# Magnetic field induced quantum spin liquid in the two coupled trillium
lattices of K2Ni2(SO4)3

Ivica Živković<EMAIL_ADDRESS>Laboratory for Quantum Magnetism,
Institute of Physics, École Polytechnique Fédérale de Lausanne, CH-1015
Lausanne, Switzerland  Virgile Favre  Laboratory for Quantum Magnetism,
Institute of Physics, École Polytechnique Fédérale de Lausanne, CH-1015
Lausanne, Switzerland  Catalina Salazar Mejia  Hochfeld-Magnetlabor Dresden
(HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat, Helmholtz-
Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Harald O. Jeschke
Research Institute for Interdisciplinary Science, Okayama University, Okayama
700-8530, Japan  Arnaud Magrez  Crystal Growth Facility, École Polytechnique
Fédérale de Lausanne, Lausanne, Switzerland.  Bhupen Dabholkar  Department of
Physics and Quantum Centers in Diamond and Emerging Materials (QuCenDiEM)
group, Indian Institute of Technology Madras, Chennai 600036, India  Vincent
Noculak  Dahlem Center for Complex Quantum Systems and Fachbereich Physik,
Freie Universität Berlin, 14195 Berlin, Germany  Helmholtz-Zentrum für
Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany  Rafael
S. Freitas  7 Instituto de Física, Universidade de São Paulo, 05508-090, São
Paulo, Brazil  Minki Jeong  School of Physics and Astronomy, University of
Birmingham, Edgbaston, Birmingham, B15 2TT UK  Nagabhushan G. Hegde
Laboratory for Quantum Magnetism, Institute of Physics, École Polytechnique
Fédérale de Lausanne, CH-1015 Lausanne, Switzerland  Luc Testa  Laboratory for
Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de
Lausanne, CH-1015 Lausanne, Switzerland  Peter Babkevich  Laboratory for
Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de
Lausanne, CH-1015 Lausanne, Switzerland  Yixi Su  Jülich Centre for Neutron
Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Forschungszentrum
Jülich, Lichtenbergstrasse 1, D-85747 Garching, Germany  Pascal Manuel  ISIS
Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Harwell
Science and Innovation Campus, Didcot, Oxfordshire, OX11 0QX UK  Hubertus
Luetkens  Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute,
CH-5232 Villigen, Switzerland  Christopher Baines  Laboratory for Muon Spin
Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen, Switzerland  Peter J.
Baker  ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton
Laboratory, Harwell Science and Innovation Campus, Didcot, Oxfordshire, OX11
0QX UK  Jochen Wosnitza  Hochfeld-Magnetlabor Dresden (HLD-EMFL) and Würzburg-
Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum Dresden-Rossendorf,
01328 Dresden, Germany  Institut für Festkörper- und Materialphysik, TU
Dresden, 01062 Dresden Germany  Oksana Zaharko  Laboratory for Neutron
Scattering and Imaging, Paul Scherrer Institut, CH-5253 Villigen, Switzerland
Yasir Iqbal  Department of Physics and Quantum Centers in Diamond and Emerging
Materials (QuCenDiEM) group, Indian Institute of Technology Madras, Chennai
600036, India  Johannes Reuther  Dahlem Center for Complex Quantum Systems and
Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
Helmholtz-Zentrum für Materialien und Energie, Hahn-Meitner-Platz 1, 14109
Berlin, Germany  Henrik M. Rønnow  Laboratory for Quantum Magnetism, Institute
of Physics, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne,
Switzerland

###### Abstract

Quantum spin liquids are exotic states of matter which form when strongly
frustrated magnetic interactions induce a highly entangled quantum paramagnet
far below the energy scale of the magnetic interactions. Three-dimensional
cases are especially challenging due to the significant reduction of the
influence of quantum fluctuations. Here, we report the magnetic
characterization of K2Ni2(SO4)3 forming a three dimensional network of Ni2+
spins. Using density functional theory calculations we show that this network
consists of two interconnected spin-1 trillium lattices. In the absence of a
magnetic field, magnetization, specific heat, neutron scattering and muon spin
relaxation experiments demonstrate a highly correlated and dynamic state,
coexisting with a peculiar, very small static component exhibiting a strongly
renormalized moment. A magnetic field $B\gtrsim 4$ T diminishes the ordered
component and drives the system in a pure quantum spin liquid state. This
shows that a system of interconnected $S=1$ trillium lattices exhibit a
significantly elevated level of geometrical frustration.

Strongly correlated systems are at the forefront of condensed matter research,
exhibiting exotic phases and nourishing novel theoretical concepts. In
magnetism, one of the most sought-after strongly correlated phase is a quantum
spin liquid (QSL), a state in which spins avoid long-range order (LRO) and are
considered entangled on all spatial scales [1, 2, 3]. To realize a QSL,
geometrical frustration and reduced dimensionality of the magnetic subsystem
have been considered vital. 1D Heisenberg chains exhibit QSL behavior even
without frustration [4, 5] while 3D cases are rare due to the significant
reduction of quantum fluctuations. Nevertheless, it has been found that 3D
lattices like pyrochlore [6, 7, 8] and hyper-hyperkagome [9, 10] support QSL
behavior.

In this Letter we provide extensive experimental and computational evidence
that K2Ni2(SO4)3 exhibits QSL behavior, based on a novel arrangement of spins
forming two interconnected trillium lattices. Previous work on compounds
featuring a single trillium lattice was mainly driven by a pressure-induced
quantum phase transition (QPT) discovered in the itinerant helimagnet MnSi
[11] and evidence of non-Fermi liquid behavior above a critical pressure [12].
Later theoretical works [13, 14] showed some degree of geometrical frustration
in the trillium lattice, nevertheless insufficient to prevent the onset of
LRO. From that perspective, K2Ni2(SO4)3 and other members of the langbeinite
family K2M2(SO4)3 (M = Fe, Co, Mn, Cr) offer an arena for testing future
theoretical developments on interconnected trillium lattices. Previous
investigations of those compounds displayed ferroelectricity and structural
transitions but their magnetic properties remain terra incognita.

K2Ni2(SO4)3 crystallizes in a cubic unit cell ($P2_{1}3$) with $a=9.81866(12)$
Å determined from single-crystal diffraction at 100 K [15]. It consists of a
network of trigonally-distorted NiO6 octahedra, coupled through SO4 groups
[Fig. 1(a)], with a Ni–O–S–O–Ni super-super-exchange mechanism mediating
magnetic interactions between $S=1$ spins. There are two crystallographic Ni
sites, distinguished by their Ni–O distances [15], each site forming a single
trillium lattice.

Mapping the GGA+$U$ total energies [15] onto a Heisenberg Hamiltonian for
K2Ni2(SO4)3 $\hat{\cal
H}=\sum_{i<j}J_{ij}{\bf{\hat{S}}}_{i}\cdot{\bf{\hat{S}}}_{j}$ as shown in Fig.
1(c) yields the five non-zero exchange couplings that are listed in Table 1
and shown in Fig. 1(b), visualizing the exchange network. The couplings within
each TL are given by antiferromagnetic (AFM) $J_{3}$ and $J_{5}$,
respectively. On the other hand the strongest coupling is found to be AFM
$J_{4}$ that inter-connects the two lattices. Interestingly, if $J_{4}$ was
the only coupling in the system it would support a Néel-type LRO. Thus, our
calculation shows that the physics of K2Ni2(SO4)3 is determined by an
interplay between $J_{4}$ induced ordering tendencies and $J_{3}$ and $J_{5}$
driven frustration.

