$ since $\theta_{hex}=0$ or negligible
$\nu^{(2)}_{tri}=-4(1)$ kHz. The positions of the two central lines

$\nu^{C}_{\alpha}=\nu_{0}(1+K_{\alpha}^{0}+K^{s}_{\alpha})+\nu^{(2)}_{\alpha}$
(4)

are then rather set by the first term of magnetic origin, where
$K_{\alpha}^{0}$ is the temperature-independent orbital shift and
$K^{s}_{\alpha}$ the spin shift arising from the polarization of the unpaired
electron of the neighbouring Cu2+ ions and therefore proportional to the spin
susceptibility. From the plots of $\nu^{C}_{\alpha}$ versus the susceptibility
$\chi_{c}$ measured along the $c$ direction in the range 100-300 K (not
shown), we extract for the configuration B${}_{ext}\parallel c$ the hyperfine
constants $A_{hex}=0.245(3)$ T/$\mu_{B}$ and $A_{tri}=-0.426(16)$ T/$\mu_{B}$
and the orbital terms $K_{hex}^{0}=120(10)$ ppm and $K_{tri}^{0}=52(10)$ ppm.

Figure 6: Evolution of the central lines of the 35Cl NMR spectra with
temperature. The spectra are shifted vertically for clarity. Note the change
of the width and the splitting of the line below 40 K which become prominent
below 20 K. The dashed vertical line indicates the reference frequency.

Figure 6 shows the evolution of the central lines upon cooling down to 5 K.
Marked changes of the spectral shape are observed below 40 K where the
structural distortions take place: the triangular site line broadens
significantly and the hexagonal one splits. Interestingly, although the
changes appear in the range 30-40 K, they really get prominent only below 20 K
where a second peak is observed in thermal expansion.

From the neutron diffraction study, the structural distortion at 33 K does not
affect the crystal symmetry and maintains only two distinct Cl sites. It
mainly involves small displacements of the protons, which naturally modify the
EFG and therefore the positions of the lines but cannot account for line
broadening or line splitting. Therefore, 35Cl NMR reveals more complex
distortions with local symmetry breaking but no spatial correlation so that
the average symmetry as detected by neutron diffraction is preserved. In
particular, the splitting of the hexagonal site line implies the occurrence of
at least two locally non-equivalent crystallographic sites and at least one
with a finite tilt $\theta_{hex}$ of its EFG so that $\nu^{(2)}_{hex}$ does
not vanish in equation 4. From the measured splitting and using Eq. 4, we
estimate $\theta_{hex}\sim 5^{\circ}$ at 20 K and $\theta_{hex}\sim
10^{\circ}$ at 10 K.

These changes of the local electrostatic environment of Cl affect even more
clearly, at first order, the satellite lines. As shown in the inset a) of Fig.
7, the hexagonal satellite acquires a complex structure below 33 K pointing at
even more than two nonequivalent sites. In order to determine the onset of the
structural transition, we have measured the width of this satellite versus
temperature as plotted in Fig. 7. Beside the abrupt change at the $33$ K
structural transition evidenced in heat capacity and neutron diffraction
experiments, a marked change of the slope is detected at 13(1) K suggesting a
secondary transition as detected in thermal expansion and also clearly in NMR
relaxation measurements as discussed further in this section.

Figure 7: Hexagonal satellite linewidth evolution with temperature which
indicates 2 structural changes (black arrows). Inset (a) depicts the line
shape changes occurring below 33 K for the hexagonal satellite. Inset (b)
shows the hexagonal NMR central lines for 37Cl and 35Cl at 10 K.

To confirm the structural nature of the 13 K transition we have compared the
35Cl and 37Cl NMR spectra at 10 K at the same reference frequency of 24.000
MHz. We have fitted the split hexagonal site central lines with gaussian
functions and compared the frequency differences $\Delta\nu^{i}$ between the
two peak maxima for the two isotopes $i$ (inset (b) Fig. 7). The ratio
$\Delta\nu^{35}/\Delta\nu^{37}=1.5(2)$ compares well to the square of the
ratio of the quadrupolar moments of the two isotopes 1.61 as expected from Eq.
4. At variance, in case of a magnetic origin with static internal fields, we
would have observed a scaling with the gyromagnetic ratios of the two
isotopes, i.e. a ratio $\sim 1.20$.