Label | Type | Distance (Å) | Exchange (K)
---|---|---|---
$J_{1}$ | Ni(1)–Ni(2) | 4.42877 | 0.42(1)
$J_{2}$ | Ni(1)–Ni(2) | 4.90057 | $-0.16(1)$
$J_{3}$ | Ni(2)–Ni(2) | 6.08379 | 1.09(1)
$J_{4}$ | Ni(1)–Ni(2) | 6.12050 | 5.38(1)
$J_{5}$ | Ni(1)–Ni(1) | 6.12695 | 2.54(1)
Table 1: K2Ni2(SO4)3 exchange energies obtained by DFT energy mapping, with
paths identified by Ni–Ni distance.

Figure 2(a) displays the temperature dependence of dc magnetic susceptibility
$\chi_{\rm dc}(T)$ and its inverse in a wide temperature range. The monotonic
increase of $\chi_{\rm dc}(T)$ with decreasing $T$, without any noticeable
features, suggests the absence of LRO down to 2 K. The linear behavior of
$1/\chi_{\rm dc}(T)$ above 50 K allows us to use the Curie-Weiss law
$\chi(T)=C/(T-\Theta_{\rm CW})$, which gives $C=1.37(2)$ emu K/mol and
$\Theta_{\rm CW}=-18(1)$ K. The value of $C$ corresponds to $S=1$ with a
slightly enhanced $g$-factor of $g=2.34$. $\Theta_{\rm CW}<0$ indicates
predominant AFM interactions, in accordance with density functional theory
(DFT) calculations. Additionally, measurements along three orthogonal
directions practically overlap, indicating no significant anisotropy.

Below 50 K, $1/\chi_{\rm dc}(T)$ starts to deviate from the Curie-Weiss law,
following the build up of correlations between magnetic moments. To emphasize
this behavior, magnetization curves obtained at several temperatures are
plotted in Fig. 2(b), together with the curves of the Brillouin function,
which describe an assembly of non-interacting $S=1$ spins, at corresponding
temperatures (dashed lines). To approximate the magnetization behavior,
classical Monte Carlo calculations employing the DFT Hamiltonian have been
performed. The resulting curves (full lines) are closer to the experimental
ones but it is apparent that with decreasing $T$ the deviation from the
classical prediction becomes more pronounced, suggesting a sizeable influence
of quantum fluctuations on this 3D lattice.

Further evidence of strongly correlated spins can be obtained from specific
heat measurements. Fig. 2(c) shows the $T$ dependence of the total specific
heat of a single crystal of K2Ni2(SO4)3, together with a non-magnetic analog
K2Mg2(SO4)3. At temperatures above 20 K the two compounds show a very similar
behavior, indicating a dominant phonon contribution. Below 20 K, K2Ni2(SO4)3
exhibits a significant deviation, with a broad maximum around 5 K and two
features occurring at $T^{*}=1.14$ K and $T^{**}=0.74$ K. Below $T^{**}$, the
heat capacity behaves according to a power-law $C_{p}\sim T^{n}$, with
$n\approx 2$. This value of the exponent differs appreciably from $n=3$ for
classical AFM and has been observed in several frustrated magnetic systems
[16, 17, 18, 8].

To extract the magnetic specific heat $C_{m}$, the phonon contribution using
the data obtained on K2Mg2(SO4)3 has been subtracted. On the high $T$ side the
subtraction works up to 50 K where K2Mg2(SO4)3 shows a kink [15], associated
to previously observed lattice related features in the heat capacity [19]. On
the low $T$ side a polynomial $BT^{3}+CT^{5}$ has been used [15]. The error of
the total entropy $S=\int(C_{m}/T)dT$ due to background subtraction is
estimated to be a few percent. As can be seen in Fig. 2(d), at 50 K more than
98% of the expected entropy for $S=1$ system is recovered, with more than 90%
being released up to 20 K. The saturation towards the $R\ln{(2S+1)}$ value for
$S=1$ indicates that no residual entropy is present at $T=0$ and that
K2Ni2(SO4)3 exhibits a non-degenerate ground state.

Application of magnetic field along the [111] direction induces little change
in the overall behavior of the heat capacity of K2Ni2(SO4)3. A small but
noticeable redistribution occurs for fields above $B=7$ T [ Fig. 2(e)] but
even with fields up to 14 T the overall shape of the curve remains unchanged.
The power law $C_{p}\sim T^{n}$ observed at low $T$ for $B=0$ is maintained
for $B>0$ without a visible crossover towards the gapped polarized state, as
seen, for example, in YbMgGaO4 [20]. The value of the extracted exponent
remains field independent up to 14 T [inset of Fig. 2(d)].

The order of transitions at $T^{*}$ and $T^{**}$ is revealed through their
overall shape. The feature at $T^{*}$ resembles a typical, asymmetric
$\lambda$-shape, characteristic of second-order phase transitions. On the
other hand, at $T^{**}$ a narrow, symmetrical peak is found, often seen in
first-order phase transitions. Although the entropy released at $T^{*}$
amounts to only 1% of the total $R{\rm ln}3$ [15], the sample purity
determined by single-crystal x-ray diffraction [15] rules out any impurity-
related scenario. Additionally, a comparison with specific-heat measurements
on a powder sample reveals that $T^{*}$ is significantly diminished while
$T^{**}$ is completely absent [15]. With a tentative assignment of $T^{**}$ as
a first-order phase transition, its presence in a single-crystal experiment
suggests that it is intrinsically related to the low-temperature magnetic
phase of K2Ni2(SO4)3.

The magnetic field dependence of $T^{*}$ and $T^{**}$ is presented in Fig.
2(f). $T^{**}$ is quickly diminished in amplitude and for $B>1$T it disappears
completely. $T^{*}$ is practically unchanged up to $B=1$ T with a subsequent
decrease and a reduction of the size of the anomaly [15]. By assuming a
quadratic $B$-dependence of the second-order phase transition the value of the
critical magnetic field $B_{c}\lesssim 4$ T can been estimated, above which a
completely dynamic and fluctuating state exists down to the lowest $T$.

To shed more light on the peculiar magnetic properties of K2Ni2(SO4)3, a
series of neutron scattering experiments have been performed. Fig. 3(a) shows
the results of polarized neutron scattering, in which a $Q$-dependence of the
scattering intensity at 0.5 K is presented. It exhibits a broad maximum
centered at $Q_{\rm max}\approx 0.75$ Å-1 followed by an attenuating
oscillatory dependence. Such a broad, liquid-like structure factor is typical
for systems with strong quantum fluctuations. This conclusion is further
supported by the fact that the diffuse scattering pattern in Fig. 3(a) is well
reproduced by pseudofermion functional renormalization group (PFFRG)
simulations of the DFT Hamiltonian. Remarkably, despite the general
difficulties in simulating a strongly fluctuating 3D spin system with complex
frustrated interactions as realized in K2Ni2(SO4)3, not only the positions of
the extrema are well reproduced but also the global amplitude variations.
Additionally, the oscillatory behavior is seen to persist at least up to 17 K
[15], clearly indicating its connection to strong correlations developing
below 20 K.