At lower temperatures ($T<4.2$ K), the NMR lines broaden significantly, likely
as a result of the building up of magnetic correlations, in line with the
FC/ZFC divergence observed in this $T$ range in the susceptibility (see Fig.
1). This magnetic broadening blurs the quadrupolar splitting and down to 2.1
K, we observe only two broad central lines, overlapping partially, for the two
different, triangular and hexagonal, chlorine sites (see Fig. 8). At lower
temperatures an abrupt broadening for the triangular site and more moderately
for the hexagonal one occurs, which represents the spectral NMR signature of
the magnetic transition detected in the bulk, $T_{1}$ or $\mu$SR measurements.
Actually, two triangular site peaks develop on both sides of the hexagonal
central line and move away from each other upon cooling down. To quantify
these evolutions, we compare in Fig. 9 the change of the linewidth $W(T)$ with
temperature for the two sites.

Figure 8: Central line spectra at 1.3 K, 2 K, 3.6 K and partial view of the
ground state spin texture proposed in Ref. [19] in the top right corner. The
spectra are shifted vertically for clarity. The blue line is a gaussian fit of
the hexagonal central line. This contribution is subtracted from the
experimental spectrum to expose the triangular site one, shown as a red line.
The triangle configurations a, b and c yield three different dipolar fields at
the triangular Cl site. Configurations a and c yield almost equal but opposite
dipolar fields whereas configuration b has almost zero dipolar field along the
$z$ axis, i.e. the applied field direction.

In the recent theoretical study Ref. [19], a $(1/3,1/3)$ long range magnetic
order has been proposed for this system with a coplanar spin configuration
reproduced partially in Fig. 8. Remarkably, for this predicted spin
configuration, the hyperfine field at the hexagonal sites is zero (hexagons
with different colors spins) or nearly zero (hexagons with same color spins).
At variance, for each triangle, two spins are nearly opposite leaving the
third one uncompensated and thus able to yield a finite field at the
triangular sites. There are actually three types of triangle configurations
labelled a, b and c in Fig. 8 corresponding to three different orientations of
this uncompensated spin. For an anisotropic hyperfine tensor, we thus expect
three different lines for the triangular Cl in the magnetically ordered phase.
Although we could not determine experimentally all the components of the
hyperfine tensor, we note that assuming a purely dipolar coupling, the shown
spin configuration yields one very small and two almost equal but opposite
$z$-components of the transferred fields at the triangular Cl site. Thus, the
low temperature spectrum supports at least qualitatively the theoretically
proposed order, both for the triangular and hexagonal Cl sites.

Figure 9: Evolution versus temperature of the linewidth $W(T)$ for the
triangular (red dots) and hexagonal (blue dots) sites.

Figure 10 shows the evolution with temperature of the spin lattice relaxation
rate ($T_{1}^{-1}$) measured using the saturation recovery sequence at both Cl
sites. Being quadrupolar active, Cl nuclei probe both the magnetic and
structural fluctuations. In the temperature evolution of $T_{1}^{-1}$ the
structural distortions at $\sim$33 K and $\sim$13 K discussed earlier are
clearly observed as a sharp and broad peak respectively, together with the
magnetic transition around $\sim$ 2.1 K.

To extract the $T_{1}$ values, the recovery curves of the nuclear
magnetization $M(t)$ at each temperature were fitted to [26]

$M(t)=M_{sat}\left(1-ae^{-\left(\frac{t}{T_{1}}\right)^{\beta}}-be^{-\left(\frac{6t}{T_{1}}\right)^{\beta}}\right)$
(5)

where $a+b=1$, $a=0.1$ and $0.23$ respectively for the hexagonal and
triangular sites, and $\beta\leq 1$ is a phenomenological stretching parameter
used to account for a possible distribution of $T_{1}$ [27]. The evolution of
$\beta(T)$ for both sites is shown as insets in Fig. 10. While a single value
of $T_{1}$ is measured at high temperature ($\beta=1$), an increasingly
broader distribution ($\beta<1$) is needed upon cooling down towards the
structural and magnetic transitions, likely pointing at the building up of
spatial correlations. During the transition regimes, we could fix the value of
$\beta$ with no noticeable reduction of the fit quality, to avoid artefacts in
the determination of the peaks of the correlated $T_{1}^{-1}$ quantity.