To investigate the system’s static component, a neutron powder diffraction
experiment has been performed well above and well below $T^{*}$. From a wide
$Q$ diffraction pattern shown in Fig. 3(b) it is found that for $Q>1$ Å-1 all
peaks are present at both temperatures, indicating their lattice origin. On
the other hand, a series of very weak magnetic peaks can be found at $T=0.1$ K
for $Q<1$ Å-1 as seen in the upper half of Fig. 3(c). They can all be assigned
to satellites of the main nuclear Bragg peaks $(h,k,l)$ in the form $q_{\rm
magnetic}=(h,k,l)\pm Q_{i}$, where $Q_{1}=(\frac{1}{3},0,0)$,
$Q_{2}=(\frac{1}{3},\frac{1}{3},0)$ and
$Q_{3}=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. The existence of three
propagation vectors indicates that even LRO is heavily influenced by
frustration, leaving several possible structures with similar ground state
energies.

Due to the complexity of the scattering pattern, including several propagation
vectors, tripling of the magnetic unit cell and very weak amplitudes, it is
not possible to completely determine the magnetic structure nor to extract the
value of the ordered moment. Nevertheless, utilizing a purely magnetic
scattering pattern from polarized neutrons [Fig. 3(a)] one can estimate an
upper limit for the static component. To this end, we envisage that the total
intensity $S(Q)$ is composed of two contributions $S(Q)=S_{\rm
static}(Q)+S_{\rm dynamic}(Q)$, with the jagged $S_{\rm static}(Q)$ roughly
following the powder diffraction profile and sitting on top of the smooth
$S_{\rm dynamic}(Q)$. Although the resultant ratio $S_{\rm
static}(Q)/S(Q)\approx 11\%$ cannot be directly related to the value of the
ordered moment, it serves as a supporting evidence that the ground state in
K2Ni2(SO4)3 is dominated by spin fluctuations.

In Fig. 3(d), we show time-of-flight (TOF) data obtained as a direct
subtraction of the background intensity obtained at 80 K from a measured
intensity at 0.5 K. Streaks of intensity can be observed at the same positions
as maxima in $S(Q)$ found with polarized neutrons. The upper limit of spin
excitations is found to be around 1.8 meV which agrees well with the
temperature at which specific heat starts to significantly deviate from a
purely phononic behavior. In Fig. 3(e), a narrow Q-integrated energy
dependence of intensity is shown, indicating a continuum of excitations down
to the elastic line. Due to the existence of the ordered component, it is not
straight-forward to assign this continuum to the QSL state. On the other hand,
the dominance of the dynamic component, revealed by specific heat data and
polarized neutron scattering, renders this conclusion very plausible, which
would then support the hypothesis of a gapless nature for the QSL.

To probe further the peculiar coexistence of static and dynamic properties
revealed in K2Ni2(SO4)3, muon spin relaxation ($\mu$SR) experiments have been
performed. As shown in Fig. 4(a), no obvious wiggles are observed down to
lowest $T$. On a phenomenological level the relaxation is often described by a
stretched-exponential function

$A(t)=A_{0}e^{-(\lambda t)^{\beta}}+A_{\rm BG},$ (1)

where $A_{0}$ is the initial asymmetry, $A_{\rm BG}$ a constant background,
$\lambda$ is the relaxation rate and $\beta$ is the stretching exponent that
in an ideal case of $\beta=1$ leads to a simple exponential relaxation.
$\beta<1$ has usually been associated with either a distribution of relaxation
times, multiple muon stopping sites, or with intrinsic disorder in the
magnetic system. As is evident from Fig. 4(b), at low $T$ the observed time
dependence of the asymmetry cannot be satisfactorily described by a single
contribution. Thus, we have extended Eq. (1) with an additional term

$A(t)=A_{0}(fe^{-(\lambda_{1}t)^{\beta_{1}}}+(1-f)e^{-(\lambda_{2}t)^{\beta_{2}}})+A_{\rm
BG}.$ (2)

and fixed $f=0.5$ and $\beta_{1}=1$ to avoid over-parametrization. We find
that it is necessary to use Eq. (2) up to 3 K while for $T>3$ K Eq. (1) is
sufficient (for the discussion of the overlapping region see [15]). In Fig.
4(c), we present the temperature evolution of relaxation rates and exponents
(see inset) extracted using Eq. (2) (green symbols) and Eq. (1) (blue
symbols).

Below $T\sim 1$ K, the extracted parameters attain a constant value, a feature
often associated with a highly dynamic nature of QSLs [21, 22, 23]. We point
out that the value of the exponent $\beta\simeq 2$ is indicative of a specific
type of a correlated spin system based on spin-singlets [24]. Within this
scenario, the Gaussian shape of the relaxation profile develops from a
sporadic appearance of unpaired spins. The time interval of their existence is
much shorter than a life-time of a muon, so for the majority of time muons
experience very small fields related to the short-lived but very distant
unpaired spins. Such a scenario is in accordance with a practically field-
independent magnetic specific heat seen in Fig. 2(e) [16]. Within this
framework the strong relaxation at low temperatures described by $\lambda_{1}$
can be associated with a partial but homogeneous order while the remaining
dynamics is due to the sporadic unpaired-spin appearances. The absence of
oscillations can then be associated with a spread of local fields originating
from complex magnetic structures given by propagation vectors $Q_{1}$, $Q_{2}$
and $Q_{3}$. Additionally, the coherent regions giving rise to magnetic peaks
in neutron diffraction are probed on much shorter time scales ($\sim 10^{-14}$
s), allowing for local fluctuations between different magnetic structures on
the time scale of muons.

We find two possible scenarios that could encompass a small value of the
static component existing alongside the dominant, fluctuating component. The
first scenario assumes the existence of a quantum critical point (QCP) between
an ordered phase and a quantum-fluctuation-dominant phase, with K2Ni2(SO4)3
being on the ordered side of QCP but “accidentally” close to it. In this case,
the ordered moment $m_{s}$ is strongly renormalized due to the prevalence of
quantum fluctuations close to a QCP, as has been demonstrated in TlCuCl3 where
a pressure-controlled QPT between a LRO AFM state and a non-magnetic dimer
phase is arbitrarily decreased ($m_{s}\sim\sqrt{p-p_{c}}$) close to a QCP
[25]. In this context, a possible control parameter could be the ratio of
intra- ($J_{3}$, $J_{5}$) and inter-trillium lattice couplings ($J_{1}$,
$J_{2}$, $J_{4}$). Given that Ni(1) and Ni(2) sites form a bipartite lattice,
the limit of dominant $J_{1}$, $J_{2}$, $J_{4}$ results in a semiclassical AFM
phase. With $J_{3}$, $J_{5}$ dominant, the system is in the limit of two
weakly coupled trillium lattices. As demonstrated theoretically for a single
trillium lattice, it is expected to form a variant of the 120$\degree$ order
[13, 14]. The case of two interconnected trillium lattices represents a novel
research direction with many members of the langbeinite family providing ample
opportunity for comparison with theory.