Figure 10: Plots of 35Cl NMR relaxation rate $T_{1}^{-1}$ versus temperature
measured at the hexagonal (top) and triangular (bottom) central line
positions. The two structural transition regimes (2) and (3) and the magnetic
one (1) are highlighted. The $T_{1}$ distribution parameter $\beta$ is plotted
in the inset. Below 5 K, when the NMR lines of the two sites overlap, $T_{1}$
was also measured at the isolated hexagonal satellite position (dark cyan
dots) and shows the same behaviour as the central line.

##  VI Muon spin relaxation

To elucidate the nature of the ground state of Y- Kapellasite, we performed a
$\mu$SR experiment on the GPS and DOLLY instruments at the PSI muon facility
from the paramagnetic regime ($>$20 K) down to 0.28 K. The 100% spin polarized
muons are implanted at time zero in the sample, where they stop at the most
electronegative positions. The muon spins evolve in the local fields until the
muons decay into positrons, emitted preferentially in the last direction of
the muon spins after a mean life time of 2.2 $\mu$s. By collecting the
positrons emitted forward and backward with respect to the muon beam
direction, the muon decay asymmetry, proportional to muon spin polarization
can be recorded versus time and allows to characterize the internal fields and
their fluctuations [28, 29, 30].

For this $\mu$SR experiment, we used two coaligned phase pure single crystals
of sizes $\sim 3\times 3\times 1$ mm3 with their $c$ axis parallel to the
initial muon spin direction. The muon decay asymmetry in zero external field
(ZF) is depicted at some selected temperatures in Fig. 11. At low temperature,
the asymmetry evidently differs from previous measurements [11] on a powder
sample shown in the inset, with a much faster relaxation and a characteristic
damped oscillation signaling static internal fields.

Figure 11: Time evolution of the zero-field (ZF) muon decay asymmetry at some
selected temperatures in single crystal samples. Lines are fit to Eq. 9.
Inset: powder sample data from Ref. [11].

Above 2.5 K, the relaxation hardly depends on temperature nor on the nature of
the sample, powder or single crystal. The electronic spins are fast
fluctuating in their paramagnetic regime, and the relaxation is dominated by
the quasi static, weak and random, nuclear fields. Following the model used
for the powder sample [11], we fitted the muon decay asymmetry at high
temperature to

$a_{0}P_{para}(t)=a_{0}\left[fP_{OH}(t)+(1-f)KT_{\Delta_{Cl}}(t)\right]\\\ $
(6)

where

$P_{OH}(t)=e^{-\frac{\Delta^{2}_{OH}t^{2}}{2}}{\bigg{[}}\frac{1}{6}+\frac{1}{3}\cos{{\bigg{(}}\frac{\omega_{OH}t}{2}{\bigg{)}}}+\\\
\frac{1}{3}\cos{{\bigg{(}}\frac{3\hskip
2.84526pt\omega_{OH}t}{2}{\bigg{)}}}+\frac{1}{6}\cos{{\bigg{(}}{\omega_{OH}t}{\bigg{)}}}{\bigg{]}}.\
$ (7)

The latter $P_{OH}(t)$ stands for the formation of $\mu$OH complexes with a
pulsation $\omega_{OH}=\frac{\hbar\mu_{0}\gamma_{\mu}\gamma_{H}}{4\pi d^{3}}$
which depends on the $\mu$-H distance $d$, the gyromagnetic ratios
$\gamma_{H}=267.513$ Mrad/s/T and $\gamma_{\mu}=851.616$ Mrad/s/T respectively
for protons and muons, and a gaussian broadening $\Delta_{OH}$ due to the
other surrounding nuclear spins. The second term in Eq. 6 is a static Kubo-
Toyabe relaxation standing for the minority fraction $1-f$ of muon stopping
sites, likely close to the Cl${}^{\textbf{-}}$ ions, where they experience a
gaussian distribution of static nuclear fields with a width $\Delta_{Cl}$. The
parameter $a_{0}=0.255$ is the initial muon decay asymmetry in our
experimental conditions.

The nuclear relaxation parameters in Eq. 6 and Eq. 7 evaluated by fitting the
ZF muon polarisation at 20 K are presented in table 2.