The second scenario dismisses the “fortuitous” constellation of parameters
describing K2Ni2(SO4)3 and considers it positioned well within the QSL phase.
Due to the presence of antisymmetric exchange coupling (the Dzyaloshinskii-
Moriya interaction (DMI)) allowed by the non-centrosymmetric space group, the
ground state gets “dressed” with a small ordered component due to the admixing
of higher lying states, similar to the admixture of triplet wave-functions
into the ground state singlet of an AFM dimer. An exciting consequence of this
scenario arises from topological aspects imposed on the QSL state. Magnetic
structures forming in non-centrosymmetric space groups are shown to support
skyrmions, topologically protected spin textures [26, 27]. Fractional wave-
numbers $Q_{1}$, $Q_{2}$ and $Q_{3}$ revealed in the diffraction experiment do
indicate a potential role of DMI in the formation of LRO.

In either case, the observed coexistence between fluctuating spins and a small
static component which vanishes in a magnetic field could be linked to already
developed concepts like field-induced spin liquids in Kitaev-type honeycomb
models featuring non-Abelian fractional quasiparticles [28]. The ability to
tune its behavior across QCP with magnetic field into a pristine QSL state is
an exciting opportunity which should stimulate further experimental and
theoretical studies.

Acknowledgments. We thank Bi Wen Hua for his help with x-ray diffraction
experiments. I. Ž. acknowledges financial support by the Swiss National
Science Foundation (SNSF) project No. 200021-169699 and 206021-189644. H. M.
R. acknowledges financial support by SNFS projects No. 200020-188648 and
206021-189644. Part of this work was supported by the Deutsche
Forschungsgemeinschaft (DFG) through the Würzburg-Dresden Cluster of
Excellence on Complexity and Topology in Quantum Matter–$ct:qmat$ (EXC 2147,
Project No. 390858490), as well as by the HLD at HZDR, a member of the
European Magnetic Field Laboratory (EMFL). R.S.F. acknowledges financial
support by FAPESP (Grant No. 2015/16191-5) and CNPq (Grant No. 429511/2018-3).
J.R. acknowledges financial support by the German Research Foundation within
the CRC183 (project A04). Y.I. acknowledges financial support by SERB,
Department of Science and Technology (DST), India through grants SRG (No.
SRG/2019/000056), MATRICS (No. MTR/2019/001042), and Indo-French Center for
the Promotion of Advanced Research CEFIPRA (No. 64T3-1). This research was
supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958, the ICTP through the Simons Associateship scheme, IIT Madras
through the IoE program for establishing the QuCenDiEM group (Project No.
SB20210813PHMHRD002720), ICTS, Bengaluru, India during a visit for
participating in the program “Novel phases of quantum matter” (Code:
ICTS/topmatter2019/12). Y.I. acknowledges the use of the computing resources
at HPCE, IIT Madras. The $\mu$SR experiments were performed at MUSR beamline,
ISIS (1710223) and LTF and GPS beamlines, PSI (2017119 and 2017119). The
neutron diffraction experiment was performed at WISH beamline, ISIS (2010010).
The spin-polarized neutron diffraction and TOF experiments were performed at
DNS beamline, MLZ (13656).

## References

  * Savary and Balents [2016] L. Savary and L. Balents, Quantum spin liquids: a review, Rep. Prog. Phys. 80, 016502 (2016).
  * Zhou  _et al._ [2017] Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017).
  * Broholm  _et al._ [2020] C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quantum spin liquids, Science 367, 6475 (2020).
  * Lake  _et al._ [2013] B. Lake, D. A. Tennant, J. S. Caux, T. Barthel, U. Schollwöck, S. E. Nagler, and C. D. Frost, Multispinon Continua at Zero and Finite Temperature in a Near-Ideal Heisenberg Chain, Phys. Rev. Lett. 111, 137205 (2013).
  * Mourigal  _et al._ [2013] M. Mourigal, M. Enderle, A. Klöpperpieper, J. S. Caux, A. Stunault, and H. M. Rønnow, Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain, Nat. Phys. 9, 435 (2013).
  * Wen  _et al._ [2017] J.-J. Wen, S. M. Koohpayeh, K. A. Ross, B. A. Trump, T. M. McQueen, K. Kimura, S. Nakatsuji, Y. Qiu, D. M. Pajerowski, J. R. D. Copley, and C. L. Broholm, Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in Pr2Zr2O7, Phys. Rev. Lett. 118, 107206 (2017).
  * Gao  _et al._ [2019] B. Gao, T. Chen, D. W. Tam, C. L. Huang, K. Sasmal, D. T. Adroja, F. Ye, H. Cao, G. Sala, M. B. Stone, C. Baines, J. A. T. Verezhak, H. Hu, J. H. Chung, X. Xu, S. W. Cheong, M. Nallaiyan, S. Spagna, M. B. Maple, A. H. Nevidomskyy, E. Morosan, G. Chen, and P. Dai, Experimental signatures of a three-dimensional quantum spin liquid in effective spin-1/2 Ce2Zr2O7 pyrochlore, Nat. Phys. 15, 1052 (2019).
  * Plumb  _et al._ [2019] K. W. Plumb, H. J. Changlani, A. Scheie, S. Zhang, J. W. Krizan, J. A. Rodriguez-Rivera, Y. Qiu, B. Winn, R. J. Cava, and C. L. Broholm, Continuum of quantum fluctuations in a three-dimensional $S=1$ Heisenberg magnet, Nat. Phys. 15, 54 (2019).
  * Okamoto  _et al._ [2007] Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi, Spin-Liquid State in the S = 1/2 Hyperkagome Antiferromagnet Na4Ir3O8, Phys. Rev. Lett. 99, 137207 (2007).
  * Chillal  _et al._ [2020] S. Chillal, Y. Iqbal, H. O. Jeschke, J. A. Rodriguez-Rivera, R. Bewley, P. Manuel, D. Khalyavin, P. Steffens, P. Thomale, A. T. M. Nazmul Islam, J. Reuther, and B. Lake, Evidence for a three-dimensional quantum spin liquid in PbCuTe2O6, Nat. Comm. 11, 2348 (2020).
  * Pfleiderer  _et al._ [1997] C. Pfleiderer, G. J. McMullan, S. R. Julian, and G. G. Lonzarich, Magnetic quantum phase transition in MnSi under hydrostatic pressure, Phys. Rev. B 55, 8330 (1997).
  * Pfleiderer  _et al._ [2001] C. Pfleiderer, S. R. Julian, and G. G. Lonzarich, Non-Fermi-liquid nature of the normal state of itinerant-electron ferromagnets, Nature (London) 414, 427 (2001).
  * Hopkinson and Kee [2006] J. M. Hopkinson and H.-Y. Kee, Geometric frustration inherent to the trillium lattice, a sublattice of the B20 structure, Phys. Rev. B 74, 224441 (2006).
  * Isakov  _et al._ [2008] S. V. Isakov, J. M. Hopkinson, and H. Y. Kee, Fate of partial order on trillium and distorted windmill lattices, Phys. Rev. B 78, 014404 (2008).
  * [15] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.XXX for additional information on sample preparation, experimental and computational methods, powder diffraction, crystal structure data, magnetization, specific heat, neutron scattering, muon spin relaxation experiments and PFFRG calculations.
  * Ramirez  _et al._ [2000] A. P. Ramirez, B. Hessen, and M. Winklemann, Entropy Balance and Evidence for Local Spin Singlets in a Kagomé-Like Magnet, Phys. Rev. Lett. 84, 2957 (2000).
  * Nakatsuji  _et al._ [2005] S. Nakatsuji, Y. Nambu, H. Tonomura, S. Sakai, O. Jonas, C. Broholm, H. Tsunetsugu, Y. Qiu, and Y. Maeno, Spin Disorder on a Triangular Lattice, Science 309, 1697 (2005).
  * Silverstein  _et al._ [2014] H. J. Silverstein, K. Fritsch, F. Flicker, A. M. Hallas, J. S. Gardner, Y. Qiu, G. Ehlers, A. T. Savici, Z. Yamani, K. A. Ross, B. D. Gaulin, M. J. P. Gingras, J. A. M. Paddison, K. Foyevtsova, R. Valenti, F. Hawthorne, C. R. Wiebe, and H. D. Zhou, Liquidlike correlations in single-crystalline Y2Mo2O7: An unconventional spin glass, Phys. Rev. B 89, 054433 (2014).
  * Boerio-Goates  _et al._ [1990] J. Boerio-Goates, J. I. Artman, and B. F. Woodfield, Heat Capacity Studies of Phase Transitions in Langbeinites II. K2Mg2(SO4)3, Phys. Chem. Minerals 17, 173 (1990).
  * Xu  _et al._ [2016] Y. Xu, J. Zhang, Y. S. Li, Y. J. Yu, X. C. Hong, Q. M. Zhang, and S. Y. Li, Absence of Magnetic Thermal Conductivity in the Quantum Spin-Liquid Candidate YbMgGaO4, Phys. Rev. Lett. 117, 267202 (2016).
  * Mendels  _et al._ [2007] P. Mendels, F. Bert, M. A. de Vries, A. Olariu, A. Harrison, F. Duc, J. C. Trombe, J. S. Lord, A. Amato, and C. Baines, Quantum Magnetism in the Paratacamite Family: Towards an Ideal Kagome Lattice, Phys. Rev. Lett. 98, 077204 (2007).
  * Balz  _et al._ [2016] C. Balz, B. Lake, J. Reuther, H. Luetkens, R. Schönemann, T. Herrmannsdörfer, Y. Singh, A. T. M. Nazmul Islam, E. M. Wheeler, J. A. Rodriguez-Rivera, T. Guidi, G. G. Simeoni, C. Baines, and H. Ryll, Physical realization of a quantum spin liquid based on a complex frustration mechanism, Nat. Phys. 12, 942 (2016).
  * Fujihala  _et al._ [2020] M. Fujihala, K. Morita, R. Mole, S. Mitsuda, T. Tohyama, S. Yano, D. Yu, S. Sota, T. Kuwai, A. Koda, H. Okabe, H. Lee, S. Itoh, T. Hawai, T. masuda, H. Sagayama, A. Matsuo, K. Kindo, S. Ohira-Kawamura, and K. Nakajima, Gapless spin liquid in a square-kagome lattice antiferromagnet, Nat. Comm. 11, 3429 (2020).
  * Uemura  _et al._ [1994] Y. J. Uemura, A. Keren, K. Kojima, L. P. Le, G. M. Luke, W. D. Wu, Y. Ajiro, T. Asano, Y. Kuriyama, M. Mekata, H. Kikuchi, and K. Kakurai, Spin Fluctuations in Frustrated Kagome Lattice System SrCr8Ga4O19 Studied by Muon Spin Relaxation, Phys. Rev. Lett. 73, 3306 (1994).
  * Rüegg  _et al._ [2008] C. Rüegg, B. Normand, M. Matsumoto, A. Furrer, D. F. McMorrow, K. W. Krämer, H. U. Güdel, S. N. Gvasaliya, H. Mutka, and M. Boehm, Quantum Magnets under Pressure: Controlling Elementary Excitations in TlCuCl3, Phys. Rev. Lett. 100, 205701 (2008).
  * Mühlbauer  _et al._ [2009] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Skyrmion lattice in a chiral magnet, Science 323, 915 (2009).
  * Seki  _et al._ [2012] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Observation of skyrmions in a multiferroic material, Science 336, 198 (2012).
  * Kitaev [2006] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  * Koepernik and Eschrig [1999] K. Koepernik and H. Eschrig, Full-potential nonorthogonal local-orbital minimum-basis band-structure scheme, Phys. Rev. B 59, 1743 (1999).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Liechtenstein  _et al._ [1995] A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators, Phys. Rev. B 52, R5467 (1995).
  * Jeschke  _et al._ [2015] H. O. Jeschke, F. Salvat-Pujol, E. Gati, N. H. Hoang, B. Wolf, M. Lang, J. A. Schlueter, and R. Valentí, Barlowite as a canted antiferromagnet: Theory and experiment, Phys. Rev. B 92, 094417 (2015).
  * Jeschke  _et al._ [2019] H. O. Jeschke, H. Nakano, and T. Sakai, From kagome strip to kagome lattice: Realizations of frustrated $S=\frac{1}{2}$ antiferromagnets in Ti(III) fluorides, Phys. Rev. B 99, 140410 (2019).
  * Mizokawa and Fujimori [1996] T. Mizokawa and A. Fujimori, Electronic structure and orbital ordering in perovskite-type 3$d$ transition-metal oxides studied by Hartree-Fock band-structure calculations, Phys. Rev. B 54, 5368 (1996).
  * Reuther and Wölfle [2010] J. Reuther and P. Wölfle, ${J}_{1}\text{$-$}{J}_{2}$ frustrated two-dimensional Heisenberg model: Random phase approximation and functional renormalization group, Phys. Rev. B 81, 144410 (2010).
  * Baez and Reuther [2017] M. L. Baez and J. Reuther, Numerical treatment of spin systems with unrestricted spin length $S$: A functional renormalization group study, Phys. Rev. B 96, 045144 (2017).
  * Polchinski [1984] J. Polchinski, Renormalization and effective Lagrangians, Nucl. Phys. B 231, 269 (1984).
  * Wetterich [1993] C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993).