Static parameters | Y3Cu9(OH)19Cl8
---|---
$f\%$ | 75.00$\pm$1.00
$\omega_{OH}$ (Mrad.s-1) | 0.55$\pm$0.02
d (Å) | 1.63$\pm$0.02
$\Delta_{Cl}~{}(\mu s^{-1})$ | 0.09$\pm$0.02
$\Delta_{OH}$ ($\mu$s-1) | 0.208$\pm$0.02
Table 2: Static nuclear parameters derived from high temperature fit of the ZF
asymmetry with Eq. 6.

Upon cooling, the relaxation from the electronic spins increases progressively
and below 2.1 K, at variance with the powder sample [11], a strongly damped
oscillation develops, conclusively indicating a magnetic transition in the
single crystal samples (Fig 11). Below 1.5 K, the asymmetry could be fitted
accordingly to

$a_{0}P_{f}(t)=a_{0}\left[\frac{2}{3}\cos(\omega_{f}t+\phi)e^{-\frac{\sigma^{2}t^{2}}{2}}+\frac{e^{-\lambda_{f}t}}{3}\right]$
(8)

which accounts for a magnetically frozen single phase with an average internal
field at the muon sites $B_{int}=\omega_{f}/\gamma_{\mu}$. The damping
parameter $\sigma$ encodes the width of the distribution of these internal
fields. The last exponential term accounts for residual fluctuations in the
frozen phase. At base temperature $T=0.28$ K, we obtain $B_{int}=8.6$ mT which
is almost twice lower than the lowest internal fields previously reported in
the closely related $x=0$ variant of YCu3(OH)6OxCl3-x [11, 31]. Although there
is an uncertainty about the exact position of the muons, from this dipolar
field value, we calculate a strongly reduced ordered magnetic moment for the
Cu2+ $\sim 0.075$ $\mu_{B}$ using the majority $\mu$OH complex forming sites.
Besides, the use in Eq. 8 of the coefficient $2/3$ and $1/3$ for the
oscillatory and non-oscillatory components in our single crystal, expected for
a spin-glass like state, suggests a rather disordered ground state. The strong
damping $\sigma/\gamma_{\mu}=3.2$ mT compared to $B_{int}$ also points to a
rather short range order and may simply reflect the absence of magnetic
correlations in the $c$ direction, as shown by inelastic neutron scattering
(see section VII).

In order to fit the asymmetry on the whole temperature range, we combined both
equations 6 and 8 in

$a_{0}P(t)=a_{0}\left[f_{f}P_{f}(t)+(1-f_{f})P_{para}(t)e^{-\lambda_{p}t}\right]$
(9)

with a switching parameter $f_{f}$ that tracks the frozen volume fraction from
fully ordered for $f_{f}=1$ to fully paramagnetic for $f_{f}=0$. All the
nuclear parameters are kept constant, so that the only varying parameters
shown in Fig. 12 are the frozen fraction, the oscillation frequency and its
damping rate, and the relaxation rate $\lambda_{p}$ or $\lambda_{f}$ for the
paramagnetic or the frozen phase. The latter show a clear peak at 2.1 K as
expected for the critical slowing down of the fluctuations at the magnetic
transition. Surprisingly, there is a singular point at 1.56K in the relaxation
rate plot but no significant changes in other variable parameters at the same
temperature. Although this feature demands further confirmation, it is
reminiscent of the spin dynamics re-entrance reported in clinoatacamite [32]
and echoes an anomaly reported in the low temperature magnetization
measurements [33].

Figure 12: Temperature evolution of the parameters used in Eq. 9 to fit the ZF
$\mu$SR data: a) frequency $\omega_{f}$ reflecting the internal field
magnitude, b) damping rate $\sigma$, c) fraction $f_{f}$ of the frozen phase
and d) relaxation rates in the paramagnetic ($\lambda_{p}$) and frozen
($\lambda_{f}$) phases.

To conclude, from our $\mu$SR measurements, Y-Kapellasite shows a completely
frozen ground state in the case of single crystals in contrast to the
fluctuating ground state observed in powder sample. The relaxation rate
variation with temperature indicates a phase transition at $\sim$2.1K where
the internal static fields set in. At 0.28 K, deep in the magnetic phase
state, we estimate a strongly reduced frozen moment for the Cu2+ of about
0.075 $\mu_{B}$.