Figure 1: (a) Unit cell of K2Ni2(SO4)3. A Ni–O–S–O–Ni super-super-exchange
path contributing to a trillium coupling is marked by dashed lines. (b)
Exchange network between nickel sites. A ten site loop formed by the strongest
exchange $J_{4}$ is marked by arrows. (c) Exchange couplings determined by DFT
energy mapping. The vertical line indicates the $U$ value where the calculated
Curie-Weiss temperature matches the experimental value. Figure 2: (a) $T$
dependence of $\chi_{\rm dc}$ along the three orthogonal directions (left
axis) for $B=0.1$ T, and the inverse susceptibility $1/\chi_{\rm dc}$ (right
axis) with $B||[111]$. The solid grey line represents the Curie-Weiss law. (b)
Magnetic field dependence of magnetization, together with the Brillouin
function for $S=1$ and $g=2.34$ (dashed lines) and Monte Carlo simulations
(solid lines). (c) Zero field temperature dependence of the total specific
heat of K2Ni2(SO4)3, together with the total specific heat of K2Mg2(SO4)3
representing the phonon contribution. (d) $T$ dependence of magnetic entropy.
An inset shows the magnetic field dependence of the exponent $n$ in the power
law $C_{p}=AT^{n}$. (e) $T$ dependence of the magnetic specific heat for
several magnetic field values. (f) $T$ – $B$ phase diagram of K2Ni2(SO4)3.
Figure 3: (a) Pure magnetic scattering pattern derived from the spin-polarized
neutron diffraction data at 0.5 K (blue diamonds) and from PFFRG (solid line),
(b) Neutron powder diffraction at 0.1 K and 10 K, (c) Combined data from
polarized neutrons (blue diamonds from panel (a)) and powder diffraction (blue
(10 K) and red points (0.1 K) from panel (b)). Blue shading indicates an upper
limit of the contribution from the ordered component, yellow shading
represents the contribution from the fluctuating component, (d) TOF data at
0.5 K, (e) $E$-dependence of the integrated intensity for two $Q$-ranges.
Figure 4: (a) ZF-$\mu$SR spectra at selected temperatures. The solid lines
represent fitted curves following the model in Eq. (2). (b) A comparison of
the applicability of two models at 0.02 K (shifted down for clarity) and 3 K.
(c) Temperature dependence of the $\mu$SR rates extracted from two models.
Circles and triangles represent relaxation rates from the model in Eq. (2),
diamonds from the model in Eq. (1). The inset displays the temperature
dependence of the exponent $\beta_{2}$ from Eq. (2) and $\beta$ from Eq. (1).

##  I SUPPLEMENTARY MATERIAL

##  II Methods

###  II.1 Sample preparation

The powder of K2Ni2(SO4)3 was prepared by solid state reaction from a
stoichiometric mixture of K2SO4 and NiSO${}_{4}\cdot 6$H2O annealed at 450∘C
for five days. The powder is quenched to room temperature and stored in a
desiccator as K2Ni2(SO4)3 is mildly sensitive to moisture. High quality single
crystals were obtained by sealing the powder in an evacuated quartz ampoule.
Millimeter sized crystals are obtained by cooling the melt from 850∘C to 750∘C
at a 1 K/h rate.

###  II.2 Single-crystal x-ray diffraction

A small single crystal of K2Ni2(SO4)3 has been glued onto the tip of a glass
needle and cooled down to 100 K with a flow of cold nitrogen gas. Data has
been collected on a Rigaku SuperNOVA diffractometer using Mo/Cu Duo source
with Atlas CCD.

###  II.3 Magnetization and magnetic susceptibility

Magnetization $M$ and magnetic susceptibility $\chi_{\mathrm{DC}}=M/B$ of
powder and single crystal samples were measured using a commercial
superconducting quantum interference device magnetometer MPMS-5T (Quantum
Design).

###  II.4 Heat capacity

Heat capacity measurements above 2 K were performed on powder and single
crystal samples using a commercial PPMS (Quantum Design). Below 2 K, a home-
made setup using a dilution refrigerator has been used to measure single
crystal sample. In both cases a short (1-3 %) heat pulse method has been
utilized.

###  II.5 Muon spin relaxation ($\mu$SR)

$\mu$SR experiments were performed on powder samples at MUSR, ISIS (UK) and
LTF/GPS, PSI (Switzerland) beamlines using the spin-polarized positive muons
($\mu^{+}$).

###  II.6 Neutron diffraction

Neutron diffraction on powder was performed on the time-of-flight
diffractometer WISH, ISIS (UK). For temperatures below 1 K, a copper can was
attached to a dilution refrigerator and filled with 15 g of powder. Above 1 K,
a vanadium can was used with 15 g of powder in a helium-flow environment.

###  II.7 Spin-polarized neutron diffraction and inelastic neutron scattering

Both spin-polarized neutron diffraction and non-polarized time-of-flight (TOF)
inelastic neutron scattering measurements were carried out at the polarized
spectrometer DNS at the Heinz Maier-Leibnitz Zentrum (MLZ), Garching, Germany.
Approximately 2 g of powder were enclosed in an annular cylinder sample holder
made with oxygen-free copper and sealed in a He atmosphere. Measurements were
taken in a 3He insert installed in a top-loading CCR cryostat. A neutron
wavelength at $\lambda=4.2$ Å was chosen for both measurements. The magnetic
scattering cross-section was obtained via the XYZ polarization analysis
method, for which the standard procedures such as flipping-ratio correction
and normalisation of detector efficiency have been applied. The TOF inelastic
neutron scattering data were taken with a disc chopper running at 250 Hz,
which yields an energy resolution at $\sim$0.25 meV at 4.2 Å. The runs for
both vanadium and empty copper sample can were undertaken under the same TOF
condition. The powder-average inelastic scattering profiles were obtained via
Mantid-based data reduction routines.

###  II.8 Density functional theory

We study K2Ni2(SO4)3 using density functional theory (DFT) calculations based
on the full potential local orbital (FPLO) basis set [29] combined with the
generalized gradient approximation (GGA) to the exchange correlation
functional [30] and with a GGA+$U$ correction for the strongly correlated Ni2+
$3d$ orbitals [31]. We employ the energy mapping technique [32, 33] to extract
the Heisenberg exchange interactions up to a Ni-Ni distance of 8.6 Å from 20
GGA+$U$ total energies of selected spin configurations in a
$\sqrt{2}\times\sqrt{2}\times 1$ supercell. We fix the Hund’s rule coupling at
$J_{\rm H}=0.88$ eV following Ref. [34].

###  II.9 PFFRG

The model Hamiltonian for K2Ni2(SO4)3 with the Heisenberg exchange
interactions obtained from DFT is further studied within the pseudofermion
functional renormalization group (PFFRG) method. [35] This approach is based
on a fermionic rewriting of the spin operators, where a spin-1 is represented
by two coupled spin-1/2 degrees of freedom. [36] The resulting fermionic
theory is then treated with many-body Feynman diagram approaches.
Particularly, via the introduction of an infrared frequency cutoff, the
fermionic vertex functions are subject to a renormalization group flow as
described within the standard functional renormalization group (FRG) scheme.
[37, 38] We solve the corresponding differential equations in real space on a
one-loop level, by taking into account spin-spin correlations up to a distance
of twice a lattice vector of the underlying cubic lattice and approximate the
frequency dependence of the vertex functions by 64 discrete mesh points. The
central outcome is the zero-frequency, momentum-resolved real part of the
magnetic susceptibility $\chi^{\prime}(Q)$ which is obtained from the
fermionic two-particle vertex. Using Kramers-Kronig relations,
$\chi^{\prime}(Q)$ is related to the dynamical spin structure factor
$S(Q,\omega)$ via

$\chi^{\prime}(Q)\propto\int d\omega S(Q,\omega)/\omega\;,$ (3)

indicating that $\chi^{\prime}(Q)$ primarily represents the low-energy
contribution of $S(Q,\omega)$. Most importantly, $\chi^{\prime}(Q)$ takes into
account quantum fluctuations well beyond mean field and is, hence, well suited
to simulate the fluctuating moments of K2Ni2(SO4)3. Furthermore, possible
instability signatures during the renormalization group flow allow one to
detect static magnetic long-range order.

###  II.10 Classical Monte Carlo

Monte Carlo simulations are performed for classical Heisenberg spins on the
bi-trillium lattice with periodic boundary conditions for a system of $8L^{3}$
spins. We employ the single-flip metropolis update with 5 over-relaxation
steps added after every sweep of the lattice, and $10^{4}$ Monte Carlo sweeps
are used for thermalization. This is followed by $10^{5}$ Monte Carlo sweeps
during which measurements are performed every $10$ Monte Carlo sweeps. The
calculations for magnetization as a function of applied field shown in Fig.
2(b) of the main textare performed for a lattice size of $L=8$ ($4096$ spins).

##  III x-ray diffraction

Powder x-ray diffraction of K2Ni2(SO4)3 at room temperature is presented in
Fig. 5. The agreement is very good ($R_{wp}=6.2$ %), with no visible traces of
impurities.

The experimental versus calculated structure factors for a single crystal of
K2Ni2(SO4)3 is shown in Fig. 6. The tight distribution of the data around the
red line $F^{2}_{obs}=F^{2}_{calc}$ indicates the high quality of the
refinement. Additional refinement parameters are given in the supplementary
Table 2. The agreement factors and the goodness-of-fit value confirm the high
accuracy of the K2Ni2(SO4)3 structure description. Atomic positions as well as
distances and angles are listed in supplementary Table 3 and 5, respectively.

Temperature | 100.01(10) K
---|---
Crystal system, space group | Cubic, P2(1)3
a = b = c | 9.81866(12) A
Volume | 946.58(4) A3
Z, Calculated density | 3, 3.395 g/cm3
Absorption coefficient | 5.589 mm-1
F(000) | 952
Theta range for data collection | 3.594 to 30.444 deg
Limiting indices | -5$\leq$h$\leq$14, -9$\leq$k$\leq$14, -14$\leq$l$\leq$13
Reflections collected / unique | 4119 / 965 R(int) = 0.0274
Completeness to $\theta$ = 25.242 | 98.8 %
Data / restraints / parameters | 965 / 0 / 59
Goodness-of-fit on F2 | 1.046
Final R indices I$>$2sigma(I) | R1 = 0.0143, wR2 = 0.0330
R indices (all data) | R1 = 0.0146, wR2 = 0.0332
Absolute structure parameter | -0.034(9)
Extinction coefficient | 0.0145(8)
Largest diff. peak and hole | 0.238 and -0.245 e.A-3
Table 2: Single crystal refinement parameters of the K2Ni2(SO4)3 structure at 100 K | x | y | z | U(eq) | Site
---|---|---|---|---|---
Ni(1) | 1645(1) | 1645(1) | 1645(1) | 5(1) | 4a
Ni(2) | 5945(1) | 945(1) | 4055(1) | 5(1) | 4a
K(1) | 1854(1) | -1854(1) | 3146(1) | 10(1) | 4a
K(2) | 4507(1) | 4507(1) | 4507(1) | 10(1) | 4a
S(1) | 2826(1) | 1233(1) | 4806(1) | 5(1) | 12b
O(1) | 2550(2) | 952(2) | 3371(2) | 12(1) | 12b
O(2) | 2581(2) | -28(2) | 5572(2) | 15(1) | 12b
O(3) | 4246(2) | 1699(2) | 4987(2) | 10(1) | 12b
O(4) | 1907(2) | 2262(2) | 5371(2) | 13(1) | 12b
Table 3: Fractional atomic coordinates ($\times 10^{4}$) and equivalent isotropic displacement parameters (Å${}^{2}\times 10^{3}$) for K2Ni2(SO4)3 at 100 K. $U(eq)$ is defined as one third of the trace of the orthogonalized $U_{ij}$ tensor.  Ni(1)-K(1) | 3.7435(3)
---|---
Ni(1)-O(1) | 2.0315(18)
Ni(2)-K(2) | 3.7977(4)
Ni(2)-O(3) | 2.0417(18)
K(1)-O(1) | 2.847(2)
K(1)-O(2) | 3.065(2)
K(2)-O(3) | 2.8094(19)
S(1)-O(1) | 1.4613(18)
S(1)-O(2) | 1.469(2)
S(1)-O(3) | 1.4782(18)
S(1)-O(4) | 1.4636(18)
Table 4: Bond lengths (in Å) for K2Ni2(SO4)3 at 100 K. O(1)-Ni(1)-K(1) | 48.72(6)
---|---
O(3)-Ni(2)-K(2) | 46.38(5)
O(1)-K(1)-O(2) | 47.00(5)
K(1)-S(1)-K(2) | 146.85(2)
O(1)-S(1)-K(1) | 49.47(8)
O(1)-S(1)-K(2) | 100.00(8)
O(1)-S(1)-O(2) | 107.72(12)
O(1)-S(1)-O(3) | 110.41(11)
O(1)-S(1)-O(4) | 112.40(11)
O(2)-S(1)-K(1) | 58.29(9)
O(2)-S(1)-K(2) | 149.72(9)
O(2)-S(1)-O(3) | 110.71(11)
O(3)-S(1)-K(1) | 124.71(7)
O(3)-S(1)-K(2) | 46.05(7)
O(4)-S(1)-K(1) | 126.40(8)
O(4)-S(1)-K(2) | 72.52(8)
O(4)-S(1)-O(2) | 106.69(12)
O(4)-S(1)-O(3) | 108.85(11)
Ni(1)-O(1)-K(1) | 98.86(7)
S(1)-O(1)-Ni(1) | 145.35(12)
S(1)-O(1)-K(1) | 107.57(10)
S(1)-O(2)-K(1) | 97.65(10)
Ni(2)-O(3)-K(2) | 101.88(7)
S(1)-O(3)-Ni(2) | 127.24(10)
S(1)-O(3)-K(2) | 111.68(9)
Table 5: Angles (in degrees) for K2Ni2(SO4)3 at 100 K.

##  IV Magnetization

Low temperature view of the temperature dependence of the inverse of magnetic
susceptibility $\chi_{dc}=M/B$. A weak deviation from the Curie-Weiss law
starts below 50 K but it is significantly visible only below 20 K.

##  V Specific heat

As shown in Supplementary Figure 8, at higher temperatures both K2Ni2(SO4)3
and the non-magnetic analog K2Mg2(SO4)3 show kinks in their specific heat.
These are probably related to the freezing of SO4 groups without a noticeable
symmetry lowering from the cubic space group.