##  VII Time-of-flight neutron scattering

Figure 13: (a-c) Evolution in temperature of the dynamical structure factor
$S(q,E)$ along the $[H,H,0]$ direction (integrated over $L=[-6,6]$) at
$T=1.55$ K, $T=5$ K and $T=60$ K . (d) Brillouin zones boundaries (gray lines)
with $K$ and $\Gamma$ points and extended Brillouin zone (black dashed line)
in the reciprocal space. The red dotted line represents the $[H,H,0]$
direction. (e-g) Evolution in energy of $S(q,E)$ in the $(H,H,L)$ plane at
$T=1.55$ K. (h) INS measurement integrated in energy over the elastic
resolution showing a weak increase below 5 K at the $K^{\prime}_{1}$ and
$K^{\prime\prime}$ positions, characteristic of the $(1/3,1/3)$ magnetic
order.

With large deuterated single crystals of Y3Cu9(OD)19Cl8 at hand, we also
performed inelastic neutron scattering on this anisotropic kagome
antiferromagnet in order to probe the magnetic excitations in reciprocal space
and to identify the magnetic order [34]. Inelastic neutron scattering
measurements were performed on the IN5 disk chopper time-of-flight
spectrometer [35] at the Institut Laue-Langevin on a collection of single
crystals with a total mass of 0.3g. These crystals were co-aligned in the
$(H,H,L)$ horizontal plane and glued on a thin aluminium plate with the
hydrogen-free CYTOP solution (CTL-809M). Measurements runs were performed at a
constant temperature with an incident neutron wavelength $\lambda=4.8$ Å
(incident energy E${}_{i}=3.55$ meV, giving a FWHM energy resolution of 0.08
meV) or $\lambda=2.6$ Å (E${}_{i}=12.1$ meV), while the sample stage was
rotated around the direction perpendicular to the beam, to map out the
reciprocal space. The detectors efficiency was corrected by a preliminary
measurement on a vanadium can.

Magnetic scattering could be identified only below 3 meV, an unexpected
feature given the large antiferromagnetic Curie-Weiss temperature of
$\theta_{\rm cw}=-100$ K [10]. We thus focused on measurements using our low
incident energy of 3.55 meV and followed the temperature evolution of the
dynamical neutron structure factor $S(q,E)$. Figure 13 shows the intensity
contour color plots of $S(q,E)$ along the $[H,H,0]$ direction. At 60 K there
is a large phonon contribution that substantially reduces at 20 K and below,
where nearly vertical columns of intensity become apparent close to
$K^{\prime}_{1}=(2/3,2/3,0)$ and $K^{\prime\prime}=(4/3,4/3,0)$, with a region
of broad intensity located around 2 meV and close to $[1,1,0]$. At our base
temperature of 1.5 K, sharp dispersive features emerge from these two $Q$
positions which we identify as spin-wave excitations. They show a pronounced
two-dimensional character, as revealed by the presence of rods of scattering
intensity along the $[0,0,L]$ direction (Fig. 13 e-g), and thus justify the 2D
Heisenberg kagome antiferromagnet model applied below. From this time-of-
flight measurement, one can isolate the elastic contribution by integrating
over twice the energy resolution ($\sim 0.15$ meV) only. Figure 13 h) shows
such contribution with the apparition of tiny $q$-resolution limited magnetic
Bragg peaks below 5 K located at both $K^{\prime}_{1}$ and $K^{\prime\prime}$
positions. This elastic scattering points to a small ordered fraction of the
Cu2+ moment, that could be easily missed in a conventional energy-integrated
neutron diffraction experiment, and strongly supports a magnetic non-collinear
coplanar phase with a $Q=(1/3,1/3)$ ordering wave vector.

The inelastic scattering in the $(H,K,0)$ plane is displayed in Fig. 14. The
energy-integrated cuts show clear intensities located at the $K$ positions of
the Brillouin zone, mirroring the sharp spin wave dispersions already observed
below 1 meV in Fig. 13 a). Above 1 meV, the inelastic scattering broadens
significantly and extends toward the Brillouin zone center.

Figure 14: Dynamical structure factor $S(Q,E)$ for the $(H,K,0)$ plane
measured at $T=1.55$ K. The four color plot maps show its evolution in energy
with the corresponding integrated energy range. Gray lines indicate the
Brillouin zones boundaries. Bright spots of intensity are observed at the $K$
points, and corresponds to the dispersion of spin-waves.

To better isolate the inelastic magnetic contribution present at 1.5 K, we
subtracted a background constructed by the replication of the 5 K dataset,
integrated in the range $H=[1.6,2]$, along $[H,H,0]$, where the phonon and
magnetic contributions appear to be minimal. The result is shown in Fig. 15
a). The dispersive excitations rising from $K^{\prime}_{1}$ and
$K^{\prime\prime}$ are clearly singled out, and then merge into a broad arch
extending from 1 to 2.5 meV. Guided by the observation of an in-plane
distortion of the kagome lattice, and by the recent ab initio DFT results
[19], we performed linear spin-wave calculations using the SpinW package [36]
of the Heisenberg model with only nearest-neighbors interactions in order to
model the excitations on such an anisotropic kagome lattice:

$\mathcal{H}=\sum_{\left\langle
i,j\right\rangle}J_{ij}\bm{S_{i}}\cdot\bm{S_{j}}\\\ $ (10)

where $J_{ij}$ can take three different values $J_{\hexagon}$, $J$ and
$J^{\prime}$ depending respectively on the bonds $d_{1},d_{2}$ and $d_{3}$ as
defined in Fig. 1. From DFT calculation, Hering et al. [19] confirmed the
relevance of such spin models in the context of Y-Kapellasite using three
antiferromagnetic couplings $J\simeq J_{\hexagon}\simeq 140$ K and
$J^{\prime}\simeq 10$ K. The spin-wave simulations assumed an enlarged unit
cell of nine sites, because of the two different copper sublattices, and a
coplanar order with a $Q=(1/3,1/3)$ propagation vector, as described in
details in Ref. [19]. The $J=J_{\hexagon}=J^{\prime}$ case leads to the well-
known $\sqrt{3}\times\sqrt{3}$ phase of the classical kagome antiferromagnetic
model which fails to reproduce our data when $J>15$ K, due to the maximum of
the band observed around 2.5 meV. Given that $\theta_{\rm cw}=-100$ K imposes
a higher exchange value, we discard the isotropic case and instead study the
excitations for the parameters $J=J_{\hexagon}$ and $J^{\prime}\neq J$. From
the evolution of the spin-wave spectrum with $J$ or $J^{\prime}$, we find our
best model using $J\simeq J_{\hexagon}=140$ K and $J^{\prime}=63$ K. However,
we also find that a single set of interactions can not account for the spread
of intensity observed at $[1,1,0]$, from 1 to 2.5 meV. Instead, we consider a
$\sim 10$ % distribution of $J^{\prime}$, tentatively attributed to some local
disorder, and that could indeed satisfactorily fit our inelastic data, as
shown in Fig. 15 c), d) and e), leading to $J\simeq J_{\hexagon}=140(10)$ K
and $J^{\prime}=[56,70]$ K. Finally, we note that the intensity observed at
low $Q$, near $Q=(1/3,1/3,0)$, and below 0.5 meV (see Fig. 15 a) is much
broader than predicted by the simulation, and may hint at a more complex
ground state in Y-kapellasite with the presence of short-range order and
fluctuations. We note that this broad intensity extends mostly around the zone
boundaries of the first Brillouin zone (Fig. 14) and not close to the
boundaries of the _extended_ Brillouin zone, as predicted for the classical
spin liquid [19].

In summary, INS gives evidence for elastic scattering below 5 K, in agreement
with the static magnetism revealed by $\mu$SR and line broadening in NMR,
consistent with a $Q=(1/3,1/3)$ magnetic ground state. Furthermore, we found a
set of magnetic interactions in Y-kapellasite which is in very good agreement
with the predictions from DFT and variational Monte-Carlo studies [19]. The
main difference is our experimental value of $J^{\prime}\simeq 63(7)$ K as
compared to the theoretical estimate $J^{\prime}=10.3(7)$ K. This locates
Y-kapellasite in the $Q=(1/3,1/3)$ magnetic order of the phase diagram, but
very close to the classical spin liquid phase reported in Ref. [19].

Figure 15: Comparison between experimental data (a,b) and spin-waves
simulations (c,d) at $T=1.5$ K. The INS data in (a) measured at 1.5 K was
subtracted by a 5 K dataset (see text) to highlight the magnetic contribution.
(e) Energy-integrated cuts along $[H,H,0]$ of data shown in (a) and (c) for
different energies $E$ with an integration range of 0.15 meV. Symbols are for
data and lines for the simulation. They are both shifted vertically for
clarity. (f) Classical phase diagram reproduced from [19]. The black square
denotes the position found from DFT calculation [19] while the red star
locates Y-kapellasite using the exchange couplings found in our work.