The phonon contribution below 2 K has been estimated by employing a polynomial
$BT^{3}+CT^{5}$ with $B=1.33(3)\cdot 10^{-3}$ J/mol K4 and $C=2.1(8)\cdot
10^{-6}$ J/mol K6 that best matches the measured specific heat of K2Mg2(SO4)3
at low temperatures, as seen in Supplementary Figure 9.

Determination of specific heat involved in the second order phase transition
at $T^{*}$ is displayed in Supplementary Figure 10. The red curved dashed line
is the measurement at 5 T where no anomaly is present, adjusted to match the
zero-field data at 0.8 K and 1.5 K. The exact position of the background line
does not change significantly the extracted value of $\sim 1$ % of the total
$Rln(3)$ entropy of spin-1 system.

Magnetic field evolution of $T^{*}$ and $T^{**}$ is presented in Supplementary
Figure 11. For $B=0.75$ T a small shoulder appears around 0.5 K, possibly
indicating another phase. Given that this is seen for a very narrow magnetic
field range, it could also reflect an experimental artifact.

Comparison of a power-law behavior $C_{P}\sim T^{n}$ for $n=2$ and $n=3$ with
a gaped behavior $C_{P}\sim exp(-\Delta/T)$ is presented in Supplementary
Figure 12. Zero field data show a somewhat varying slope, possibly influenced
by the presence of the static component. $\Delta=0.5$ K corresponds to the
value of a spin-triplet gap for a static dimer on the $J_{4}$ bond.

Comparison between results of specific heat obtained on single crystal and on
powder samples is shown in Supplementary Figure 13.

##  VI Neutron scattering

In Fig. 14 we show the temperature dependence of the scattering profile of
polarized neutrons. At 17 K the period of oscillations is still visible,
although with a smaller amplitude. At 80 K the profile is practically
featureless. The spikes occur at positions of nuclear Bragg peaks and are
related to imperfect subtraction of large numbers in spin-flip and non-spin-
flip channels.

Fig. 15 displays the diffraction data at 90 mK together with a LeBail fit
using three propagation vectors $Q_{1}=(\frac{1}{3},0,0)$,
$Q_{2}=(\frac{1}{3},\frac{1}{3},0)$ and
$Q_{3}=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. For $Q>1$ Å the satellites
appear as shoulders of strong nuclear Bragg peaks which together with a
diminishing form factor makes them very hard to distinguish.

##  VII Muon spin relaxation

Taking into account that the whole data set is measured with two different
experimental setups (a dilution refrigerator and a variable temperature
insert, implying different backgrounds and different initial asymmetries), the
temperature evolution of relaxation rates and exponents are presented in two
segments. The low temperature segment, from 20 mK up to 4 K, is modeled using
Eq.(2) from the main text, while the high temperature segment, from 100 K down
to 3 K, is modeled using Eq.(1). In the region around 3 K both approaches can
be used so if the low temperature segment is modeled using Eq.(1), the
extracted relaxation rates overlap, as shown in Fig. 16 with green and blue
diamonds.

It is rather simple to understand why two approaches work equally well. In
this overlapping region the exponent $\beta$ acquires values close to 1,
rendering two contributions in Eq.(2) identical and effectively becoming
Eq.(1). For two approaches to smoothly transform from one to the other it
would be necessary to allow for the fraction $f$ and $\beta_{1}$ to be freely
varied or that a microscopic model is developed which could meaningfully
constrain other parameters.

Fig. 17 shows longitudinal-field $\mu$SR relaxation at 1.7 K, well within the
correlated region. The system remains dynamic with fields up to 0.78 T.

##  VIII Additional PFFRG information

In PFFRG the magnetic susceptibility depends on the renormalization group
parameter $\Lambda$ which is implemented as a sharp infrared frequency cutoff.
Despite the artificial nature of $\Lambda$ it shows various similarities with
the temperature $T$, particularly, kinks or cusps in the $\Lambda$-dependence
of the susceptibility signal the onset of magnetic long-range order. Most
importantly, the identification of either magnetic long-range order or a
magnetically disordered phase does not rely on any prior assumption on the
system’s ground state. To illustrate the identification of magnetic order,
Fig. 18 shows the maximal susceptibility in momentum space as a function of
$\Lambda$ for various different systems. The orange curve corresponds to a
spin-1 Heisenberg model on the lattice network of K2Ni2(SO4)3 but with
$J_{4}>0$ interactions only. The antiferromagnetic Néel order in this system
manifests in a strong peak. On the other hand, the green curve is a typical
example for a smooth non-magnetic renormalization group flow as given for the
spin-$1/2$ nearest neighbor antiferromagnetic Heisenberg model on the
pyrochlore lattice. The PFFRG data for K2Ni2(SO4)3 is presented by the blue
curve and shows an intermediate behavior: A small kink at $\Lambda\approx
0.45$ is observed which, however, does not develop into a pronounced peak
(note that small oscillations below $\Lambda\approx 0.45$ are typically
artifacts of the discretization of continuous frequency variables within our
numerics). This indicates that our PFFRG results are in accord with a small
ordered moment in the absence of an external magnetic field.

Figure 5: Powder diffraction of K2Ni2(SO4)3 (black circles) with the result of
a Rietveld refinement (red line). The difference between the measured
intensities and the fit is given with the blue line. Peak positions are marked
with green vertical lines. Figure 6: Experimental structure factors are
plotted against the calculated structure factors obtained by single crystal
structure refinement of K2Ni2(SO4)3. Figure 7: Deviation from the Curie-Weiss
law. Figure 8: Kinks in the specific heat of K2Ni2(SO4)3 and K2Mg2(SO4)3.
Figure 9: Specific heat of K2Mg2(SO4)3 with a low temperature extension based
on a polynomial $BT^{3}+CT^{5}$. Figure 10: Second order transition at
$T^{*}$. The shaded area carries an entropy of $\sim 1$ % of the total $Rln3$.
Figure 11: Magnetic field dependence of specific heat below 2 K. Individual
curves are shifted vertically for clarity. Figure 12: Low temperature specific
heat for magnetic field values $B=0,0.5,1.5,7$ and $14$ T (top to bottom).
Individual curves are shifted vertically for clarity. Figure 13: Comparison
between single crystal and powder measurements. Figure 14: Temperature
dependence of spin-polarized neutron diffraction. Figure 15: Low-$Q$
diffraction profile of K2Ni2(SO4)3 at 90 mK (points) with a Lebail fit (line).
Figure 16: Temperature region where two segments are equally well described
with both equations (see main text). Figure 17: Longitudinal-field $\mu$SR
relaxation at 1.7 K. Figure 18: Maximal susceptibility in momentum space as a
function of the renormalization group parameter $\Lambda$ from PFFRG. The
green curve is an example for a non-magnetic system (spin-$1/2$ nearest
neighbor antiferromagnetic Heisenberg model on the pyrochlore lattice). The
orange curve represents a magnetically ordered system (spin-1 Heisenberg model
on the double trillium lattice with $J_{4}>0$ couplings only). The blue curve
represents the data for K2Ni2(SO4)3.