##  VIII Discussion and Conclusion

Our experimental investigation of Y3Cu9(OH)19Cl8 reveals an unusual
bidimensional coplanar $Q=(1/3,1/3)$ long range ordered ground state, overall
in very good agreement with the theoretical study in Ref. [19]. In particular,
the proposed spin arrangement in the ground state is found to be compatible
with the Cl NMR lineshape at low temperature. However, at variance with first
principle calculations where the third interaction of the model $J^{\prime}$
was found to be negligible, we determine experimentally a finite
$J^{\prime}\lesssim J\hexagon/2$. This finding of a sizeable antiferromagnetic
$J^{\prime}$ seems nonetheless consistent with the Cu-OH-Cu bond angles which
vary only by 4.6∘ among the three different bonds. The smallest angle
$113.1^{\circ}$ corresponding to $J^{\prime}$ remains significantly larger
than 105∘ which is known to yield small $\sim-15$ K ferromagnetic interaction
in Zn-kapellasite [37].

Besides, to reproduce the broad magnetic excitations in INS, a distribution
$\sim 10\%$ of the $J^{\prime}$ interaction was imposed. This suggests some
kind of disorder. Due to the large difference in the ionic radii of Y3+ and
Cu2+ we expect no significant intersite mixing at variance with Zn-kapellasite
or other Zn and Cu based kagome materials and indeed we found no evidence of
such disorder in the magnetic lattice above 33 K, neither in neutron
diffraction nor in NMR which on the contrary exposes well resolved narrow
lines. Therefore, the distribution of $J^{\prime}$ may arise only at low
temperature, as a result of the structural instabilities observed at 33 K and
13 K. The slightly different local environment detected below 33 K by Cl NMR
would then reflect in the moderate distribution of the interaction, while
preserving on average the high temperature magnetic model, since the symmetry
and lattice parameters remain unchanged on the whole temperature range, down
to 65 mK.

The experimentally determined interactions locate Y-kapellasite in the
classical phase diagram close to the boundary between the (1/3,1/3) long range
ordered and the ”classical spin liquid” phases (see Fig. 15 f). Although the
effect of quantum fluctuations has not been investigated so close to the phase
boundary, we may anticipate that they are responsible for the strongly reduced
value of the Cu2+ moments $\lesssim 0.1\mu_{B}$, which could not be detected
in standard neutron diffraction but yield static internal fields detected both
in NMR and $\mu$SR. The proximity to a phase boundary may also help
understanding the striking difference in the ground states of the present
large single crystal samples and formerly studied polycrystalline ones [11].
Indeed in the latter, the moments were found to remain fluctuating in the
ground state and also no structural transition was detected. We may assume
that some additional disorder in the polycrystalline samples or slightly
different interactions, due to the absence of structural transition,
destabilize the fragile ordered state observed in large single crystals. We
note that such a sensitivity to the sample crystallinity seems to be a common
feature of anisotropic kagome compounds like volborthite [38, 17] or
vesigneite [39, 40, 18]. The fragility of the ground state in all these
compounds seems to originate from a subtle interplay between structure and
magnetic frustration.

Beyond the proposed anisotropic nearest neighbor model, and despite its
apparent success to reproduce the physics in Y3Cu9(OH)19Cl8, one important
perspective to complete our understanding of this material is to quantify and
address the role of Dzyaloshinskii-Moriya interaction. Let us indeed recall
that this anisotropy of the interaction was pointed out as the main ingredient
driving long range order in the sister compound YCu3(OH)6Cl3 [41] with a
perfect kagome lattice. Quantum fluctuations and especially their role at the
phase boundaries and how they impact the ”jammed” classical spin liquid phase
is another key perspective of this work. Finally we note that a Br counterpart
[42] was recently synthesized, showing no sign of ordering, and likely
allowing to span further the phase diagram of the anisotropic kagome model.

## Acknowledgements

The authors acknowledge the support of the French Agence Nationale de la
Recherche, under Grant No. ANR-18-CE30-0022 ”LINK”, and funding from Deutsche
Forschungsgemeinschaft (DFG) through TRR 288-422213477 (project A03). We thank
Samir Hammoud for carrying out the ICP and gas extraction experiments, and A.
Razpopov, R. Valenti, H.O. Jeschke and J. Reuther for useful discussions and
for sharing INS simulation data.

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